All right, good afternoon everyone just a couple things at the top and we'll get right to your questions First up secretary Austin Currently remains hospitalized at Walter Reed National Military Medical Center and is in good condition For the statements we released yesterday Secretary Austin was transported by a security detail at approximately 2 20 p.m. Eastern time yesterday to Walter Reed National Military Medical Center to be seen for symptoms suggesting an emergent bladder issue At approximately 4 55 p.m. The same day the secretary transferred the functions and duties of the office of the Secretary of Defense To Deputy Secretary of Defense Kathleen Hicks Deputy Secretary Hicks continues to retain the functions and duties of the Secretary of Defense at this time According to Secretary Austin's doctors after a series of tests and evaluations He was admitted yesterday evening into the critical care unit at Walter Reed for supportive care and close monitoring Now shortly before today's briefing we released an update from the Secretary's doctors at Walter Reed regarding his status and To ensure everyone here today and those watching have the same information I will read that full statement. This is a statement from Dr. John Maddox trauma medical director and Dr. Gregory Chestnut Center for prostate disease research of the Murtha Center Director at Walter Reed National Military Medical Center in Bethesda, Maryland Beginning the statement quote the Secretary of Defense Lloyd J. Austin the third underwent non-surgical procedures under general anesthesia To address his bladder issue. We anticipate a successful recovery and will closely monitor him overnight a Prolonged hospital stay is not anticipated. We anticipate the Secretary will be able to resume his normal duties tomorrow Current bladder issue is not expected to change his anticipated full recovery. His cancer prognosis remains excellent end quote Moving forward. We will continue to provide updates as new information becomes available regarding Secretary Austin status And we'll certainly we all certainly wish him a speedy recovery in the meantime Secretary Austin will no longer travel to Brussels this week as originally scheduled However, Wednesday's Ukraine defense contact group will continue it albeit virtually While Secretary Austin currently intends to participate in the virtual UDCG He will remain flexible depending on his health care status Assistant Secretary of Defense for International Security Affairs Dr. Celeste Wollander who will be in Brussels this week is prepared to represent the Secretary is required and The chairman of the Joint Chiefs of Staff General CQ Brown Jr. Will also be participating in the UDCG virtually from Washington, DC Ministers of Defense and senior military officials from nearly 50 nations will convene virtually to discuss the ongoing crisis in Ukraine and the continued support from the international community to provide the Ukrainian people with the means necessary to defend their sovereign territory Separately US permanent representative to NATO Ambassador Julie Smith will represent Secretary Austin at the NATO Defense Ministerial Scheduled for Thursday. We will be sure to keep you updated on outcomes for from both of these important meetings Switching gears this month marks the two-year anniversary since the White House released its Indo-Pacific strategy In the last two years the Department of Defense has done more than ever to advance a shared vision for a free and open Indo-Pacific region. We've made historic advancements and alliances and partnerships with the Republic of Korea, Japan, Australia, Philippines, India And others across the region and even as we confront challenges elsewhere The United States will continue to prioritize our efforts in this critical region in support of international security stability and prosperity For more information, please see our fact sheet on the DOD website And finally our prayers are with the people of the Philippines as our long-standing ally faces severe flooding and landsides landslides from a series of storms US Marines from the third marine expeditionary force have assisted with two C-130s to help the Philippine government and USAID In delivering 15,000 food packs to affected families For more information, please contact Marine Corps public affairs and with that be happy to take your questions We'll go to Associated Press Tara Thanks general writer. Um, I want to get a sense of how serious this is. This is Secretary Austin's second uh trip to the intensive care unit at Walter Reed since his surgery He's canceled his trip. Can you give us a sense of where he's at in his recovery? And is this latest hospitalization part of ongoing cancer treatments? What can you give us as a status on his prostate cancer? Yeah, thanks Tara. I I'd point you back to the the statement there That I just read, you know in in terms of the secretary's condition as highlighted. He's in good condition according to his doctors He will remain in the ICU ward for the duration of his stay in order to provide appropriate privacy And will continue to receive critical care support in order to closely monitor his progress for now And and as I highlighted in the statement from his doctors Uh, his current bladder issue is not expected to Change his anticipated full recovery from cancer and his prognosis for cancer remains excellent Is the bladder issue a complication from his surgery to treat prostate cancer? Or is it part, you know, is it related to his prostate cancer? Uh, Tara, I don't have anything to provide at this point beyond what I've shared with you But certainly again as we have more information we'll be sure to pass it on Last, um, you know since his hospitalization in an effort to be more transparent Can you give us a sense of what stage his prostate cancer was? How early was this detected? Is this something that? You know hospital you're expecting future hospitalizations because of this ongoing fight Yeah, I think as you've heard us say before, uh, his cancer was detected very early And so as his doctors have previously highlighted He is expected to make a full recovery from His cancer diagnosis There is no further treatment associated with his cancer expected other than the physical therapy that he's been undergoing To deal with some lingering leg pains, which again, he highlighted during his press briefing So as you highlight, uh, we will continue to keep you updated with as much information As possible regarding the secretary and his health status and be as transparent as we can or Has secretary austin spoken to the president since his most recent hospitalization since yesterday afternoon Has he expressed any reservations about his his own ability to serve to the president or to anyone else in the administration? And then when he is released is he expected to work from home or we'll be straight back in the pentagon Yeah, thanks orant. Uh, he has not spoken to the president to my knowledge Again, you went into the hospital yesterday to get checked out and again I'd refer you back to our statements and and what I just read out I can assure you the secretary is continued or continues to be eager to perform his duties And and I have no doubt of that But in terms of uh, what's next regarding his health care again, we'll keep you updated jennifer European allies have expressed a great deal of consternation in recent days Due to the ongoing debate in congress about whether the u.s. Will continue to fund ukraine Recent comments by former president trump over the weekend about What he would do to nato and messaging to prudent in russia what he would allow them to do to nato Why isn't deputy defense secretary kath hicks or somebody of a senior stature going to brussels and participating in these Meetings. Yeah, again. First of all our support for ukraine continues to be Ironclad as you heard, you know me highlight You're going to have nearly 50 countries come together as they have on a monthly basis to talk about how we support ukraine And and as I also highlighted is the secretary's intent to participate in the virtual udcg again We'll keep you updated on that front and dr. Wallander as you know has been at the at the center of Working for the department of defense to represent the united states and our support for the udcg So she is well experienced well equipped To represent the secretary should she need to do that And she as I mentioned will also be in brussels To support there both virtually but also support the ambassador as she represents the secretary at the nato defense minister Idris just to follow up on that obviously Celeste and julie smith are well qualified, but deputy secretary hicks would be a representation of how seriously The administration takes ukraine and its needs So does the secretary not have confidence in his deputy to send her while she's performing his duties? Yeah, I reject that notion the secretary of course has full confidence in the deputy secretary of defense He has transferred his functions and duties to her as evidence of that And again anyone who questions our commitment to ukraine Just needs to look at the facts in terms of the amount of support and the amount of assistance We have provided and continue to provide ukraine and will continue to provide going forward How long was the secretary unconscious or under general alice easier force since yesterday? Is it throughout is it a couple of hours here and there do you have a sense of that? Idris at this time I don't have anything to provide beyond whatever relate to you But again, we'll continue to keep you updated as new information comes in David so when in that sequence from 220 being taken to the hospital 455 turning over his responsibilities. Did he inform the white house? The white house was informed that he was being taken to the hospital So that happened before he actually departed to go to the hospital And in addition in that period congressional notifications were made as well so The statement says we'll be able to resume his normal duties tomorrow, but you just Told us that he's cancelled A planned trip. How do you square those two? David again, uh, we'll keep you updated on his status This is a statement from his doctors explaining that they expect that he'll be able to resume his normal duties Again, we'll keep you updated on whether that includes from the hospital or if that is from home But again, all indications based on the on the doctor's statements is that uh, you know, he's gonna recover well He's in good condition and we'll keep you updated Yeah, secretary has uh the secretary received any radiation treatment and what caused him What symptoms did he have before going into the hospital? Was he In a lot of pain was he bleeding and then can you say exactly the procedure that was done when he got there? And again, I'd point you back to the statement from his doctors As we get more information, we'll certainly share that with you again as I understand it This was related to a emergent bladder issue. It was not related to his cancer treatment Again, and as the statement that I just read out to you highlights his prognosis for that is good And and the bladder issue should not affect his recovery from that And you can't say whether or not he's gotten radiation therapy To my knowledge, he is not Again, I'd point you back to the statement in terms of the the treatment that he's received Not related to his cancer treatment, but this is a complication of the surgery, correct? Again, I'm not a doctor I'm not going to try to Speak medical speak. I'm just going to point you back to the statement and as we get more information to provide with Certainly will have a I told you this is a Or anybody on his staff that this is a Complication he was admitted. He went to the hospital yesterday for an emergent bladder issue His doctors have said previously That he is expected to make a full recovery from his cancer diagnosis Which is what I just highlighted in the the top of the statement So again as we have more information to provide we'll certainly do that Courtney That's my question too. So just to be clear, you're you're leaving open the possibility. This is a totally separate issue from his complication from his Prostate cancer surgery in december What I'm saying is what is in the statement that the current bladder issue is not expected to change his anticipated full recovery His cancer prognosis remains excellent. So again as we get more information from his doctors to provide We'll be sure to share that if it's possible to get the question I mean, I think there's an assumption that that this is that this this emergent bladder issue is related to the Complications that he had on new year's day. So if that's not the case and this is something completely separate I'll take the question. Thank you. Thank you. Fadi two questions on on raffa yesterday The israeli is attempted like a rescue mission. It seems and up until now information Still being updated, but it seems like Maybe up to 100 Palestinians were killed In in that operation the u.s. Have assets in terms of UAV Doing some work to help the israelis recover Some of the captives was the u.s. Involved in that operation And you have any comments about the civilian death toll from from that the aftermath of it So fadi when it when it comes to israeli operations, I'll refer you to them in terms of talking about that And when it comes to civilian casualties as you've heard me say many times We do not want to see civilian casualties or innocent civilians killed whether they be palestinian or israeli And and that will continue to be our view going forward and But did the u.s Play any role in that operation? Fadi i'm not aware of any role the united states Yeah, and as you probably heard from israeli prime minister netanyahu seems like he's planning to Invade a ground invasion of raffa where more than 1 million people Are taking refuge amnesty international is warning of an imminent risk of genocide Does the pentagon support an israeli ground invasion of raffa at this moment? Again, you know, i'm going to refer you to the israelis to talk about their operations I think mr. Kirby has addressed this in his press briefings. You know, we certainly Do not want to see Any more suffering human suffering among those in gaza, which is why again we continue to communicate actively With our israeli partners on the importance of taking civilians into account in planning and conducting their operations.  I know there was a point where general glenn was there and we kind of got a little bit of fidelity on that Well, yeah, I mean, I think I think All echelons, Courtney. I mean you've had the president speak to his counterpart You've had secretary austin speak to his counterpart You've got us central command speaking to their counterparts the chairman of the joint chiefs speaks to his counterpart So there's ongoing discussions as it relates to israel's defense of its nation As well as the support we're providing and the importance of ensuring that humanitarian assistance gets through Fadi's question about rafa, though I mean, is there is there a level of are there are there people in the military right now or the defense department or cent com or Wherever it is who are providing like technical. I'm not aware. I'm not aware of us Understand their plan again. What he's talking about is public comments by the prime minister saying what they're going to do And again, that's something that they need to talk about. Let me Okay, and then let me move over to the other side of the room Did the israeli is share any planning with the pentagon about this Ground invasion Again, fadi. I'm not going to have anything to provide beyond what I've talked about already and I'd refer you back to mr Kirby's comments joseph Can I just ask do you have any updates to provide on yesterday's hmc? iraqi's What I would tell you is that You know yesterday's meeting marks another important step along the path of transition as military technical talks as you highlight We're held in bagdad As I understand it they discussed three areas of assessment the threat of isis the operating environment and iraqi security forces capabilities and so You know, we'll continue to keep you updated as appropriate But these talks are important again as we look to discuss the transition of the coalition military mission To the longer term u.s. iraq bilateral security Do you have any updates on the tower 22 investigation? I do not I'd refer you to sankham for any updates on that missy Just to follow up on joseph's question the um iraqi military spokesman said that Giving a readout of that of those talks yesterday said that based on the discussion yesterday The timetable will be set for the gradual reduction of international coalition advisors and an end of the coalition mission Is that correct that there was that the outcome was an agreement that there would be a timeline set for that and then separately could you give us an update on The review the internal review about austin's hospitalization Did you have a chance to review that yet and could you update us on the publication of that? Yeah on your first question missy? I I don't have any information on that so I'll take that question As far as the review no changes from what I provided on thursday The secretary does have that review He will continue to take a look at it Again, our commitment is to try to provide as much information about the review to you when we can So certainly we'll continue to endeavor to do that lar Thank you. Um, first of all is capyx acting sec def now and at what point does she become acting sec def? Can you just clear up what her role title is right now? Yes, she has uh Assumed the duties and functions of the secretary defense. So in that uh, since she is acting as the secretary defense And I still don't understand why they couldn't have had the ukraine contact group meeting without secretary austin I mean, what was what was the reason for making it virtual? I'm not sure I understand your question. So why I guess what was the reason for making the contact group? Meeting virtual was it solely because the secretary couldn't attend because there's many other people that attend that meeting, right? You know lar, I think the as I highlighted the secretary intends to participate. Um, and again, uh, given the The importance as we've already discussed about we want to ensure that this continues and You know as we've done in the past, we're going to be flexible in terms of the format and so Yeah, we have attended for I'm not going to get into hypotheticals lar Yeah, I'm not going to get into rationale for that Well, I've read to you what we're doing and there you go. All right. Let's go to the phone here. Jeff shogle task and purpose Thank you. Uh, on what day did secretary austin undergo this procedure on when he was under general anesthesia? And uh, secretary austin is sixth in presidential line of succession. Does that mean that secretary hicks is temporarily Until the the secretary assumes duties that she is now sixth in line in succession Yeah, thanks. Jeff on your on your ladder question there I'll take that uh, and we'll come back to you in terms of my understanding is uh, that that procedure would have been today Um, so just leave it there. Okay Follow-up question about the secretary's health. So I wonder if Uh, it is difficult for the secretary to travel to the in the past week for a while because In general, the traveling to the region takes longer than any other region Uh, are you asking me why he hasn't traveled to Because of the secretary's health program I wonder if it is difficult for the secretary to travel to the region to the in the past week region for a while Yeah, thanks roe. So at this point, you know, clearly, uh The secretary's focused on getting better As I highlighted, he's not traveling to brussels But I certainly expect that he will travel to the asia pacific region in the future When he is able to we don't have any travel to announce And so again, he continues to I'm sure remain eager to perform his duties to include travel when when he is able to do that So thank you. Time for one more. Yes, sir. Thank you, but Regarding hmc discussions If you know how long the discussions will continue and Just bracket brahmanistah has said that we are ahead of ending The mission of alliance forces like even the united states force in iraq So I don't have a timeline to my knowledge that hasn't necessarily been bounded. Of course that will be decision between The you know within the hmc to include in consultation with the government of iraq And then as far as us presence in iraq again, I'd point you back to what I said earlier the hmc will will discuss the Transition from a military coalition mission to the What the us presence will look like as it relates to a bilateral us iraq relationship Our forces continue to be there at the invitation of the government of iraq I'm not aware of any specific official requests at this point in time for us forces to leave And we'll continue to keep you updated. So all right. Thanks very much. Everybody appreciate it We have a request to hear from the deputy defense secretary now acting defense secretary cat hicks We've never had a chance to question her at the podium or hear from her. Okay. Thank you. Jennifer. Appreciate it Doctors just to get an update on his status. I will take that request. Thank you very much everybody. Thank you


Good morning. You will recall in the last class we were interested in determining if a blast wave hits an object how much of the wave gets transmitted into the object and how much gets reflected. Therefore let us continue on that. We define something known as an impedance z which we said is equal to the change in pressure divided by the change in velocity. We call velocity as u prime, the change in pressure as p prime, the change in velocity as this. And in the limit that the wave is an acoustic wave, we determine an expression for this impedance which we called as mechanical impedance as equal to the density of the medium into the sound speed in the particular medium. You will tell me that well it is for in the limit of sound waves that means a very weak shock waves. Is it really applicable for strong shock waves? We will take a look at that but we will keep this assumption clear. It so happens that it can still be used and I will justify it a little later. But let us proceed with this impedance and the problem which we are considering is supposing I have let us say a medium, some medium and into this medium let us say a wave is propagating and at the end of it it meets some other medium let us say over here. How much of this wave gets transmitted into the second medium and how much gets reflected back into the first medium. Therefore before we do that I think we should have some feed for this impedance value which we called as mechanical impedance or impedance and let us put the values for some particular medium or some particular materials of construction. Let us say if the medium is let us say air we found that the value of the impedance or we called as a mechanical impedance the unit B you will recall it was Newton second by meter cube and the value for air was determined to be something like 380 plus the actual value is something like 440 Newton second by meter cube. Let us put down a few of these things it will act as a guidelines to develop further on this. Let me give a few values for water the value is around 1.5 into 10 to the power 6 Newton second by meter cube for something like the tissues in our body that is the fact tissues in the body the value is 1.33 into 10 to the power 6 something like a muscle muscle in the body that means we are talking of human beings we are talking of tissues and muscle the value is around 1.7 into 10 to the power 6 Newton second by meter cube when I consider a bone in the body bone is something harder the value is around 6.6 into 10 to the power 6 if I consider a brick which is used for a building well the value is 7.4 into 10 to the power 6. Let us take one or two more materials the glass is somewhat higher let us say Pyrex glass the value is around 13 into 10 to the power 6 well if I consider steel which is very much harder than brick the value is around 46 into 10 to the power 6 and the last material which I consider is a hard material like tungsten the value is around 101 into 10 to the power 6 therefore you see that the value changes from a few hundreds to several millions and it is these values which really decide whether the wave gets transmitted whether how much of the wave gets reflected and we will keep this as a reference with which we can understand something more about the reflection and transmission of waves but to be more clear about what this impedance really represents we can tell ourselves well in the limit of let us say weak shock waves nearing an acoustic wave what is happening well u prime is something like maybe I can call it as p square p prime square divided by u prime square we are talking of molecular velocity fluctuations how much of the molecular velocity fluctuations represent as pressure fluctuations or pressure changes associated with molecular velocity changes is what is in the limit what we derived from the other day therefore let us keep this as standby let us now put the problem together let us do how much of it gets reflected therefore I say let me say I have a medium two medium separated by a let us say an interface over here let us say on the left of this interface I have a medium which I say is medium a and right of this particular one let us say this is medium a over here let us let me hatch the whole thing for you this is in yellow color the right side of this medium is let us say B medium B which I show like this this interface separates the two medium over here well one could be air one could be solid one could be water one could be solid it could be different phases or may be different gases one could be hydrogen which is a very light gas the other could be air which is a little denser or could be a still denser gas and so on this is the difference between these two medium well I am interested if a wave is travelling let us say a pressure wave pressure wave is travelling into this medium a over here and it is incidence on this surface how much of this pressure wave gets reflected and how much of this pressure wave gets transmitted into this particular medium into the second medium well let us put some numbers to it let us say the pressure wave has let us say an over pressure P prime over here I say well it is P prime this is incident on the surface I say incident on the surface in medium a therefore a over pressure PIA travels towards this particular interface that is towards this medium what happens at the surface let us presume what gets reflected is P R in the medium a this is what is getting reflected and what gets transmitted into the medium B is equal to let us say P B prime over here therefore we are considering this problem in medium a wave of amplitude or let us say a pressure wave of over pressure P prime a comes over gets reflected over here and in medium B because of this wave something gets transmitted I want to determine this value that means I am interested in determining P B value I am also interested in determining the reflected value therefore to do this problem immediately I tell myself well I can do this problem because there is an increase in pressure over here because I have P prime over here it further increases over here therefore the net pressure P B prime at the interface is equal to the sum because at this interface I have some increase here further some increase over here it is equal to PIA over here prime plus the what is reflected over here P R of a prime over here this is my equation one well associated with these pressure changes or pressure fluctuations or let us say over pressures there is also a velocity component and let us now presume that the medium a has an impedance Z A and the medium B has an impedance let us say Z B now what is going to happen because of this P prime over here you know because of this pressure fluctuation there will also be a velocity fluctuation let us say that the velocity fluctuation associated with this incident wave is equal to U prime into I incident of A as well along with this that means I have a velocity perturbation shown in LO similarly the velocity perturbation associated with this reflected wave is going to be U prime incident you know this is reflected therefore R of A over here similarly I also have a velocity perturbation associated with P B here P prime B here and that is equal to well I have velocity perturbation U prime associated with B this is equal to U B prime now at the interface what is happening some velocity comes here some velocity is removed from here and therefore the velocity perturbation for B I can write as let us write it out U prime B is equal to the value of U prime incident at A minus U prime reflected that is what is pulling it back into A over here these are the two equations which I get now I want to solve these equations to determine the value of P R A and P B but then I note something I note yes I said there are two medium this is the interface between these two media over here therefore I can write the value of is a day is equal to the impedance of A is equal to in terms of pressure and the associated velocity as equal to P prime I have incident of A divided by U prime incident of A and similarly I can also write the value for this reflected pressure this is also equal to the value of reflected pressure at A divided by U prime reflected of A over here because in this case P prime by U prime is equal to this impedance P prime divided by U prime for the incident wave is equal to impedance I have this equation similarly I have the value of is it B that is the impedance of B is equal to I have P B prime divided by the value of U prime of B now from this impedance equation for A I can determine the value of U I A is equal to if I write it here it becomes P prime A divided by is a day I also get U prime reflected at A is equal to P reflected A fluctuation divided by is a day or change here divided by is a day and therefore and what do I get U B U B prime is equal to P B prime by is a B and therefore equation to now becomes I can write it as equal to P B prime divided by is a B is equal to I I write U prime A is equal to P I prime A divided by is a day minus I have P R which is equal to U prime reflected A is equal to P R prime divided by is a day and therefore equation to now reduces to the form let us say equation to A over here therefore let us put these two equations together and what is it I get from equation one I get P I A P prime plus P prime reflected A is equal to that is this is equal to P prime B and from this particular equation what is it I get well I get P prime A incident A divided by is a day plus I get P prime reflected A divided by is a day is equal to P prime B divided by is a B over here this becomes my this is my first equation this is my second equation which I called as 2A I want to solve these equations to determine the value of P R P prime R A that is the pressure over pressure of the reflected as a function of the over pressure of the incident I also want to determine what is transmitted what is the pressure transmitted over here therefore to be able to solve this well I find it is P prime plus P prime for incident and reflected I have is a day why not multiply the second equation by is a day in which case I get P prime I A plus P I think we have made a mistake over here it is equal to P prime A by is a day minus P R A by is a day therefore the value here is equal to minus over here therefore the value is if I multiply by is a day I get P I A minus P prime R A is equal to P B prime into I have multiplied by impedance of medium A divided by impedance of medium B therefore this becomes my equation 2 this becomes my equation 1 this is my equation 1 well I add the two together if I add the two the reflected component gets knocked down I get 2 P I A prime is equal to I get P B prime into 1 plus is a day by is a day and if I subtract from this equation I subtract this well it becomes minus P I plus is equal to minus therefore I get 2 P reflected A prime is equal to P I A prime is equal to P B prime well I take P B prime outside and I get 1 minus is a day by is a day over here. Now I want to solve these two equations for the reflected components and therefore let us first get one value let us get the value of P B prime how can I get the value of P B prime well I use this equation and what does it tell me from this equation I said P B prime is equal to twice the incident value into Z B divided by Z B plus is a day or rather I get from this equation I can write the value let me write it over here P B prime is equal to the value of 2 P I A to P incident value I into I get the value what comes over here is Z B divided by Z A plus Z B that is Z A plus Z B or rather I take the 2 inside and if I have to express it in terms of P I A I can write this as equal to P I A incident in A into 2 Z B divided by Z B plus Z A and now I substitute it in the second equation to get the value of the reflected value I get P prime reflected in medium A is equal to I get the same value here P in P prime of the incident value and if I were to substitute it well I have I have taken this this is 2 divided by 2 that means P R A is equal to P B by 2 2 cancels off and in the denominator I have Z B by Z A now it becomes Z B minus Z A divided by Z B plus Z A therefore what is it we have done we have been able to get the pressure which propagates in medium B as a function of the pressure in medium A in terms of the impedance of B and the impedance of A therefore this is this is pressures because now we are able to get a value of the of the pressure which gets transmitted into the medium therefore in this particular problem I tell myself well I can determine this value in terms of the incident value of the pressure over pressure and I also get the reflected value which is equal to Z B minus Z A divided by Z B minus Z A well this is all about the derivation we are able to get the value of the reflected of the transmitted pressure in medium B we are able to get the reflected value in medium A let us try to discuss this result and see what it really represents. Let us consider this medium again let me erase out some of the things over here such that I can use the portion below this to be able to represent some physical conditions supposing I say well this medium is something like let us say air for as an example for which the Z A is around let us say 440 Newton Newton second by meter cube this I say something this I say something like steel or some material for which the value is around very much higher almost of the order of 4 into 10 to the power 7 therefore in this case what is we are talking of is Z B very much greater than Z A what is going to happen let us take a look at the expression what is going to happen let us say now in this case I have medium A over here which is let us say air the second medium is let us say steel over here for which the impedance is very much higher now let us say I have a shock wave a shock wave is like this I have an over pressure and this is my magnitude of the pressure which is going towards the wave and now what is going to be the wave which gets transmitted into this medium if I have this which is incident let us say this is my P prime A over here what gets transmitted is this magnitude multiplied by if I were to take a look at this it is going to be 2 that is we are telling 2 Z B into Z A plus Z B that means the magnitude of the pulse which gets transmitted into steel is going to be higher and this is what gets transmitted into my particular medium and is there something what is going to be reflected well what is reflected we saw is equal to Z B minus Z A let me write it down Z B minus Z A divided by Z A plus Z B we told ourselves well B is very much greater than A therefore this is a strongly positive number of the similar magnitude because Z A is very small and therefore I am going to get the same magnitude which is reflected back and therefore I tell myself well this is the reflected component this is the transmitted component and now I am in business I am able to find out the transmitted pressure wave I am able to get the reflected pressure or the reflected shock wave and this is how we do problem but this was for the particular case of Z B greater than Z A can I do this problem let us take the other example let us take the example of Z B that is the second medium being less than Z A what is it I will get well in this case let us consider the reverse let us still consider the case of steel let us consider the case of air as medium B now this is A as the medium therefore what is going to happen in this case let us again sketch out what is going to happen this is the particular interface at which the two media come in this is media A and you have the second media this is B over here this is the interface between the two media and now into the media I have something like a shock is coming over here this is the amplitude over here what gets transmitted well I find that Z B is smaller and whereas Z A is very much higher therefore what gets transmitted is little lower this is what gets transmitted into the medium and what is reflected well if I take a look at reflected I have Z B is less than Z A it is negative that means what goes as a compression gets reflected as an expansion rather what is going to happen it is an expansion wave behind the expansion wave well it is something like the opposite of this I have compression behind the expansion wave it gets reflected as a expansion wave I will repeat this again because this is important you know you have a compression wave which is moving towards the interface and what gets reflected is not the compression but just the expansion that means this is the pressure I have still a drop in pressure and what gets reflected is something like an expansion wave what do we mean by this you know why should why should something which travels as a compression that is a strong over pressure get reflected as an expansion is you know you have this particular surface and when it meets the surface over here well the surface relaxes because it is very much rare and when surface relaxes what happens is well the surface relaxes and therefore what expands is going to be much it is an expansion process over here and therefore what we get is an expansion wave or rather we tell ourselves well at the surface I cannot really have anything over here but the media expands over here and what reflects back is something like an expansion wave over here and therefore we tell ourselves well based on these expressions I also tell myself that when the impedance of two media A and B are such that when A is greater than B I have a compression reflected as a compression back over here and something is great which gets transmitted is even of greater value greater pressure which gets transmitted over here but the moment when I talk of the impedance of medium A being less than or impedance of of let us say medium of the second medium B being very much less than this what happens is well a compression gets reflected as an expansion and I get transmission of a compression or a shock into the medium that is the shock reflection becomes an expansion and therefore now if I slightly the problem and if I talk of a problem in the following context like let us say I have a medium over here some medium here which let us say has a high value of impedance let us say I have ZB over here I have ZA over here which is small impedance that means ZA is less than ZB and I put this small impedance again over here ZA over here what is going to happen well this could be some material like steel or bone or anything which is heavier let us say I am talking in terms of some heavier material over here some material and therefore when the pressure wave comes over here what gets reflected is some it gets reflected because ZB being greater than A it reflects as a compression wave that means I have compression wave coming over here compression wave getting reflected over here and what gets propagated is larger magnitude of the compression wave let me show it in ordinary white chalk well I have a compression wave over here and when compression wave this is one interface let us say interface one I have the second interface here because I am considering this media to be restricted that means I have interface two when this pressure wave comes over here it gets reflected as an expansion wave that means it is something like this it is expanding over here and what gets transmitted is something like a smaller compression wave over here let us take a look at this what will this result in I have a wave compression wave going I have over pressure here behind this there is an expansion that means behind this initial compression wave I have expansion and particles over here are now moving away because it is an expansion behind this particles are moving here and then what happens the reflected wave comes over here when the reflected wave comes over here I have expansion particles are moving like this therefore an particle here which is subjected to the shock wave is pulling it in this direction a particle which is the same particle or a similar particle then adjacent particle when it is subjected to the expansion wave is pulling in this particular direction and therefore what is going to happen the material gets pulled like this and the material fails and this type of failing is known as spall or spalling of the material therefore we tell ourselves whenever I have a material whose impedance is higher than the impedance into which it is traveling well and what could happen is when the value of the medium here is less than medium over here I get an expansion and the particles therefore one is pulling another is pulling I have a tensile failure and therefore we say the material spalls spalls and therefore the material fails therefore using this example that means we are now able to say something about we are talking of the of the transmitted wave we are talking of the reflected wave and using this let us try to extrapolate a little bit and talk little more about the different applications of this scheme or transmitted and the reflected waves. Let me use this table and again give you one application which all of us are familiar this we will keep using let us record it in our minds we tell ourselves well when Zd is such that it is less than Za that is the second medium is less than the first medium well I have this problem and therefore let us see how it can be applied you know all of us are familiar we talk of kidney stones you know when the calcium in the body is large or some may be the foodstuff we eat is such that we do not drink sufficient amount of water in the kidneys we have stones being formed and therefore let us take a look at this in the kidney well a stone is formed it is more like calcium may be a stone is formed let us say over here let me now enlarge it and say well the small stone is formed I say this is a small stone which is formed in the kidney over here it is in the medium of fluid and the fluid is something let us say water therefore it is in water I have something like a stone which is formed in the kidney and the kidney and may be it attaches itself to the to the material of the kidney which is something like a tissue let us say therefore we are talking of a tissue over here which has a impedance let us say about 1.3 into 10 to the power 6 we are talking of a stone which is something like a bone has a value of impedance that is Z over here is equal to 7 into 10 to the power 6 we say we have water which is around 1 into 10 to the power 6 therefore we are talking of a system wherein I have a stone over here which is in a medium of that means the stone has a higher value of impedance compared to the impedance of water and the impedance of this you will also recall over here may be water fat tissues and muscle have almost similar values of impedance around let us say 1.2 to 1.7 type of situation whereas if I look at bone the value is higher if I look at air well it is very much less than these quantities and therefore what is it happening I have let us now label these things together in some form let us say this is material A let us say this is material B medium B this is medium C well you know what is going to happen let us now say I sort of direct shock wave into the body that means I focus the shock wave onto this particular stone over here what is going to happen well shock is going to go through this it gets transmitted well it gets reflected also yes I know it gets reflected and it gets transmitted into this well our shock gets transmitted into this but the moment it comes to the other surface what is going to happen well I have the impedance which is less and therefore it gets reflected as a as a expansion wave that means something like this it is an expansion wave this is a compression wave this is an expansion wave and now in this particular stone if I consider and then of course something is further transmitted into this medium it is quite small what is getting transmitted that means I have something which is coming here it comes here gets reflected into this therefore when I when I look at the picture of the stone what I get is well considering this as a body in which we are focusing our interest well this is the incident wave and what is getting reflected is an expansion behind the behind the wave which is propagating in this direction I have expansion process taking place I have expansion taking place therefore the the particles in the stone are being pulled like this because of spalling that is is a tensile failure the stone fragments and therefore by passing a shock wave into your kidney I sort of fragment the stone make it into powder and it gets expelled out and this is one of the application that means I use the impedances to be able to crush the kidney stone well it is not only kidney stone you know anything you know there are different applications I can have therefore but what is central is well the values of the impedances is what decides how much gets reflected what is the type of reflection what I have will a compression get reflected as a as a expansion or rare faction or will it still persist as an expansion and what gets transmitted into the system well we will pursue on this a little later but just to say where does this energy come from if I say well energy is equal to if I say energy which is associated with a wave is equal to let us say the pressure fluctuation into the velocity fluctuation is something like a energy associated before therefore this is sort of the power because we are talking of meter per second we are talking of Newton per meter square this is equal to the power density if I say dt and if I integrate well this is the type of energy which is coming and therefore what happens I can I can write u prime as equal to p prime or rather I can write it as p prime square into dt one over the impedance of this particular medium what what is associated let us say rho zero into a zero is this and this energy is what is dissipated what what is used for crushing the stone as it were therefore we tell ourselves well I can use this rare faction to my advantage in some particular applications supposing we were to use some some material to shield us suppose I say there is a human being over here and he he uses for as a shield some particular material say iron or tungsten or a heavy medium having a high value of impedance over here something like let us say an armor you will recall if we I am sorry the spelling you use an armor you know if we read the novels like like maybe Arthur and his knights and all that you know they they ride on horseback with armor and all that let's say you have an armor like this and supposing let's say blast wave strikes the armor what is going to happen well the armor may spoil but we must also remember that the armor is going to transmit the wave into this medium over here therefore the armor what is used may really not be blast resistant we have to keep this in mind we will take a look at it in the in the next lecture but an armor may not really be blast resistant what it resist this maybe if a fragment from the blast comes and hits it well if the impact is not too high well it will rebound back it will help against let us say a secondary effects of a blast rather than the primary effects that means if I say well I am going to have a bulletproof thing over here which is going to save my car or save my skin well it may not be very effective for a blast wave only thing what it does is well it it it protects you against the fragment effects well this is all about impedances we have been talking but you know I also told you that well these expressions are in the limit of weak shocks can I really use impedances for strong shocks and I also told you well it is not very wrong to use it the reason being let's go back and see how this expression got derived what did we tell ourselves well I have the medium over here which is propagating at a velocity a0 sound speed and what happens I wrote the equation when the wave is stationary I have the medium rho 0 coming towards it with a velocity a0 and what happened in this case the velocity behind was u that is the particles are following this in this case the velocity we called as u1 which is equal to in this frame of reference it is a0 minus u corresponding to this u over here now if we tell ourselves instead of an acoustic wave something like a shock wave is propagating rs dot into a medium could be solid could be liquid could be anything over here and then the particles here are following with u and therefore in the frame of reference of the shock stationary I have rs dot and this medium is now moving with a velocity u1 you know what did we do in the earlier classes we got an expression for u1 as a function of rs dot and what did we get you got u1 by rs dot when the shock wave is quite high let us say a large number what did we get the value as equal to gamma minus 1 plus 2 over ms square divided by gamma plus 1 ms square is high therefore this this term gets knocked off this comes out to be for air it is around let us say typically around 6 or so for different materials the values will be smaller or larger depending on the value of gamma if gamma is small it tends to be a little larger therefore you have us by rs dot and now if I am taking if how did we get this value we said from mass balance equation we were able to be neglected the value of rho prime into u prime that is under this condition we got z is equal to 1 o z is equal to rho 0 into a0 that means we said that the particulate velocity and the density fluctuation is small and now what does it we find we see that the product of the changed components namely the velocity u prime the density rho prime the product of them are still very much smaller than the product of the initial density and the shock velocity therefore I can use the same derivation which I used for acoustic impedance again and therefore I tell myself well I can I I now can specify the impedance as something like initial density into something like the shock speed however we also note the following when we derive this expression let me rub it out and put this again when we derive the expression for the impedance and we related the reflected pressure what did we get it as equal to we got the reflected pressure as equal to zb minus za divided by zb plus za into the incident value of the pressure which I can also write as equal to dividing by by za I get zb by za minus 1 divided by zb by za plus 1 into the incident pressure that is the initial shock speed initial shock pressure and we related it to the reflected pressure so also if I look at the transmitted pressure that is pb prime what is it we get now we get instead of getting 2 zb I get 2 of zb by za divided by zb by za plus 1 now what is it we observe the transmitted pressure and the reflected pressures are ratios of the two impedances zb by za and not the absolute values and therefore I can still continue to use z as equal to rho 0 into a0 for the given medium and solve my problem having said that let me come back to the last part which I want to do today and therefore let us visit this problem of formation of craters again in the context of what is being transmitted and what is being reflected let's revisit this problem of let's say crater let's put everything together what is a crater we said well a crater could be formed on the earth well it could be a sandy soil it could be a rocky soil I could have either a surface burst or I could have an in-depth explosion let's not consider these two things let us consider the case where in some energies or an explosion source of explosion is in the air above the surface of the earth let us say it is at some height and the moment an explosion gets originated what happens well a wave gets transmitted as time progresses the wave travels forward this is at small time larger time and let us assume that the explosion is quite powerful well a wave comes and strikes over here now the impedance of the earth is going to be higher than the impedance of the air above it and therefore well a shock is transmitted into the medium let's say well I have the shock getting transmitted into the medium the blast wave travels into the medium but more importantly well we talk in terms of reflected waves and let us show these reflected waves in red let us say reflected waves that is from here a reflected wave forms over here a reflected wave travels forward over here this is the way a reflected wave travels this is the direction in which the incident wave is traveling now there are a few things at this particular point when I have a strong shock or a strong blast hitting the surface we told ourselves yesterday yes the value of the reflected to the value of the pressure behind the incident shock is almost around 8 for that is the over pressure behind the reflected shock is 8 times the over pressure behind the incident shock therefore the pressure over here is extremely extremely high and because of this high pressure well this blast goes here stronger strength we also found yes the impedance of this is higher therefore the pressure wave is quite strong and this because of this high pressure over here I get this reflected wave you also will remember that in yesterday's class we defined we determined the temperature behind the in the wave behind the incident wave for strong values for Mach number of 10 the value was around something like 5 8 4 5 Kelvin let's say around 6000 Kelvin and therefore you know the reflected shock is now traveling in a medium whose temperature is already increased because of the incident wave because of this temperature increase the sound velocity in the medium is high and therefore this reflected wave now travels quite fast it travels faster than the incident waves which are traveling forward these are the incident waves which are traveling forward therefore the reflected wave travels faster than the incident wave and since it travels faster as the incident wave is coming over here what happens is that the reflected wave travels faster and in fact it overtakes the incident wave and what a person sees far away he sees the signature of the reflected wave and not the incident wave which strikes the earth and now let's take a look at what is going to happen now let me extrapolate this over here let me put these lines in a better fashion in more more spherical I have this line coming over here I have this line coming over here and still it goes further and it comes over here when I have these high pressure coming over here I also get some waves in this that's something like seismic waves which travel along the surface of the earth I have these transmitted waves which are traveling over here and now I find that well the reflected waves are overcoming over here they are strong and because of the expansion behind the reflected wave well it takes the particulates out over here it it sort of sucks the particles and the particles are pushed over here something like this and therefore the situation what I get above the surface of the earth is well the particulates are taken out over here I get the reflected wave signature over here this looks something like that mushroom it looks like a mushroom cloud it is in the shape of a mushroom something like this and therefore I get something like this the things are taken care of and I have a cloud over here from the reflected wave and what the signature what gets is from the reflected pressure the the destruction is much more than from the incident wave which one does not even see point one second is let us take a look at what happens here the transmitted wave goes over here the transmitted wave well compresses the material over here behind it there is an expansion wave and you know the the the this this material fragments over here and because of the expansion the material gets thrown over here a crater is formed that means I have a crater the the material gets thrown over here comes over here a lip of the crater is formed over here and therefore I have a diameter of the crater which is formed with a lip over here and this is how in the earth I have spalling and ultimately I have a crater of diameter d d not form which we said is proportional to the explosion length therefore an explosion above the surface of the earth looks like a mushroom cloud and this is a signature of an explosion we must keep it in mind therefore with this background if you had to put things on impedances together you know there are certain parts in our body which contain air water and let's say the flesh around it let's say it contains the air it contains some water it contains fat tissues and bones and all that therefore you know whenever the the the material contains air in addition to water and tissues if I say water and I have tissues well the impedances is not that that different because we find that impedances of the same order but if I have something like air over here there is radical change in this and because of this what would happen I could have destruction of the materials because of the expansion and pulling that is the small type of failure and therefore you know the organs of the body which contain air are maybe here you have a drum and you also have lung our lung which contains air and water you also have something let let me try to put the third thing you have the gastrointestinal tract which also contains air in addition to the liquids what it contains and therefore whenever you have this air in the medium you know it is most susceptible to blast failure because of the expansion waves associated with the reflection after the compression waves which result in failure and therefore these you know the damage of air the damage of lung the damage of gastrointestinal tract in our human body is something which blast wave is very capable of doing in fact you know if you look at a human being you know he doesn't seem to be materially affected you know he is still there as one single piece but these these particular organs get drastically affected therefore what is it we have done today let's quickly summarize what we have done before we we tell what we must be doing in the next class in today's class we started off with impedances of different bodies we looked at how impedances will affect how much of the of the shock wave gets transmitted into the second medium and how much of it gets reflected we also found if the impedance of the second body is less than the impedance of the first body well what gets reflected back is not a compression wave or is not a shock which is which can compress the body but which expands it and this in a in a particular body of dimensions a given dimension is capable of causing damage by the tensile failure point 1 point 2 yes we now know how to calculate the transmission and the reflection and we applied it we said well it has been used for for fragmenting the kidney stones it in in practice if I have something supposing I want to I want to protect my house over here let's say if I want to protect my house well it's not sufficient for me if I just put some material over here which is a heavy material and which will withstand because the wave can still transmit over here and I can still have blast damage for the inner things that means the choice of blast resistant materials require some choice and we also find that materials which have low impedances like I use sponge material let's say the sponge material is something similar to a muscle material you know it can it can fail but it can absorb the blast wave therefore with this I stop here and in the next class what we do is we will look at mechanism of how a blast wave destroys objects we will look at maybe the over pressure impulse and also a little bit on the impedances we will summarize what we have learnt in blast waves and try to do one or two small model problems well thank you thank you