Yes. Terrific. Well, look, my name is Morgan Curran. I work for FFW. I'm here from Cambridge, Massachusetts today, so really happy to be in Nashville. I'm joined by my client and partner, Mr. Greg Verudis, from National Audubon Society, and we're really excited to be here with you today. We're looking forward to sharing a project that we have been working on together over the last six to nine months and that we're continually working on. But essentially, it's data visualization for a better world, National Audubon Society, FFW, and Drupal. So we've had a fantastic time over the last six months working on this engagement and we wanted to share it with you. So before we dive in, a little bit about the team. Greg, why don't you come on up here and talk a little bit about your history. Thanks, Morgan. I'm Greg Verudis. I'm a data scientist and design technologist and what that means is my background is in geography. I like to make maps and other tools to visualize and communicate data to diverse audiences. My name is Morgan Curran. As I said, I work for FFW in Cambridge, Massachusetts. My background is in user experience design. When I started out in design, there was no user experience. There was no digital. So I'm aging myself a little bit there. Over the last 15 years or so, I've worked exclusively in digital experience design, starting out with large agencies. I've worked for Digitas for a long, long time and then more recently with children's K-12 education, Houghton Mifflin Harcourt, working on helping them transform from a book publishing company to a digital learning company with distributed content. So that work actually prepared me very well for the engagement here at FFW with Greg and the Audubon team because we worked a lot on maps and visualizing data for children and making it so that they could easily understand what they were looking at without having to have it be interpreted for them. Additionally here, as with all things Drupal, Greg and I are on stage, but Joanna Grand is not here today but was very much part of the project, as were these folks right here and many others as well. So Greg and I are telling the story, but really it was a large distributed team effort. Okay, so before we dive in, we're going to talk to you a little bit about Audubon. You may have heard of the National Audubon Society. We're going to talk a little bit about it and how it came to be. We're going to give you some background on this engagement and why the Audubon Society was looking for a conservation tracker. We're going to talk you through our project process as we went through discovery, design, development, and then launch. And then we're going to demo the product in its current state. It's continually evolving, but we're going to share it with you as it exists today. Once we've done that, we'll talk a little bit about the technology and the data layers that underpin it and support it. And then finally, we're going to talk to you a little bit about lessons learned and the things that we're going to do moving forwards to continually improve it. Make sense? Cool. Okay, so over to Greg. Thanks, Morgan. I represent National Audubon Society, and some of you might be wondering what that is. The group began around the early 1900s, and it's the oldest such organization that blends science and education for the purpose of conservation. And in our case, it's avian conservation birds. What's really powerful about the Audubon Network is that it's mostly volunteer-driven. So I was able to attend our biannual convention this year, and you have volunteers, members from our different chapters, who are doctors, lawyers, and other professions by day, and then they spend their free time, and sometimes even their money, on supporting bird conservation because they are avid birders or they just really care about the species in general. The network also has these sanctuaries and centers, which are these scientific research nodes on the ground, local work to advance the mission, and then national and state offices, which I'm a part of. But you see this picture here of some fashionistas, and the origin of Audubon is actually a very interesting story, and I'll try to tell it well. These feather hats, there was a frenzy in the early 1900s people murdering up to five million birds a year to create these plumage, these hats. And in the early 1900s, we saw women conservationists, mostly based in Florida, coming to the forefront and basically saying, this is not right. And so it was the first time a popular movement coalesced around the defense of the environment, and not surprisingly, in this case, it was to save birds. So how do we fast forward 100 years? How do we go from this early beginnings to a high-tech, leading-edge, science-based conservation organization, which is what I would classify us as now? Audubon's unique in that it has this powerful network, as I mentioned, these three tiers, national, state, and local. But really, to be honest, the local piece is what drives the group, because it's volunteer-based and so much more. Some of you may have heard of the Christmas Bird Count, the CBC, it's the oldest-running community science program in the world. People have been going out after Christmas to their backyards and their neighborhoods and counting birds. But if you break down Audubon and many other conservation, environmental conservation organizations, they're really pretty similar. You might have heard of groups like the World Wildlife Fund and the Nature Conservancy. Very similar missions, but the one thing I've learned in having worked with a few of them is that they all operate under slightly different currencies. And as you might imagine, the currency with Audubon is birds and their habitats. Interestingly, probably the most cutting edge or new program that I'd like to share a quick story about is our Climate Watch program. And Climate Watch, what it really tries to do is just move beyond the issue of politics when talking about climate change and instead, trying to let birds tell the story of how climate change is affecting birds and the environment. So many Audubon members, like I mentioned, are from traditionally red states. And so bringing up the topic of climate change is not generally recommended. So what we do is we say, you are our members. You care about birds. Go out to your backyard. Go out to your neighborhood park and tell us, are you seeing the same birds you've been seeing over the last 10, 50, 100 years and let the birds tell that story and validate some of our scientific models and other tools that we're using to understand this? So they go out. Maybe they see new species that they never have before because these birds are colonizing areas that traditionally they hadn't. Or maybe something that they're used to seeing, a bird that they're used to seeing is disappearing. And we can tell that story and we can share it with other people in a very agnostic sort of apolitical way. And then not to belabor this point, but just the network being very powerful, very adaptive, and volunteer driven allows us to do a lot of things that other organizations won't, including investing in technology and what we're about to show you today. So that's why I believe Audubon's unique and now sort of the ask, the reason why we started this project, which we call the conservation tracker. My boss is boss, Audubon's chief conservation officer, asked a very fundamental basic question to the organization and it was, how are we doing? You know, what are our vitals? Are we actually having an impact? Are we moving the needle? Are we just spending donor and foundation dollars on this mission that is not having any result? And that's actually a really difficult question to answer when you're talking about science-based work and advocacy. So we knew pretty quickly who our audience was. We wanted to be accountable to our funders and our board, but the purpose evolved over time and we started to see some challenges. Because the Audubon network is so broad, as I've mentioned, there's really no centralized data management, no authoritative data that we could draw upon. Everybody had their own orphan Google docs floating in the ether and nobody really was talking to anybody else. And then these challenges you see here on the right, hopefully you can see, I don't think they're unique to Audubon. I think every organization, whether they're private, public, for profit, struggles with this. Experts working in silos, technical and tech infrastructure under development, and so forth. But the one I want to highlight and was really a big barrier to entry for us with this tool was resistance to change. You have groups that have been doing their spreadsheets and their finances and their data management the same way for so many years, that this is really why we designed the tool and what something we knew we were going to have to struggle with from the start. And the last thing I'll say is that the literature says time and time again that co-developing, co-designing something so that it's not done in isolation, but with others, with stakeholders, we're going to essentially drive the process and produce a tool that's more meaningful and salient to those involved is really important. So we set out to engage people from the very beginning, our board and our donors, leadership, conservation staff, everybody in the network. And it became pretty clear that the first use case for this tool, which you'll see in a moment, was for fundraising, so that we could fundraise the development of our back end, which is still under development. Nobody wants to fund a back end. They want to see a sexy front end and use that to fund the back end. So that became pretty clear early on, as well as this whole accountability piece and priority setting and so forth. So that's where we are. And fortunately, we asked FFW to help us out in this venture. And they said yes. And Morgan's going to tell you about it. Terrific. Thank you. So Greg has given you guys a background on Audubon and what the need was there. I'm going to talk a little bit about the project itself and the engagement and the process that went into it. So first of all, a little bit about the ask. One of the things that we talked a lot about during our discovery process was this crossing point, this nexus between conservation and technology. How can we use one to leverage the other and amplify in the process? Similarly, as Greg mentioned, there were many different audiences for conservation information and conservation tracking information. How could we build an experience that worked well for those various audiences who sometimes have very different needs? Some are scientific and professional, while others are grassroots and non-professional, if you like. And we decided that we wanted, decided very early on, that we wanted to have a destination we drove people to for individual bird species and individual landscapes that was a content and data-rich report card. So a consistent experience that was data visualization that brought together all the various disparate content and information sources that were sprinkled throughout the organization and that were available through various different sources. How could we bring all those together in one place under a great experience? And then lastly, use site metrics to drive most popular information. We see this every day on consumer retail sites. You go to Amazon. You go to eBay. You might also like. People like you also looked at X. So how can we bring that type of experience to the conservation tracking world, where there are literally thousands of species and thousands of landscapes? Instead of asking people to start at the top every single time, how can we use their behavioral data to drive the experience and narrow it a little bit when they come in the front door, so that they're seeing species from their location or that other people like them have also looked at? From a user experience point of view, we want us to make it interactive, make it easy, make it flexible, and make it compelling. You know, sometimes when a lot of data comes together in one place, you get this kluge experience that's not really that useful, that you kind of have to wade through. We wanted to try and be clean about it and really deliver a beautiful experience that showcased all the data, but not in a way that was overwhelming. We wanted to visualize rather than list data wherever possible, so show it rather than tell it. Have map information talk to chart and graph information and vice versa. One of the things that we talked a lot about was making the data manipulable, allowing very tactile interactive control of data, so that as folks moved around and interacted with the maps, so charts and graphs would update and vice versa. Don't let the data overwhelm the experience. There's so much information about some of these species and landscapes that it was difficult to say, okay, look, how can we put this in a small space on a screen and make it work well? And then lastly, integrate it seamlessly with the existing autobahn.org site. I don't know if you've been to autobahn.org, but it is a beautifully designed, incredibly visual, very clean design experience, and so we wanted to make sure we mapped right into that, so we adopted a lot of the look and feel conventions from it. Specific requirements that we were given when we went through our discovery process, concentrate raw conservation data together in one place and visualize it so it tells a compelling story about what's happening with a particular species or landscapes in terms of threats to it, actions being taken to mitigate those threats, responses based on those actions, so not only snapshotting the information but showing progress over time. And then against those actions, what conservation efforts are in place to protect that species or landscapes in terms of goals, specific initiatives, partnerships with outside organizations, and individual priorities within a larger goal. Be able to answer those common questions and address consistent needs, such as Greg's boss is boss, how are we doing? So be able to use the data to actually deliver an insight, not just to deliver the data for the sake of it. And then obviously communicate trends over time and that's the tracker part of the experience. So that was the ask, if you like. I'm gonna switch gears now and talk a little bit about the process that we went through. So many of you in the room here will be very familiar with what you're seeing here. It's similar to many other software development processes. We like to think that we over-index on the early part of it that we call discovery because really focusing on understanding and clarifying with our client what the business goals are at the same time that we're looking at user needs and wants, at the same time that we're looking at the technology underpinnings that have to support the experience is where you get to that sweet spot that is what we will design and build. We've all been in situations where a design has happened that just can't be built because the technology doesn't exist. Or we've built something that's fantastic from a technical point of view but the experience is not that great or it doesn't meet the business goals. We focus really, really carefully on understanding that sweet spot where business, user and system interact to get to an optimal solution. From the point of view of the conservation tracker we went through that process working with Greg and his team. We then wire framed it out so that we could see the progression or the user journey through the experience. We built out the interactive reports seeing what the data actually looks like once it showed up because sometimes it looks good and sometimes it's not so cooperative and you have to work with it. We tested it with various audiences to make sure that it was working and we revised based on that before working with our technology team in a very agile way. And that's not to say that the technology team was not involved throughout because they were. It's very much a part of the way we work that we're highly collaborative between business and strategy, design and development. Okay, so as Greg said our experience strategy was to really connect with people that are passionate about conservation. Primarily funders and donors on the Audubon board but secondarily Audubon conservation staff, scientists working on conservation, the Audubon network so that loose network of connected organizations and of course the general public. And what you're looking at here below is what we developed as the primary user journey or happy path if you like. We wanted to make sure that there was a warm welcome when people landed in the conservation tracker world because there's a lot of data in there. There's a lot of information, a lot of content. So that first glimpse where you established the mental model for the experience is very, very important. We then wanted to make sure that we put in place a powerful search mechanism so that whether folks wanted to search by keyword, by hunt and pack if you like, clicking through and browsing or by another means we gave them multiple avenues into the data. Once they'd down selected to either a subset or a single species or landscape that they wanted to learn more about, make that data interactive. Allow people to interact with it, touch it, move it around, play with it to make it the way they want. Make it easy to download. So as Greg has mentioned, various different audiences had multiple needs for the content. And so we wanted to make it easy to deliver on that. And then lastly, make sure that people leave having had a good experience. You know, from a user experience point of view, which is what I focus on, that takeaway experience is very important because if they come in, they have a good experience and they leave happy, they'll say something about it or they'll come back. And we intend for this tool to be used as, you know, not just a one time thing but as a continual resource for people interested in tracking conservation. Okay, just before we look at the prototype itself, we had some stuff to work with when we started. There was an initial prototype that had been built to connect some of the data sources to visualization tools, whether those were bar charts or mapping experiences. But what we were finding is that when all of those things showed up on the screen, they weren't really telling a story and they weren't really working together to deliver insights. It was just an accumulation of data. And so we worked very hard with Greg and Joanna and the team to really map out the real estate because don't forget on the web, you're dealing with a low resolution sequential environment. So there's only so much you can bring together in one space and you have to balance and compromise and make decisions about what to show first, second, third, I call that eyeball management, but it's very important, especially when you're bringing together a lot of data, a lot of content in a tight space. So once we talked through that and done quite a bit of virtual whiteboarding, we went through a process of building out wireframes. Are you guys familiar with wireframes? Have you worked with those before? Pretty straightforward. We worked in desktop and mobile because this is a fully responsive experience and we have break points in between so it will work on tablets, both landscape and portrait as well. But you can see that we worked through the welcome experience, landing pages and then specific deeper dives so as that we understand how the information presents at multiple levels of depth from welcome right through to interactive report card at the most detailed level. Once we built out the wireframes, we worked on visual design comps and we built a clickable prototype. Now, this is not code. This is not developed in Drupal. This is essentially static comps that are daisy chained together in order to simulate the experience of the developed code. And what that allowed us to do was it allowed us to take it and share it with various target audiences and get feedback, which was hugely important because things that we think, oh, it's very straightforward. You can get from A to B, no problem. It's pretty easy. You find that other folks might get hung up on and vice versa, things that you think weren't such a big deal. People say, wow, that's really cool. So this process of learning as you design continuous iteration is really, really important when it comes to good data visualization experiences. And you'll see as we look through the actual live site now in a moment, things that you see here didn't make it into the design or things that you see here have been refined and enhanced in the actual design. Okay, let's switch gears one sec. The reason I asked is because we put FW on a pretty tight timeline and we started this in about October and now we have a working prototype, so all this is a Patrick and Morgan and others who probably really appreciate their dedication and pushing this out quickly but also at a high level of quality. Thanks, Greg. So we're gonna take a quick look at the actual live experience right now but before we dive in, I just wanna say a little bit about autobahn.org, the conservation tracker is based on the look and feel of the live autobahn.org site which is just spectacular from a visual point of view. One of the things that Greg and his team are really lucky is that people take photographs of birds all the time and they submit them for competitions to autobahn. So they have this incredible library of just fabulous imagery that is used throughout the site experience and is constantly being refreshed because there's annual photo competitions for both professional photographers, amateur photographers, youth photographers with the result that the site is visually stunning. So we wanted to steal with pride all of that that had already been done and that was working really, really well as we thought through the conservation tracker. So switching over to the conservation tracker here you can see that the global mass had and the welcome image convention is adopted 100%. You have this parallax experience here where you scroll down to reveal the page that you're on. And so this is the welcome experience for the conservation tracker. You can see that there's three major areas within it, strategies, species and landscapes. Essentially those are three buttons. So you can choose to search by keyword if you would like to get to something specific if you know what that is. Or you can click on one of the main areas and we'll go into each one of those in a little bit more detail in a moment. Or you can click on some of the information that's popular right now. And this is what I talked about a few moments ago in terms of using site usage metrics to drive experiences on the homepage, to shortcut the journey for folks within a certain geography with a known interest, with a previous visit that we know something about. So to the extent that we can use data of actual behavior on the site to then drive a shortened experience in the future, we'll try to do that whenever we can. While Morgan's showing this, I realize 99 maybe 100% are conservation, but maybe just think about it through the lens of your own work, whether it's for profit or whatever it is, because we really, we hope this site might be useful for other groups that want to take this down in the layout and just swap in different content or different controls. So I think it's pretty modular, pretty flexible. And that's just one thing we're thinking about. So yeah, keep that in mind as you. Yeah, we've done a lot of work with maps and manipulating map data as part of this process. So again, that's transferable to any application where you need to show data connected to topography in some way. Okay, so let's dive in. We're gonna click through to the strategy section first. Woo, we got bandwidth. So again, beautiful imagery, just stunning imagery of birds available to us all the time. And this is the strategy section. So it's kind of small, can I make it any bigger? That's a little bigger. So each strategy is broken down into coasts, working lands, water, bird friendly communities and climate. And within each one of those categories or sections, there's an overview which you're seeing displayed right here. You can look at goals associated with that strategy, whether those are conservation goals or engagement goals. You can see the partners that Audubon is working with and who are either contributing to or benefiting from the experience. You can see specific initiatives associated with that strategy. And then the priority bird species and landscapes that it touches. And you have the ability to move things around. The map is fully interactive. You can click on any one of the nodes here to see a little bit about what's happening in that particular area relative to the strategy that we're looking at right now. So this is where the data visualization aspect of this comes into play. It's highly interactive, highly manipulable. And while not all the data is in place for every single landscape or every single species yet, the goal ultimately is to be able to click through from any of these nodes to what we talked about a few moments ago, which is the interactive report card for that landscape or that species, which is that rich data environment on a single screen. We're gonna look at that in a few moments. The mapping sometimes can be a little bit too much in terms of the limited real estate with mobile. So I think in one overview, we made the call to cut the map just because it wasn't adding enough value where it was worth taking up that real estate. But for the most part, the mobile and the desktop versions are identical except for the layout. Yeah, go ahead. The data demo, the CSB, the BDS, are those static? Where does the data go on the fly? On the fly. And how are you doing that in my master? I mean, right now, I mean, we have plans to have the data up in this behind it, but it's the idea of because we just didn't have the data, so we didn't build the data up. So right now, we're actually tracing possibly by a little bit. I'm sure it's a bit of a mistake but once they're tracing files hard, it just gets, how do I get this to get across? Hello. What did you use to make those? The files themselves, the flat files were created in Sketch and the clickable prototype was created in InVision. There's a hundred different ways you could do it though, but InVision is just fast and it's also very shareable. So we could essentially share the link with Greg and his team and they can make comments right in the file. So it's one of those like Google Docs. It's the Google Docs version of prototyping essentially. Yeah, I liked InVision a lot as my first time using it. We would get on call sometimes, remote team collaboration. We'd be talking to FFW and they would be taking notes in InVision and it was all, we could see it with screen share. So it was just very easy to annotate, make changes and then make sure we knew who was gonna be making those edits. And I think when you've got remotely located teams, that ability to see the same thing at the same time and to annotate and make comments on the file itself while the discussion is happening, it's huge because if somebody takes notes and then they have to go back and relate that to a designer who then has to make the updates, share the link with the client who then says, well, that kinda wasn't what exactly I was looking for. Bang, all that goes away. So that process, and we talked about it like collaboration, it's, there's so many decisions to be made that that process of thinking it through together is hugely beneficial in terms of getting to a good result in the end. Hey. From the beginning of discovery to actually reaching the final design with some element of actual music that's in between and four leads. So tell me a little bit more about your, I mean, you talked about the tools now. How, especially the whole of you distributed, how on earth did you manage to get, I mean, it's clearly, it's more than just like, but tell us a little bit more about your user testing process. Well, the project took more than four weeks. It was more like four months. But the user experience, the testing, well, right, the design portion was pretty quick. Essentially, once we have the clickable prototype in InVision, you have something to test that represents the primary journey, not every single screen, but just enough to make it feel real, if you like. What we did is we put together a script based on our understanding of the target audiences and what their needs and wants were. So we asked them, you know, what are the types of things you wanted to get done? Can you do them? Can you find this? Are you able to find a specific species? Can you tell us what the threat is to a specific landscape and so on? And so we sort of probed for people's ability to intuitively get to the information from the highest level. And from the most part, they could. It was surprising. You know, there was very few things that tripped people up, a couple of labels. So we went back in and we put in instructional copy here and there to just help people if there was a kind of a hesitation moment, because we were watching at the same time. We did the user testing remotely, but with video. So you had a video of the person doing the test, you had a video of what they were doing on screen, and then there was the questionnaire moderator guide that we went through as we went through the process. That was where we had one sacrifice based on timing. We had very little time, so we used designated experts that Audubon nominated for us. But we were at least able to go to different divisions, so it wasn't just our science division, it was across. I'll just add that something that's kind of stressing me out now is the next step, which is we did the user testing, the user acceptance testing, and now it's like, okay, we have a prototype, and next week I go to New York City to work with our development team. So development is the fundraising unit of our group and get them actually interacting with the world. Trying to convince them that this tool can be useful for their work. If this can save them time, it's not just another tool. And I don't know if you folks familiar with like the human-computer interactions literature, but this divide between the developer, us, and the user, them, is what we're trying to narrow. And I don't know if anybody has thoughts, suggestions, words of wisdom, I'd be happy to talk afterwards because you can develop something as beautiful as it is and as solid and bug-proof as it is on that end, but are people actually gonna use it? That's always the issue with technology at the end of the day. So we're trying to unwrap to that, and we have a whole digital onboarding exercise, but we'll see. Are you doing anything with authenticating users? Is it always an anonymous user experience, or are you getting information for fundraising or planning on blanking data for this? Yeah, that's a great question. We had some ideas on that in the beginning, but it evolved into right now, but because some of the information is still pretty sensitive, and we still have to work that out through different committees that we're organizing, it's password protected and for internal use only. But let's fast forward. Once we're through that and we have a body of reports and things that we want to share and then we want to make it public-facing, we'll remove the password screen and then we would talk with FFW to develop a user login and that sort of thing so that we could have information about them if that's the path we want to take. So we wanted to do that upfront, but we decided it was probably best to wait until we were ready to make it public-facing for that sort of credentials and whatnot. I don't know if Patrick would be able to. I mean, other ideas we had, just like you location, just because they're birds, so I think we don't have to get like close to you to see what kind of birds you have in the neighborhood. I mean, the data they have is just insane to be honest. I mean, if they have a lot of attitude, so you can actually look at the yard or something like that, so I guess like 500 birds or something like that. So, I mean, that was the idea, but when you get to the species stage, it's like if you have quite a few birds in the neighborhood. I think there's also, there's a crawl, walk, run aspect to it. I think you have to get something out there first. And then we've talked a lot about passive personalization and active personalization. Passive personalization is what do we know about you when we know nothing about you? How did you arrive? What search term did you use? What site domain did you come from? You know, what do we know about you that we can use to sort of passively personalize the experience, as Patrick just mentioned? Seasonal information, geographic information, demographic information, and then over time, the run aspect becomes the integration of, is it Aquialift or some sort of a personalization experience that would allow us to really, really understand who that visitor was because they had a profile. And then you can return them. You can return them. People would like that. Exactly right. Yeah. But also, you were talking about like the annual collection of data as well. Can't you collect that now online, giving it to their local groups or how does that work? It runs the gamut, so yeah. I mean, there's, like the Christmas bird count once a year, we have the final watch program which is twice a year, and then sometimes it's even more frequent and sometimes it's less frequent, so it's just so diverse and that's why we wanna try to centralize all this stuff. And so we're having, you know, developing forms where people can add data and it'll all seamlessly talk to each other, but going back to your previous question, I think it's a good reminder about demographics and just understanding our user a little better and that's something we'll have to look into once it's a little bit more advanced. There was one other question here and then I think we're gonna have to wrap it up. Hi. Great question. So right now it's my colleague and I, Joanna, who are gonna be kind of maintaining it and because it's just these JSON files, it's pretty manageable, but we're developing...


It's still plus politics. Now, the British High Commission has announced that the United Kingdom would not reverse its travel ban on Nigeria because of a threat of retaliation by the Nigerian government. The spokesperson for the British High Commission, Dean Hurluck, said this in a response to an inquiry by Pressman over a statement by the Minister of Aviation Hadi Sirika that the United Kingdom, Canada and Saudi Arabia would be put on Nigeria's travel ban on Tuesday. Hurluck stated that the UK government has been clear that the travel abroad will be different this year and countries may impose border measures to, at short notice, in line with their own coronavirus policies. We're joining us to discuss this is Ambassador Joe Keshe, former Permanent Secretary of Ministry of Foreign Affairs. Thank you so much Ambassador Keshe for joining us. Thank you, Dean and thanks for having me. Great. So, let's talk about this politics of reciprocity. You know, it's been the conversation that has lasted all through last week into this week and I mean we've seen it happen before within embassies and fees that need to be paid and should not be paid. We've seen that happen even between the Nigerian Embassy and the US Embassy. That's happened. But at this level we've not necessarily seen something like this happen. But why is this particular issue especially the one that surrounds the Omicron virus very critical? Well, I think what we all need to understand that a number of countries all over the world if there was any less in the length in early 2020 when the pandemic started, it was a fact that it did not act early enough. And as a result the pandemic began to spread worldwide. So what you see happening now whether the South African variant Omicron is mild or not mild, every country is reacting to the best of the knowledge they have on how to stop the variant from coming into their territory in order to prevent what happened when they acted too late when COVID-19 initially started. And I think that is what we all need to understand that they are acting in the best interest of their country. They are taking steps to ensure that they do not suffer the kind of disaster, loss of human lives that characterize the early part of COVID-19 I think in 2020. Well, when the High Commissioner of the UK to Nigeria spoke on this issue she continuously harped on the fact that it was based on science and it was not based on any form of politics or as the Nigerian government made reference to it they called it some form of a pathide. They thought that this was the UK and other countries shutting out Africa from their borders and letting other countries continuously travel toe and fro their countries. So I'm wondering the energy, the kind of reciprocity that we're giving to the United Kingdom, Canada and Saudi Arabia, is it really worth it? Because again many people support the Nigerian perspective saying that we need to match that energy that these countries are giving to us. So someone who's worked as a diplomat is it really necessary for us to expend this much energy? I'm not sure it's worth expanding this sort of energy. I think we can spend all the energy we have now on the UAR that is burning Nigerian flights and Nigerians are even getting to Dubai. That's where I think we need to spend considerable energy. But on this, again I make the point, number one I think we are overreacting and those who are calling it apartheid are not being realistic to start with. The first thing I said was that if there was any lesson learned from when COVID broke out some two years ago, it was a fact that it took quite some time before the Chinese were honest in admitting that they had a problem. And by that time COVID had arrived in Europe, it has arrived in America and then was just traveling around the world. But the argument is that when COVID was found in China, and apologies for speaking over you, the issue or the argument basically here is that when COVID was discovered to have come from China, it wasn't called the Chinese virus, neither was it called the Chinese variant. But then when it was found by scientists in South Africa, this is not my argument. This is the argument that has been had. It was now called the South African variant. Let me make my case. I think we have been unnecessarily emotional about this as far as I'm concerned. Look, when the COVID-19 broke out in China, China did not inform the world early enough. And what happened was that it began to spread phenomenally around the world. And we all are aware of the consequences, the loss of life. Take what happened in Italy. The whole of Italy, I mean, Italy suffered a lot, for example. And the number of, we lost about, how many millions of people have we lost as a result of what happened, you know, when COVID started. And the failure of the Chinese to report early enough. So when the South Africans reported early enough, every country began to rush to take necessary measures to protect their citizens. We must recognize this that there's a difference between what happened the three years ago. It has nothing to do with China or the South Africans. It has to do with the fact that the governments are reacting in order to protect their citizens. That's number one. Number two was the fact that when Canada first showed us, for example, that two people came to Canada with only corona and things like that. We again did not react early enough. Now I'm not blaming. Look, we've done everything I believe we need to do. But we need to understand why some of these countries are reacting the way they are reacting. They are doing this to protect their citizens. And that is what governments are supposed to do. Let's be honest with ourselves. That is what governments are supposed to do. You don't wait and say, oh, until people start dying, until you know how to say they're on their own. No, they're learning the lessons of 2020. Let's talk about the aftermath or the effect that this reciprocity by Nigeria, if it be done by tomorrow, would have on us as a country and what it would do to the relationships that we have with these different countries. I'm talking about Canada, the United Kingdom and, of course, Saudi Arabia. Is it going to affect us economically and relations-wise? Of course. Look, it will affect us economically. It will affect them economically also. Everybody is going to pay price one way or the other. But look, in favor of the federal government, if they want to retaliate, fine. They need to do that out of national pride. There's no harm in doing that. But I think that what we need to focus in my view is how to ensure that we address the issue of COVID-19 in this country. We need to begin a new campaign to strengthen the procedures in place. We need to talk about wearing a face mask. We need to talk of all the protocols that we had almost virtually discarded. I think it was only the legal statement last week that came up with some announcement on some new protocols. I can't remember what the federal government has done or what any other states have done. Well, the federal government is insisting that every civil servant, I guess, from December 1, has to be vaccinated. This country is not made up of civil servants alone. And it's not only civil servants that live in Abuja, for example. It's not only civil servants that live in Abuja, for example. But let me say this while I say how do you earn. Look, the way forward, as far as I'm concerned, is let us begin to take measures to address the COVID-19, the Omicron issue right now. Part of what we can help everybody around the world, including Nigeria, is if the threat of Omicron begins to decrease, the world will reopen again and now everybody will be able to travel. Alright. Ambassador Joe Keshe, we appreciate your thoughts. Thank you very much for speaking with us. Thank you for having me. Thanks. Alright. Well, thank you all for staying with us. It's been Plus Politics. We'll leave you with what Nigerians have to say about the travel ban clash between Nigeria and other countries. I'm Mary Anacone. I'll see you on Tuesday. The federal government is right because Nigeria is a sovereign nation. And as a sovereign nation, we also need to make a decision to protect the interests of Nigerian citizens. Because if we look at this issue of variants, some of them come from other foreign countries. And at the end of the day, we need to prevent with the black nations from being racially discriminated. Like now they say Omicron variants is Satab Khan variants. And next thing, it's not a banning Satab Khan countries. I believe that it wasn't right because it's a form of racial discrimination. So in my own opinion, the federal government is right. They are doing the right thing because the rate at which other countries, let me just say that like every time they talk to Nigerians and it seems they take Nigeria as if they are nobody and Nigeria is a place anyone can just barge in and barge out anytime. I think if they are also placing a ban on other countries to come in, then it's also a good decision. I believe Nigeria is a bigger country. It depends on Dubai, it depends on the UK, it depends on America, Australia and all other western countries. So I don't see any reason why we should put a ban on any other country. What we should do is to resolve the issue for now. So that's what I think. Thank you very much. The government should place a ban on any country that plays a ban on Nigeria. Because no country is greater, I mean no country is greater, so I think it's a right decision.


Hello, I'm Simon Benjamin and this is lecture 8, the very last one in my course on Fourier series, Fourier transforms and partial differential equations. Being the last one I'm wearing a waistcoat, thought I'd make a bit of an effort and I think you'll find that this is a, I hope, a pretty interesting lecture. The maths level is gone back to a much more modest one after we really went deep in the last lecture and had to sort of really reach for some sophisticated solutions. This I think will be a gentler presentation, so let's have a look. I want to wrap up the whole diffusion topic by showing or looking at an overview of all the different things we've done and summarizing them. Then it is onto some fresh material deriving the wave equation, which is perhaps the most important equation in physics. There's something called de Lambert's formula, again excuse my accent, which is a very important insight that comes immediately after we've derived the wave equation and then we'll get down to solving a particular scenario, an extremely interesting one that captures a lot of the ideas of waves, which is a vibrating string and that then we'll be able to see if we can actually use our ideas to predict successfully how something like a vibrating string behaves in the real world, so a guitar string for example. As always, the notes for this course are available at SimonB.info and elsewhere. So what I have here is a table that I've pasted in from my notes, the brand new version of my notes that I just uploaded, by the way, in case you have an older version, and I scribbled onto these printed, these typeset notes, some little diagrams to remind us of all the different things we've done, so let me just very quickly run through it. We have thought about what happens if you either have matter diffusion, which of course is fixed second law, or heat diffusion from the heat diffusion equation, where you start off with a distribution that can be described by some kind of periodic function, such as the infinite square wave. It doesn't have to be a square wave, you would use the same techniques for any kind of periodic distribution. The square wave is a good one, it could be a series of thin layers, films forming a stack, which then blend into each other if it were matter diffusion. And the technique that we use there is that we used a Fourier series to build the initial condition, and then we inserted a term e to the minus k squared t, also with a diffusion or either matter or temperature diffusion coefficient up in there, unless it's equal to 1, and the result is that the higher order terms in the Fourier series decay the most rapidly, and so we end up smoothing off our initial distribution, and then over time it becomes more and more smooth until eventually it reaches just a flat distribution. Then the second thing we looked at was, or the second thing I want to summarize, was what happens if we have a scenario where there's only a finite width to our problem, like a length of a bar or the width of a sheet of material, and the endpoints are held at two different temperatures or concentrations. So there what we needed to do was make our function periodic, and we had to do that carefully because we had to think about how to extend a finite function into a periodic form so that we don't introduce any discontinuities or other anomalies. We've talked about that a fair bit, and what we found is that if we've done it properly the long-term solution would just be a straight line between the two fixed points and intermediate timescales we would just get again a softening and a reduction in the initial distribution. Pretty much intuitive stuff. Well then I would like to talk about actually the thing we did first, which was what if we have a Gaussian distribution in the first place. For example, we've heated up a region of a bar with a blowtorch, and it can be described reasonably with a Gaussian. And again there's a symmetry argument here that meant that mathematically that's the same as if we had a heat distribution at one end of an infinite bar. It would be the same mathematical treatment. So in that case we ended up with another Gaussian as we had guessed we should, but the interesting point is that the thing spread out in time with a speed if you like. The characteristic width increased according to the square root of time. So the more it evened out the more slowly it continued to spread. And that in fact is what we saw again and again in all these solutions is that there would tend to be a root T behavior. Okay then we thought about winding back the Gaussian solution in time until all the energy was focused in one point or all the material was just focused in one infinitely thin layer. And we realized that that was just going to give us a Gaussian again, but this time we were able to formally show it. And there we used the Fourier transform method rather than the Fourier series. But we used our same trick, which was to simply put in this decaying time factor. But now we put it into the integral just like we put it into the sum previously. And that allowed us to in fact re derive the Gaussian solution. And then the final and most impressive thing that we did really quite tough was to consider the innocent sounding situation of two blocks of material that were next to each other or at different temperatures and next to each other or one block that was in contact with a reservoir. Again, mathematically we can show that these received the same treatment. And a physical example would be the carburization of steel where carbon penetrates into the surface of steel until we've got just the concentration that we want. And that was another Fourier transform scenario. This time Fourier transform of a step function. Same trick to put in that time dependent exponential decay. It's simply that now the mathematical treatment was more challenging. We ended up with an error function. And again, we saw that the shape of the error function or how rapidly it softened was characterized by the square root of t. So a lot of stuff that we've looked at there are a lot of analysis. And I think it's fair to say that we've comprehensively looked at the diffusion equation in many different forms. We've usually made it 1d, but very often the three dimensional extension to this is very straightforward because the dimensions just separate and you get a one dimensional problem in each direction, so to speak. Moving on. Okay, so now I would like to talk about waves. What we'll do first is derive the governing differential equation that tells us how a wave behaves. And to do that we'll think of the case that we have an elastic string that's fixed at two points along the x axis and we want to know how it vibrates. The equation that we get, however, will be much more general than that and will apply to electromagnetic waves and waves on the oceans and all sorts of things. But this is how we will get to it. So let me draw a sketch of what I want to think about. All right, let me pause in my drawing now and talk about what I've drawn so far. By the way, the analysis we're going to do here is very reminiscent of how we got the diffusion equation in terms of taking small pieces and thinking what's happening at the boundaries of a small piece. So here in the purple line I've drawn, so this path here is actually our string. So it's a hugely exaggerated sort of oscillation that looks like a string that could be the case, I guess. And I've just drawn an arbitrary curve. You could draw any curve you like, except that it has to be fixed at x equals 0, our string is fixed at the x-axis and at x equals L. It's the same thing. So the y-direction in our diagram is the displacement. So this could literally be a photograph with a sort of fast acting camera that just is able to take a crisp shot of a rapidly moving thing of a vibrating string. Now, what we're focusing on is a little bit of string, just like in the diffusion equation we focused on a little bit of a little slice of the region so that material was diffusing in and out. Now we want a little piece of string and, of course, we're going to think of it as being infinitesimal really, but to draw it we have to give it some finite width to it. So I'll sort of bolding it in here, that's the piece we're interested in. These two extra lines I've drawn on here are supposed to be tangents to the curve at point A and point, let's call this point A. Maybe I'll draw below. So that's, we'll call that point A and this is point B. So these are tangents to that curve that are point A and point B. So let me carry on now and just write on a few more elements. Right, so I've added in there a few more features. I've put on a couple of forces, force one that acts at point A and force two that acts at point B. And these forces are pulling in different directions. They are the forces which are acting on our little segment of our elastic string. So remember it's an elastic string so it has a tension and that little piece of string that I've identified by drawing it in bold there, let me circle it, this little piece of string here clearly has forces on it. We're just interested in the forces on it due to the fact that it is elastic. So the string is stretched and it's trying to pull itself back to a shorter length. And so we can see that this little piece of string is going to have a force that's pulling it to the left and a force that's pulling it to the right. But they're not going to be in the horizontal direction. Those forces are going to be along the tangent of the string at those two points. So we have F1 and we have F2 and crucially those forces are not quite in opposite directions. And so we need to care about that slight difference. These forces are by the way not small because they are the magnitude of them it relates to the tension in the string, which isn't necessarily a small quantity. But the things like the difference in their angle will be small. So that's the kind of thing we now need to explore. I've also drawn on here alpha and beta. And these are the angles that those two forces make with the horizontal. So I hope it's pretty clear from the diagram what all those ingredients are. Now we can begin our analysis. Now we're going to start by making an assumption which if we were worried about it we could actually watch some strings vibrating with a sort of high speed video camera and we could verify that this is true unless you go out of your way to sort of set up an initial condition that would break it. In general a vibrating string will have the property that at a particular point let me mark one away from where all the other clutter on my diagram is will just vibrate up and down in the center. So what I mean by that is it won't vibrate from side to side. So that point that I've marked in red will not go either left or right it will just go up and down. So there will be no lateral movement of elements on my string. With that assumption I can now go ahead and start balancing out some forces. So if our little piece of string is not moving left or right and it's not even going to start moving left or right what that means is that the forces must balance each other in terms of the component that is in the horizontal direction. So we can immediately write down a constraint on the horizontal component of those forces. So I'm actually switching to using the symbol T just to remind myself that in the context of an elastic string we can talk about the tension within the string itself. So the tension at point one has a magnitude T1 and the tension at point two is a magnitude T2 but crucially we want to equate the horizontal components of that and so that should be cause and we can confirm it's caused by realizing that if the angles went to zero then all the tension would be in these two directions. Okay so that's really helpful now we need to think about resolving in the vertical direction. Well I've just added in there that what we can think of is that since these two quantities must be equal to each other let's just give them the symbol T and that will help us in simplifying. Okay what's this second equation I've got here well here we have the difference between the forces that act vertically on the two ends of the string. So what we're now thinking about is will the string be in cut will the forces on the string the net forces make it want to accelerate in the up direction in other words speed up or accelerate in the downward direction and sort of come back towards the relaxed horizontal axis. So we can find that out just by looking at the diagram and seeing that F1 wants the string to the our little piece of string to go down. So it supports an acceleration in the down direction and F2 supports an acceleration in the up direction. So we can just take the difference of these two and whichever one wins that will dictate whether the acceleration is positive or negative. Now so that is simply the force that's on at the net force in the vertical direction and then on the other side of our equation here what do we have nothing else but mass times acceleration. So we know from Newton that force equals mass times acceleration. Our mass is the density per unit length that's what I'm using the row symbol for here multiplied by the length of our little component which must be delta x from the diagram and then acceleration is of course just the second derivative of the position with respect to time and so there we can write that down immediately. So that's captured the question of how forces are acting to course our little component of our string to move. Now before I go any further I want to combine these lines above into a form that will be more useful for us. I want to eliminate these quantities t1 and t2 so that I've only got some kind of constant tension which we've been calling just the simple symbol t and our angles and then all we have left to deal with is the angles. So how will I do that? Well I'll just substitute for t1 and t2 like this. We can see that t2 must be t the overall tension divided by cos beta and t1 is the overall tension divided by cos alpha just from the previous equation. So now we won't need the diagram for a bit we can simplify. There we are and that is going to be helpful because we can immediately see that these quantities are tan so let me write that in. How can we get any further? Well we have to think what tan actually means so now we will go back to the diagram. What does the tan of an angle anywhere on this curve mean? In fact because it's so cluttered up in the focus area of our little piece of string I could draw a line at some other point. It's the same answer anywhere so have a think about this. Here I've tried to suggest the tangent to my curve at this orange point. What does the tan of that angle there which we could call I don't know gamma or something we won't be using it of course because it's not part of the region that we're interested in. What does the tan of that mean? Well the tan in a triangle the tan is of course the ratio of the two sides right so if this side was x and this side was y then and we had a little angle which we'll keep on calling gamma in there then tan of gamma is the ratio x over excuse me the there we are. So what does that mean when we go to our curve? Well we realize that it the tan of the angle is simply the gradient. It's how fast is y changing with respect to x at that point. We could if we put little delta symbols here then we can see immediately what we mean as we zoom in to the curve and we consider a little bit of y a little bit of x we must obtain the tan so we can immediately substitute where we have our tan alpha and tan beta for the gradients of the curve at those points let's do it. Oops I see that I'd written tan beta twice apologies for that I don't know how I did that so that should of course have been tan beta minus tan alpha so let's put those gradients in. Well we're getting there what I've emphasized in this way of writing things is I want the gradient dy by dx at the point which is the the right hand end of our piece of string of course that's the beta point or that's point b where the angle is beta and then I'm subtracting off the alpha point which is back at where x is equal to x so that's what I'm putting in brackets here just to emphasize that we're dealing with two slightly different gradients and then all we have to do is think how can we write that difference in a more elegant way we know that that although the gradient might be quite strong the difference in the gradient will be very small because we're only considering a tiny piece of string and so there can only possibly be a small change in the gradient between those two points the trick is and this is the same as we used in the diffusion problem what we'll do is we'll say well the that first element in fact let's just focus on this first one if I want to write that in a different way I could say well that must be the gradient just at x plus a little bit of a change which would be the rate at which the gradient is changing with x let's write it out explicitly times that little shift so it's the second derivative the gradient of the gradient times however far we're going but then we can see of course what the trick is going to be is that the gradient x now appears twice and so we'll be cancelling it out and that gives us a very simple equation let me write it out there I've written out I've also taken the opportunity to correct uh sneakily correct another error I'm sorry I'd written d2y by dx squared over on the right whereas of course that side is by dt squared so there we are this gives us then quite a compact form it would do if I remember to put the dx there we are so that equation is what we've reached and we can see that we're going to be able to cancel the small the the distance delta x from both sides and we may as well move all the constants to one side and even give it a new symbol so that we so that our final equation has the most elegant possible form let's do that there we are we're done let me put a box around it because it deserves it now I've introduced a symbol c squared because I know it's going to be helpful to just define it as the constant squared in this way it's simply that ratio between the tension in the string and the density per unit length of the string okay so of course we want to try this out we want to build an example give ourselves a reasonable initial condition for a plucked string and watch it go and see if we can solve it but before that there's an observation we can make straight away from the very structure of this equation this was the innovation due to the alemba which let me check when the year was okay sometime in 1700s delta and let's see what we get when we simply make a very smart choice of change of variables something then will come immediately out of these equations that's quite profound so this is the variable change that I want it's simply mixing together the space and time parts and because space and time have different dimensions we'll certainly need a constant that makes that work and that constant is c which by the way tells us that c is a speed right so c the dimensions of c must be a distance over time in order to make these statements dimensionally correct so with these variables what happens when we try and transform our equation instead of being with respect to x and t make them with respect to these new coordinates so let's write down the rule that we'll use to change a derivative say with respect to x to instead be derivatives with respect to the new coordinates so this is the chain rule type structure we come up with we see that the way that function change changes with respect to x is can be expressed as how fast is it changing with respect to u times how fast is u changing with respect to x plus how fast is it changing with respect to v times how fast v is changing with respect to x so it in this is hopefully a familiar rule for you and it makes a lot of intuitive sense and now of course we can actually write down what some of these things are so du by dx holding t constant is just going to be one and dv by dx holding t constant is also going to be one okay so what we really want of course is to take the second derivative so let's write out what we're after but now we can just regard the dy by du and dy by dv as just functions and we win we made no special assumption about the function y in the first place so we can just reapply the rule to quickly jump to what we need so in what I've written here these first two terms come from just taking the second derivative of the first term there and similarly of course the second two terms come from taking our d by dx derivative of the second of those terms and just following the same principle that we used the first time but now we can collect things up so that's just tidying things up slightly but now we have a nice further observation to make which is that when you for a continuous function like y so we're assuming some things I said we weren't assuming anything about why I guess we're assuming that it's not mathematically pathological it is a function which is capable of describing a string so it's continuous and you know it has this kind of sane behavior that we expect from our function which means that when we take the derivative with respect to one coordinate and then the other doesn't matter which order we do that in and so these two terms that I've highlighted in the middle must be the same so I can finally write that as there we are that's about as neat as it will go so that is what d2y by dx squared becomes when we put it in our new coordinate system now of course we have to do the same thing for d2y by dt squared but we can go quite fast because it's an extremely similar process let's just start it off and see how it goes there we are exactly the analogous starting expression and so the the difference oops excuse me this is of course for the first derivative and what we see is that this time we have of course du by dt and dv by dt because we're considering the time derivative but these what we what we'll find then is that we need to put in the constant c for both these things because of course we know that u is equal to x plus ct oh and we should have a minus sign hang on and v is equal to x minus ct so this one is minus c fine so let's just need that up a little bit and now we need to take the second derivative let me just do that without wasting much more time there we are and as before you can see that we just apply the the principle we we found above is just apply the second time to get the second derivatives and now we can tidy up again there it is and we find that we have a c squared constant out in front and again I've taken the two terms which are with these ones and just combine them they are both they're equal to one another so we just get a factor of two there now let's summarize well let's insert these two statements so our original equation just up to a factor of c squared that we will certainly find is in the right place we were saying that we just set the d2 by dx squared and the d2 by d times squared equal to one another let's do that and see what we're saying in the new coordinates there we are so I've just substituted substituted in the equations that we discovered and we can see that yes the c squareds are in the right place and we can just cancel them out and it looks like the right hand side is equal to the left hand side fine fine fine except then suddenly we notice that this term the cross derivative appears with a plus two on one side and a minus two on the other all the other terms can just be cancelled so what this is telling us just by change of coordinates is that that cross derivative is equal to minus itself and the only way that that can be true is if it's actually equal to zero so the discovery is that d2 y by du dv is equal to zero what does that mean well it means that if we were to write our function y in terms of these coordinates u and v then taking the partial derivative with respect to one of these things holding the other one constant and then taking the other derivative must give us zero what does that tell us about the form of the function y when written in these new coordinates tells us that the following must be the form let me write it down so what we're saying now is uh-huh if we write down our solution to the wave equation as a part that only depends on u and a part that only depends on v and add them up it's an addition unlike in separation variables where we generally have a product this is simply an addition of two separate terms then that will satisfy our equation because clearly taking the derivative with respect to v holding u constant will kill the f term and then the other derivative will kill whatever has become of the g term if these things were a product if there was even so much as a let's say a times u over on this side then our rule would no longer be followed because it would mean that one of these two terms would survive at least in the form of a constant after we have taken both our derivatives so that imposes this form which is a pretty you know that's a pretty strong conclusion if we must be able to think of a solution to the wave equation as a solution that involves only one of these coordinates added to a solution in terms of the other there must be something special about these coordinates what are we saying so the way to understand the implications of this is just to consider one of these two functions and then we'll see what it means to have both so let's just consider the g function actually so i'm going to say that my wave shape which so back in the coordinates space and time is going to be equal to just one of these two things so that if i'm going for the v then that means it must be a function of x minus ct that's no longer the general case that's just one of the that's a special case that only one of these two things that i'm allowed to use i'm actually gonna use and i'm setting i'm just choosing to set f is equal to zero for now what does that mean so let me draw out a wave or a distribution of y and we'll assume that this is at some particular moment in time there we are i've just drawn a curve nothing special about it and this is at some particular time i will call it time t z or the time t one now what i'm interested in is what will this line look like at some later time t2 well let me just draw on what we're going to get at t2 and let me just then i'll justify why i can get the function at some time t2 just by moving on a bit like let's move on a little bit like that so what you can see i've done here is just take the take it and move it a bit right so that's just a shifted on version yeah maybe i can make that more clear by drawing over it hang on and now if i erase underneath it there we are that's kind of nicer so the pink curve is the t2 line that's at t2 and the green curve was at t1 so what i'm saying is that if my function my solution for y can indeed be written as some function of just the difference between x and ct then at later times the shape of my curve will be the same as at any particular moment that i care to look at it it will always keep the same shape but it will just move along the x-axis and i wonder if you can see why so if i just consider that so let's choose a particular point i think this is the way to do it just focus on some point of interest let's take that particular point there and we'll call that x one and then the analogous point later on if my rule is correct will be x2 so provided that x2 is equal to x1 plus c times the difference in the two times as a positive thing so time two is later then we find that if we compute the thing that actually goes into the g function well for the green curve it was x1 minus ct1 and for the pink curve it's x2 minus and then well sorry ct2 but that if i use the substitution above there is in fact just going to give me x1 and then i'll have plus ct2 minus ct1 minus ct2 running out of space let me move up a bit ct2 and of course we've got a cancellation there let's change color cancellation pen we get to cross out that cross out that and we find that we're feeding exactly the same number into our function g and so of course we get the same number back so maybe stop and stare at that a bit to convince yourself of what i'm saying but because the function g can only see this hybrid quantity that is x minus ct then if it has if we can draw the shape of our curve for some particular fixed value of time and sweep x then if we consider a later fixed value of time we just need to move our entire function along a bit so what this means is that our function g which was one of only two parts that were allowed to use just the f and the g parts if we are writing our solution in terms of these new coordinates our function g is simply describing a wave that keeps its shape but moves in the positive x direction and then you can guess that if we were to have a careful think about that other function that we're allowed to use that f object that would be a wave that keeps its shape but moves at speed c in the negative x direction and that a general wave it's already interesting that those things exist right that that a solution to the wave equation or a solution allowed by the wave equation is a thing that keeps its shape and propagates in in the x direction in either direction but more than that what this is telling us is that we should always be able to think of a solution to the wave equation as something that moves in something that moves in the positive x and some potentially some completely different function could be a different shape but it moves at the same speed speed c in the negative x direction so that's I think a really nice observation to make just from looking at the fundamental equation itself and doing a coordinate change and this of course is exactly what we see right so when we throw a rock into a pool then we see ripples and those ripples move with a fixed speed if we throw a bigger rock in we get bigger ripples but they will move with is to first approximation the same speed of course there are higher order order effects in any particular medium like water dissipative processes and so on but to a first approximation in any medium and in mediums like electromagnetic radiation it's then strictly the case that we see waves propagating keeping their shape and moving with a fixed speed and they can move backwards and forwards across the medium order can be waves that go one way waves that go the other way and they interfere with each other so we the waves add up but that capturing the idea of something that keeps its shape and just moves is what we immediately discover just from this very elegant argument so I think that's really nice but now but now we're going to think about a scenario where we're fixing a string at two points and while we could try and express this in terms of waves propagating one way and propagating the other way that's no longer the natural way to think of it and so let's see what we find out when we investigate that by the way briefly going back to our original diagram if you were being very alert you might have said why did it matter that that I insisted that our string was fixed at x equals zero and x equals l where did that come in and you'd be right it actually didn't come in at all it was part of our story because it's helpful to then think of a string that has a certain tension because we imagine that it stretched to some extent even when it lies flat along the x axis so it's under tension and then when we pluck it it vibrates but we didn't actually use that property and in fact any it could be an infinitely long string if there could be such a thing as long as it had a certain tension along the string but now we will be using that fact but really for the first time because we haven't used it yet in our analysis so there I've repeated the diagram we had before but I've cleaned up all the annotation which we don't need now all we need is the conclusion we reached which is the wave equation but now I want to think how would we set about solving this in the coordinates x and t well as before when we were looking at the diffusion equation in general it's difficult to solve an equation of this form for all possible solutions which may mix together the x and the t coordinates in a complex way but what we can do is we can start to make some progress by saying let's suppose that the way my string is behaving can be described as just a product of a function that depends only on the spatial coordinates x and a function that depends only on the time coordinate so in other words we say let's limit ourselves and it's a huge limitation but we know from experience that we're going to be able to break free of this but initially we will say let's imagine that we have some function let's call it capital x of x and some function capital t not temperature but time of time so if we have a solution of that form we know that what we can do is then greatly simplify our equation because the partial with respect to x doesn't care about the time function and vice versa and so we can immediately turn our function our partial derivative has actually become full derivatives let me jump to the punchline because we have done this before what do we get so we get this function immediately or this equation and we can we've turned our partial derivatives into total derivatives because the thing they're operating on doesn't even mention the other variable and then our trick is that we divide both sides by the whole solution y and that leads us to and then the next line very similar to what we did before is to divide throughout by the whole function y and actually we'll divide by the constant c squared as well so that will just turn out to be a bit more handy so what they will then find is one over x capital x our function d2x by d coordinate x squared is equal to one over c squared one over t d2 capital t by coordinate t squared so here I've completely put all the x related quantities on the left all the time related quantities and the constant by the way on the right and as before we then have the argument that well since this is true for all values of x and t I could for example hold t constant and sweep x but then the left hand side that depends on x isn't actually allowed to change because I'm holding t controlling the right hand side constant and so it must be that the left hand side is in fact always equal to a constant regardless of the value of x same argument works the other way around these two things are both equal to some constant and now I just get to make up a symbol I go to call it minus k squared in the notes I think I use psi and that's just harder for me to write so I'm going to use k squared why minus k squared because I know that that will work out neater in the next few lines but I could have used any symbol that doesn't have to be squared and it doesn't have to be minus but then I'd find myself putting in extra stuff in the next few lines so what do we get as our separate our separate equations two similar very similar looking expressions now remember that when we were doing the diffusion equation because we only had the first derivative with respect to time our space and time equations at least were different to each other at this point but now they're practically identical the only difference really is that c square term on the right hand side and the solutions to these things are both both of them are solved the same way which is cause and sign like solutions so let's write down the general solution for each of these equations there we are that's the general solutions and I've introduced just temporarily a capital c symbol just so that I can go a b c d for my unknown constants don't worry we'll get rid of that in a minute because it is confusing to have two kinds of c around although I guess I've also got two kinds of x and two kinds of t anyway those are the sign and cause like solutions that will solve our separated differential equations what does that mean it means that a general solution that is separable must have the following form so there we are I've just multiplied the line above so I've multiplied the x and t explicitly so we can see this is the general form of our solution where capitals a b c and d convey anything we like there's even you could argue a bit more freedom than we need there in the magnitude of a you know we could move some of the weighting between a and b and c and d but that's fine we'll leave it in that form that is a separable solution but our equation is a linear one the full equation when we go back here this is a linear differential equation just like it was for diffusion and that has the very very powerful consequence that any solution we write down call it solution one and any other solution we write down solution two can always be added together to make a new legitimate solution we can keep on doing this if we thought of many different solutions we could add them all together and that would be a new solution what does that imply for our purposes well of course this solution it has freedom in the capital letters a b c d but the most interesting thing is this constant k which actually changes the form of the function changes the frequency of our signs and causes that can be any constant we like once we've set one of these then the other ones must use the same k but we can add we can we can add together an entire solution like this to a second entire solution using a different value of k and we can keep doing that for as many different k values as we like so what we find is that the general solution has to be of the following form or can be anything of the following form so what I've done there is I've just copy pasted the line above and put n subscripts all over the place so now my a symbol is in fact one of a family of constants a n a 0 a 1 a 2 and so it's my b and so it's my capital c and so it's my d and moreover this k quantity is also a set of different k values so for each particular k value there must be a cos like or sine like pi it's a sine of sodium part that varies with that k frequency multiplied by a time part that varies with the same k frequency multiplied by the constant c that pairing must always be respected but I can add together as many such pairings as I like and so this is very similar to the situation we were in with the diffusion equation of course except that there instead of this sum of signs and causes in time we had if you recall an exponential that was something like minus k squared and then some constant such as alpha for our diffusion and then t so instead of that on this occasion both the space like and time like parts are oscillatory but it's still the same basic idea because what we can do as we're about to see is plug in time t is equal to zero and bam we've got a boundary condition expression which is essentially an invitation to use Fourier series to describe our time t equals zero arrangement but let's let's step through that now what I'd like to do is think about anything further we can say about these constants a b c d given that what we're going to be interested in for the rest of the lecture is a vibrating string that is pinned like a guitar string at two points what can we say well the first thing that we can say is that when we set time t equals zero or in fact any fixed time it must be the case that at x equals zero we have no displacement at all and so that makes me come and focus on this term here and decide that I need it because what that will give me is a finite value for the display for the x is equal to zero but I am never going to need that I'm always happy that my x is equal to zero point has zero displacement a different way to say it is that I could imagine mirroring my string which I will just draw sort of with some dotted lines here I could make it into an odd function and mirror it's not really mirror so much as it is rotating it around in the usual way of an odd function and so that would be extending my solution and of course we do that when we use Fourier series to describe a finite problem a solution of that kind will always be fine that's in fact the correct way to extend this fixed function into the negative x direction and also into the direction of great x is greater than y we should extend it on by essentially considering what an odd function would look like where the origin is at that point the reason being that then there will never be any force any net force in the up down direction if you zoom if you imagine zooming in fact I can zoom in contact so having zoomed in on the x coordinate if I mirrored my function I keep on saying mirrored what I really mean is rotated 180 degrees so if I if I've enforced the principle that it's an odd function at that around the origin then okay I haven't drawn it perfectly but if I had drawn it perfectly you could see that the forces on the point at the origin would always perfectly balance out which is just what we want because we don't want the origin to move and as long as we were considering a an extended solution that by the way happens to have the property that the origin never wants to move then we're safe to stick a pin in it and say well I've pinned the origin so we've we've sort of constructed the scenario where the origin never wants to move by using the appropriate extension to our function all right and same argument works out at x is equal to l so what does that tell us about our constants well it it said we've I've just argued that I want an odd function in terms of the x coordinate just by thinking about the x equals 0 case so that allowed me to immediately say that all my an constants for this kind of solution for this fixed string should be zero all of them so that was easy got rid of one whole family of constants can I keep going in the same spirit and get rid of some other stuff well the next trick is actually to think about the velocity of the string so what I would like is to limit myself to cases where the velocity of the string at time t equals zero is zero so now I'm being very specific I'm saying this is a plucked string at time t equals zero so I am holding the string at time t equals zero and I release it so it starts from zero velocity if I'm willing to further constrain myself to those scenarios then I can take a look at the time part and just think about what that looks in fact I can write it out what does it look like in terms of a velocity oops so if I put a little dot above the y to show that that's the derivative of y with respect to x well of course that wouldn't affect the x part at all although by the way we've already deleted all the causes we've argued that they should go away but the cos terms here will become sines and have a minus sign in front and the sin terms here will become causes and we'll also oops not chalk cos and we'll also by the way have picked up a factor of lowercase c in front but that doesn't matter all I all I actually want to focus on is the fact that there's a family of sine like and cos like terms but at time t equals zero I want the velocity to be zero where that leaves me is we've argued we don't want the cos like spatial terms because of the pinned at zero condition we don't want these cos terms but they actually came from sine terms in the original expression of course so in fact what I should really do instead of me applying my deletions here I should go back to the expression above and say up here I don't want the causes and I don't want the sines because they become causes and I only want the terms that become sines and give me a zero velocity at time t equals zero so that simplified it a lot as long as I'm only thinking about a plucked string then I I reckon I should be able to do the job just using a solution of the following form that I reckon should do the trick I've introduced a symbol e as the constant that can it's just a pure number of scalar that is just telling me how much of each one of those pairings I want and they do have to be pairings now so the same k n must occur in these two things but of course I can choose to have as many different k n's as I like okay so there we've written down an expression that I'm claiming is going to give us all the generality we need to solve plucked string problems and in fact there's still one more thing we can do to simplify this expression or put a further restriction on it and that comes from our choice of these k constants do we need to consider absolutely every possible k no we actually only need to consider ones that are consistent with the periodicity of the function we're trying to build and we're going to use our trick as we've already said of extending a non-periodic function to make it periodic let's zoom out in fact the risk of making this look really silly and small let me try and extend this function okay that's the best sketch I can do it's not a great sketch but what I hope I've managed to communicate is that in the region from x is equal to l to x is equal to 2l I've just got the the same shape as between 0 and l but rotated 180 degrees around so this kind of mirror inversion that I was talking about and it's consistent with what I've drawn over in the x is negative region by the way because that would just match this region here of my overall repeating function and just for the same as the argument at x is equal to 0 the reason I like this is if I zoom in at the x is equal to l point I can see that the forces will forever and at all times be perfectly balanced so there is no force that's trying to pull my string away from its anchor point and that's why this periodic solution that is for all values of x is going to align with the scenario where we have just a finite function that's why it's the correct continuation but then the period of this thing is clearly 2 part a 2l the overall period is 2l so we should be able to use values of k oh let's go for a thicker pen again that our that are 2 pi over 2l times n and we can cancel the 2s of course so finally that gives us the form that we need to you we need to fit a Fourier series 2 we will set time equal to 0 that will cause the cos terms to disappear for our boundary condition and we essentially have the task of using a sine Fourier series to build any initial condition that we want for a plucked string but we know already what we want because a plucked string is essentially where the string makes a straight line between anchor point 1 and anchor point 2 and where we have our plectrum or thumbnail or whatever plucking it so let's draw that well and oh and we've already argued quite carefully how we're going to extend this Fourier excuse me extend this function which is that this is this is the up half and what will happen in the x is equal to l to 2l it's the the down part of our triangular wave so this is a triangular wave that's centered on the x axis and has a frequency of has a period of 2l now we can just write down what the Fourier series is for that because we've already worked with triangular waves before I'll do that now this that I've just written down is the Fourier series we want it may look a little bit different to the Fourier series we wrote down before for the triangular wave but the reason for that is that last time we considered the triangular wave we made it centered on the origin so the peak of the triangular wave was centered on the origin so in other words it was an even function made out of causes this is the shifted version that creates the same triangular wave but now as we want it now passing through zero at the origin so that's why we have a slightly different form with this minus 1 to the power of n that's a consequence of that change you can get from one Fourier series to the other just by introducing a shifted x coordinate for example now what I've written down here this is the the more primitive Fourier series actually isn't quite matching our scenario yet because it has a period of 2 pi and it has a it goes goes between plus 1 and minus 1 whereas we want to go between plus a and minus a and we want to have a period of 2l so now we need to just adjust that and that will be our solution at time t equals 0 there we are that's just made those adjustments so that should be our time t equals zero solution what then is our solution for all times well we just have to go back to our rule here and see that for every sign that's varying with some frequency k subscript n we must pair it with a cos that has the temporal frequency c times k which that's the same k so we just add that in now there we are as is often the case with these Fourier series type solutions we end up with quite a a big expression especially since here we we didn't want to have a period of 2 pi we wanted it to be realistic so that we could type in an l number so we have those constants floating around here but conceptually it's a pretty easy thing once we're familiar with Fourier series it's just our Fourier series that solves the boundary condition with the appropriate time like factor inserted for each term so to be very clear about it this quantity here is the same as this quantity here and you can see the extra c factor that's speed of our wave is the is correcting between the spatial frequency and the temporal the temporal frequency i guess angular frequency so there we are that's our full solution so here we are in Mathematica i've set the amplitude equal to 2 the length of our the distance between the fixed points on our guitar string is equal to 4 i've set a value for the speed and then as you can see here i've defined our y function exactly as we derived it just now i've also introduced a symbol kn just to keep things a little bit not too crazy and you can see that the the key line is this one here which is exactly copied across and i'm considering 20 or 21 terms so a pretty good but not infinite number of terms in our Fourier series and i'm already plotting out for us here what it looks like at time t is equal to zero over a range which is slightly bigger than the full range so remembering that the way we extended our function it actually has period two l even though we're any interested in the part from zero to l this is exactly what we expect to see right it has a period of two l that's eight and it has the right amplitude and it has the right triangular waveform so now let me tighten this up a bit now i'll zoom in to say i don't know negative 0.5 to l plus 0.5 just so that that's the region we're really interested in just going a little bit beyond the fixed points and let's start running time forwards and see what happens so let's run it forward a little bit uh what do we get oh that's easy quite a large bit all right let's run it by and it's quite strange shape right so this is a moment after we've released our pluck string and it seems like the prediction that we're getting from our equations is that we have actually a straight line it's quite strange as we've released it we've got a flattened off triangular wave what happens if we run on so we were already looking at this one but this is what we see later so that flat region just widens and the triangular wave just kind of sits there waiting for this flat region to reach it quite strange and then if we go to maybe a further 1.5 okay so let's maybe go to 2 then and maybe 2.4 it's doing a mirror image of what it did before it's starting to build that same triangular form but now in the negative uh wide direction so if we go out to I don't know 4.4 or something like that we see it's it's it's nearly finished building that and then we can guess what's going to happen it's going to oscillate backwards and forwards in fact uh Mathematica has a function that allows me to um to to to control time just with the slider let's use that so here's exactly the same uh thing again but this time I've got a little slider that I can move and I'll do that and here we see there we are that's what Mathematica predicts let's do it again so we should see this collapsing uh ripple essentially and then it builds itself up in the negative y direction now and then it would come back again of course and uh it will keep on doing that now that do we believe that do we think that that's actually what would happen if let's not say a guitar actually a guitar string I keep on saying guitar string guitar string has a very it's not very stretchy so if we wanted to see this um you know in a real experiment we would actually want to use something more like a piece of elastic because it's more stretchy so we would be able to deflect it much further and and see and you know see something like what we see literally on the screen can we do that experiment well I can't do it for you because I don't have the the gear here but fortunately I found a YouTube video and I should put the reference to it into this video because it um someone else has done the work for us and we can see whether what happens in the real world is a bit like this or have we really just made so many approximations and assumptions that what we've ended up here is an artificial solution okay so here we are the moment of truth has our maths been any good here is someone has done exactly the experiment for us there's that stretch string it's being held by some kind of little release mechanism and I can sweep sweep forward in this video just like I was on the Mathematica but what kind of shape will the wave have let's have a look look at that pretty much exactly what we predicted and it even inverts down and starts to build the same thing so we're seeing seeing some slight curvature but wow look how perfect that is okay so I think with that success story I think it's a nice moment for me to sort of wrap up the lecture so there we are that that I think is a really nice vindication of the material certainly in this lecture but of the techniques we've used in general now at the beginning of the course I promised that we would introduce some tools Fourier series and Fourier transforms that would allow us to describe real world situations that otherwise would be very very tough and indeed even something like the triangular wave it's not a very natural thing to describe mathematically unless you understand that it can be broken down into the natural sign and cause entities so there we are it it works and uh let's leave it there and I I hope you've enjoyed the course thanks a lot