CHAPTER XXII For three-quarters of a minute he talked swiftly to the Eskimo. Philip knew that he was giving the Cognmolic definite instructions as to the manner in which his rescue must be accomplished. But he knew also that Blake would emphasize the fact that it must not be an open attack, no matter how numerous his followers might be. He hurried Blake through the door to the Sledge and team. The Sledge was heavily laden with the meat of a fresh caribou kill, and from the quantity of flesh he dragged off into the snow Philip surmised that the cabin would very soon be the rendezvous of a small army of Eskimo. There was probably a thousand pounds of it. Retaining only a single quarter of this he made silly comfortable and turned his attention to Blake. With Babishkord he re-secured his prisoner with the manacle hitch which gave him free play of one hand and arm, his left. Then he secured the Eskimo's whip and gave it to Blake. Now drive, he commanded. Straight for the copper-mind, and by the shortest cut. This is as much your race as mine now, Blake. The moment I see a sign of anything wrong you're a dead man.  Good God, what a fool!  Blake snapped his whip and gave a short angry command in Eskimo. The dogs sprang from their bellies to their feet and at another command were off over the trail. From the door of the cabin the Eskimo's little eyes shone with a watery eagerness as he watched them go. Saley caught a last glimpse of him as she looked back and her hands gripped more firmly the rifle which lay across her lap. Philip had given her the rifle and it had piled upon her a mighty responsibility. He had meant that she should use it if the emergency called for action and that she was to specially watch Blake. Her eyes did not leave the outlaws broad back as he ran on a dozen paces ahead of the dogs. She was ready for him if he tried to escape and she would surely fire. Getting close to her side Philip observed the tight grip of her hands on the weapon and saw one little thumb pinched up against the safety ready for instant action. He laughed and for a moment she looked up at him, flushing suddenly when she saw the adoration in his face. Blake's right, I'm a fool! he cried down at her in a low voice that thrilled with his worship of her.  By going the other way I'd have you forever. They wouldn't follow far into the south, if at all. Maybe you don't realize what we're doing by hitting back to that father of yours, do you? She smiled. And maybe when we get there we'll find him dead, he added. Dead or alive, everything is up to Blake now and you must help me watch him. He pantomimed this caution by pointing to Blake and the rifle. Then he dropped behind. Over the length of Sledge and Teen he was thirty paces from Blake. At that distance he could drop him with a single shot from the colt. They were following the trail already made by the meat-laden Sledge and the direction was northwest. It was evident that Blake was heading at least in the right direction and Philip believed that it would be but a short time before they would strike the copper mine. Once on the frozen surface of the big stream that flowed into the arctic and their immediate peril of an ambuscade would be over. Blake was surely aware of that. If he had in mind a plan for escaping it must of necessity take form before they reach the river. Where the forest thinned out and the edge of the barren crept in Philip ran at Saley's side. But when the timber thickened and possible hiding places for their enemies appeared in the trail ahead he was always close to Blake with the big colt held openly in his hand. At these times Saley watched the back trail. From her vantage on the Sledge her alert eyes took in every bush and thicket to right and left of them and when Philip was near or behind her she was looking at least a rifle shot ahead of Blake. For three quarters of an hour they had followed the single Sledge trail when Blake suddenly gave a command that stopped the dogs. They had reached a crest which overlooked a narrow finger of the treeless barren on the far side of which possibly a third of a mile distant was a dark fringe of spruce timber. Blake pointed toward this timber. Out of it was rising a dark column of resinous smoke. It's up to you, he said coolly to Philip. Our trail crosses through that timber and you see the smoke. I imagine there are about twenty of Oopy's men there feeding on caribou. The herd was close beyond when they made the kill. Now if we go on they're most likely to see us or the dogs get wind of us and Oopy is a bloodthirsty old cutthroat. I don't want that bullet through my gizzard so I'm telling you. Far back in Blake's eyes there lurked a gleam which Philip did not like. Blake was not a man easily frightened and yet he had given what appeared to be fair warning to his enemy. He came a step nearer and said in a lower voice, Rain, that's just one of Oopy's crowds.  You've still got the chance I gave you back there.  Give me the girl and you hit out across the barren with the team. We're going on," replied Philip, meeting the others gaze steadily.  If anyone can get past them without being seen it's you. And you've got to do it.  The Eskimos may get us after that but they won't harm her in your way. Understand? We're going the limit in this game and I figure you're getting up the biggest stake. I've got a funny sort of feeling that you're going to cash in before we reach the cabin. For barely an instant the mysterious gleam far back in Blake's eyes died out. There was the hard, low note in Philip's voice which carried conviction and Blake knew he was ready to play the hand which he held.  During the next half hour Philip's eyes and ears were ceaselessly on the alert. He traveled close to Blake with the big colt in his hand, watching every humic and bit of cover as they came to it. He also watched Blake and in the end was convinced that in the back of the outlaw's head was a sinister scheme in which he had the utmost confidence in spite of his threats and the fact that they had successfully got around Oopy's camp. Once or twice when their eyes happened to meet he caught in Blake's face a contemptuous coolness, almost a sneering exultation which the other could not quite conceal. It filled him with a scarcely definable uneasiness. He was positive that Blake realized he would carry out his threat at the least sign of treachery or the appearance of an enemy and yet he could not free himself from the uncomfortable oppression that was beginning to take hold of him. He concealed it from Blake. He tried to fight it out of himself. Yet it persisted. It was something which seemed to hover in the air about him, the feel of a danger which he could not see. And then Blake suddenly pointed ahead over an open plain and said, There is the copper mine! End of Chapter 22 Recording by Roger Milleen Chapter 23 of The Golden Snare This LibriVox recording is in the public domain. Recording by Roger Milleen The Golden Snare by James Oliver Kerwood Chapter 23 A cry from Célie turned his gaze from the broad white trail of ice that was the copper mine, and as he looked she pointed eagerly toward a huge pinnacle of rock that rose like an oddly placed cenotaph out of the unbroken surface of the plain.  She's telling you that Bram Johnson brought her this way! He chuckled.  He seemed not to expect a reply from Philip, but urged the dogs down the slope into the plain. Fifteen minutes later they were on the surface of the river. Philip drew a deep breath of relief and he found that same relief in Célie's face when he dropped back to her side. As far as they could see ahead of them there was no forest. The copper mine itself seemed to be swallowed up in the vast white emptiness of the barren. There could be no surprise attack here, even at night. And yet there was something in Blake's face which kept alive within him the strange premonition of a near and unseen danger. Again and again he tried to shake off the feeling. He argued with himself against the unreasonableness of the thing that had begun to oppress him. Blake was in his power. It was impossible for him to escape, and the outlaw's life depended utterly upon his success in getting them safely to the cabin. It was not conceivable to suppose that Blake would sacrifice his life merely that they might fall into the hands of the Eskimos. And yet. He watched Blake, watched him more and more closely as they buried themselves deeper in that unending chaos of the north. And Blake, it seemed to him, was conscious of that increasing watchfulness. He increased his speed. Now and then Philip heard a curious chuckling sound smothered in his beard, and after an hour's travel on the snow-covered ice of the river he could no longer dull his vision to the fact that the farther they progressed into the open country the more confident Blake was becoming. He did not question him. He realized the futility of attempting to force his prisoner into conversation. In that respect it was Blake who held the whip hand. He could lie or tell the truth, according to the humor of his desire. Blake must have guessed this thought in Philip's mind. They were traveling side by side when he suddenly laughed.   I guess it's because I'm so cussed sorry for you.  And they're open. Wide open. And you? This time Blake's laugh was harsher. I don't count now, he said. Since you've made up your mind not to trade me the girl for your life I've sort of dropped out of the game. I guess you're thinking I can hold Yuppie's tribe back. Well, I can't. Not when you're getting this far up in the country. If we split the difference and you gave me her, Yuppie would meet me half way. God, but you spoiled a nice dream. A dream? Blake uttered a command to the dogs. Yes, more than that. I've got an igloo up there even finer than Yuppie's, all built of whalebone and ship's timbers. Think of her in that rain with me. That's the dream you smashed. And her father and the others? This time there was a ferocious undercurrent and Blake's guttural laugh, as though Philip had by accident reminded him of something that both amused and enraged him. Don't you know how these Cogmolic heathen look on a father-in-law? he asked. He's sort of walkin' delegate over the whole Bloomin' family. A god with two legs. The others. Why, we killed them. But Yuppie and his heathen wouldn't see anything happen to the old man when they found I was going to take the girl. That's why he's alive up there in the cabin now. Lord, what a mess you're heading into, Reign.  There is a half-breed in the tribe and she'll probably go to him.  He surged on ahead, cracking his whip and crying out to the dogs. Philip believed that in those few moments he had spoken much that was truth. He had, without hesitation and of his own volition, confessed the murder of the companions of Saley's father, and he had explained in a reasonable way why Armin himself had been spared. These facts alone increased his apprehension. Unless Blake was utterly confident of the final outcome he would not so openly expose himself. He was even more on his guard after this. For several hours after his brief fit of talking Blake made no effort to resume the conversation, nor any desire to answer Philip when the latter spoke to him. A number of times it struck Philip that he was going the pace that would tire out both man and beast before night. He knew that in Blake's shaggy head there was a brain keenly and dangerously alive, and he noted the extreme effort he was making to cover distance with the satisfaction that was not unmixed of suspicion. By three o'clock in the afternoon they were thirty-five miles from the cabin in which Blake had become a prisoner. All that distance they had traveled through a treeless barren without a sign of life. It was between three and four when they began to strike Timber once more, and Philip asked himself if it had been Blake's scheme to reach this Timber before dusk. In places the spruce and bangchen pine thickened until they formed dark walls of forest, and whenever they approached these patches Philip commanded Blake to take the middle of the river. The width of the stream was a comforting protection. It was seldom less than two hundred yards from shore to shore, and frequently twice that distance. From the possible ambiscades they passed only a rifle could be used effectively, and whenever there appeared to be the possibility of that danger Philip traveled close to Blake with the revolver in his hand. The crack of a rifle, even if the bullet should find its way home, meant Blake's life. Of that fact the outlaw could no longer have a doubt. For an hour before the gray dusk of Arctic night began to gather about them Philip began to feel the effect of their strenuous pace. Hours of cramped inactivity on the sledge had brought into Saley's face lines of exhaustion. Since middle-afternoon the dogs had dragged at times in their traces. Now they were dead tired. Blake and Blake alone seemed tireless. It was six o'clock when they entered a country that was mostly plain, with a thin fringe of timber along the shores. They had raced for nine hours and had traveled fifty miles. It was here, in a wide reach of river, that Philip gave the command to halt. His first caution was to secure Blake hand and foot, with his back resting against a frozen snow-hummock a dozen paces from the sledge. The outlaw accepted the situation with an indifference which had given Philip more force than philosophical. After that, while Saley was walking back and forth to produce a warmer circulation in her numbed body, he hurried to the scrub timber that grew along the shore and returned with a small armful of dry wood. The fire he built was small and concealed as much as possible by the sledge. Ten minutes suffice to cook the meat for their supper. Then he stamped out the fire, fed the dogs, and made a comfortable nest of bearskins for himself and Saley, facing Blake. The night had thickened until he could make out only dimly the form of the outlaw against the snow-hummock. His revolver lay ready at his side. In that darkness he drew Saley close up into his arms. Her head lay on his breast. He buried his lips in the smothering sweetness of her hair, and her arms crept gently about his neck. Even then he did not take his eyes from Blake, nor for an instant did he cease to listen for other sounds than the deep breathing of the exhausted dogs. It was only a little while before the stars began to fill the sky. The gloom lifted slowly and out of darkness rose the white world in a cloud-shimmering glory. In that starlight he could see the glisten of Saley's hair as it covered them like a golden veil, and once or twice, through the space that separated them, he caught the flash of a strange fire in the outlaw's eyes. Both shores were visible. He could have seen the approach of a man two hundred yards away. After a little he observed that Blake's head was drooping upon his chest and that his breathing had become deeper. His prisoner, he believed, was asleep, and Saley, nestling on his breast, was soon in slumber. He alone was awake and watching. The dogs, flat on their bellies, were dead to the world. For an hour he kept his vigil. In that time he could not see that Blake moved. He heard nothing suspicious, and the night grew steadily brighter with the white glow of the stars. He held the revolver in his hand now. The starlight played on it in a steely glitter that could not fail to catch Blake's eyes should he awake. And then Philip found himself fighting, fighting desperately to wake. Again and again his eyes closed and he forced them open with an effort. He had planned that they would rest for two or three hours. The two hours were gone when, for the twentieth time, his eyes shot open and he looked at Blake. The outlaw had not moved.  Even then Philip was conscious of fighting against the overmastering desire to sleep. It seemed to him that he was struggling for hours and all that time his subconsciousness was crying out for him to awake, struggling to rouse him in the nearness of a great danger. It succeeded at last. His eyes opened and he stared in a dazed and half-blinded tray toward Blake. His first sensation was one of vast relief that he had awakened. The stars were brighter. The night was still. And there a dozen paces from him was the snow-hummock. But Blake, Blake, his heart leapt into his throat. Blake was gone. End of Chapter 23. Recording by Roger Milleen. Chapter 24 of The Golden Snare. This LibriVox recording is in the public domain. Recording by Roger Milleen. The Golden Snare by James Oliver Kerwood. Chapter 24. The shock of the discovery that Blake had escaped brought Philip half to his knees before he thought of Célie. In an instant the girl was awake. His arm had tightened almost fiercely about her. She caught the gleam of his revolver and in another moment she saw the empty space where their prisoner had been. Swiftly Philip's eyes traveled over the moonlit spaces about them. Blake had utterly disappeared. Then he saw the rifle and breathed easier. For some reason the outlaw had not taken that and it was a moment or two before the significance of the fact broke upon him. Blake must have escaped just as he was making that last tremendous fight to rouse himself. He had had no more than time to slink away into the shadows of the night and had not paused to hazard a chance of securing the weapon that lay on the snow close to Célie. He had evidently believed that Philip was only half asleep and in the moonlight he must have seen the gleam of the big revolver leveled over his captor's knee. Leaving Célie huddled in her furs Philip rose to his feet and slowly approached the snow-hummock against which he had left his prisoner. The girl heard the startled exclamation that fell from his lips when he saw what had happened. Blake had not escaped alone. Running straight out from behind the hummock was a furrow in the snow like the trail made by an otter. He had seen such furrows before where Eskimos had wormed their way foot by foot within striking distance of dozing seals. Assistance had come to Blake in that manner and he could see where, on their hands and knees, two men instead of one had stolen back through the moonlight. Célie came to his side now gripping the rifle in her hands. Her eyes were wide and filled with frightened inquiry as she looked from the tell-tale trails in the snow into Philip's face. He was glad that she could not question him in words. He slipped the colt into his holster and took the rifle from her hands. In the emergency which he anticipated the rifle would be more effective. That something would happen very soon he was positive. If one Eskimo had succeeded in getting ahead of his comrades to Blake's relief, others of Yuppie's tribe must be close behind. And yet he wondered, as he thought of this, why Blake and the Kogmalak had not killed him instead of running away. The truth he told frankly to Célie, thankful that she could understand. It was the gun, he said. They thought I had only closed my eyes and wasn't asleep. If something hadn't kept that gun leveled over my knee, he tried to smile, knowing that with every second the end might come for them from out of the grey mist of moonlight and shadows that shrouded the shore. It was a one-man job sneaking out like that, and there's sure a bunch of them coming up fast to take a hand in the game. It's up to us to hit the high spots, my dear, and you might pray God to give us time for a start. If he had hoped to keep from her the full horror of their situation, he knew, as he placed her on the sledge, that he had failed. Her eyes told him that. Intuitively she had guessed at the heart of the thing, and suddenly her arms reached up about his neck as he bent over her, and against his breast he heard the sobbing cry that she was trying hard to choke back. Under the cloud of her hair her warm, parted lips lay for a thrilling moment against his own, and then he sprang to the dogs. They had already roused themselves, and at his command began sullenly to drag their lame and exhausted bodies into trace formation. As the sledge began to move he sent the long lash of the driving whip curling viciously over the backs of the pack and the pace increased. Straight ahead of them ran the white trail of the copper mine, and they were soon following this with the eagerness of a team on the homeward stretch. As Philip ran behind he made a fumbling inventory of the loose rifle cartridges in the pocket of his coat, and under his breath prayed to God that the day would come before the Eskimos closed in. Only one thing did he see ahead of him now, a last tremendous fight for Céline, and he wanted the light of dawn to give him accuracy. He had thirty cartridges, and it was possible that he could put up a successful running fight until they reached Armin's cabin. After that fate would decide. He was already hatching a scheme in his brain.  With Blake out of the way there might be another way of dealing with Yuppie and his Kugmalaks. It was Blake who wanted Céline. In Yuppie's eyes there were other things more precious than a woman. The thought revived in him a new thrill of hope. It recalled to him the incident of Father Brio, and the white woman nurse, who farther west, had been held for ransom by the nanomalutes three years ago. Not a hair of the woman's head had been harmed in nine months of captivity. Olythe Anderson had told him the whole story. There had been no white man there, only the Eskimos, and with the Eskimos he believed that he could deal now, if he succeeded in killing Blake. Back at the cabin he could easily have settled the matter, and he felt like cursing himself for his short-sightedness. In spite of the fact that he had missed his main chance, he began now to see more than hope in a situation that five minutes before had been one of appalling gloom. If he could keep ahead of his enemies until daybreak, he had a ninety percent chance of getting Blake. At some spot where he could keep the Kugmalaks at bay, and scatter death among them if they attacked, he would barricade himself and Céline behind the sledge, and call out his acceptance of Blake's proposition to give up Céline as the price of his own safety. He would demand an interview with Blake, and it was then that his opportunity would come. But ahead of him were the leaden hours of the grey night. Out of that ghostly mist of pale moonlight through which the dogs were traveling like sinuous shadows, Yuppie and his tribe could close in on him silently and swiftly, unseen until they were within striking distance. In that event all would be lost. He urged the dogs on, calling them by the names which he had heard Blake use, and occasionally he sent the long lash of his whip curling over their backs. The surface of the copper mine was smooth and hard. Now and then they came to stretches of glare ice, and at these intervals Philip rode behind Céline, staring back into the white mystery of the night out of which they had come. It was so still that the click-click-click of the dog's claws sounded like the swift beat of tiny castanets on the ice. He could hear the panting breath of the beasts. The whale-bone runners of the sledge creaked with the shrill protest of steel traveling over frozen snow. Beyond these sounds there were no others, with the exception of his own breath and the beating of his own heart. Mile after mile of the copper mine dropped behind them. The last tree and the last fringe of bushes disappeared, and to the east, the north, and the west there was no break in the vast emptiness of the great Arctic plain. Even afterward the memory of that night seemed like a grotesque and horrible dream to him. Looking back he could remember how the moon sank out of the sky and utter darkness closed them in, and how through that darkness he urged on the tired dogs, tugging with them at the lead trace, and stopping now and then in his own exhaustion to put his arms about Céline and repeat over and over again that everything was all right. After an eternity the dawn came. What there was to be of day followed swiftly like the Arctic night. The shadows faded away. The shores loomed up and the illimitable sweep of the plain lifted itself into vision as if from out of a great sea of receding fog. In the quarter-hours phenomenon between the last of darkness and a wide day Philip stood straining his eyes southward over the white path of the copper mine. It was Céline huddled close at his side who turned her eyes first from the trail their enemies would follow. She faced the north and the cry that came from her lips brought Philip about like a shot. His first sensation was one of amazement that they had not yet passed beyond the last line of timber. Not more than a third of a mile distant the river ran into a dark strip of forest that reached in from the western plain like a great finger. Then he saw what Céline had seen. Close up against the timber a spiral of smoke was rising into the air. He made out in another moment the form of a cabin and the look in Céline's staring face told him the rest. She was sobbing breathless words which he could not understand, but he knew that they had won their race and that it was Armin's place. And Armin was not dead. He was alive, as Blake had said, and it was about breakfast time. He had held up under the tremendous strain of the night until now and now he was filled with an uncontrollable desire to laugh. The curious thing about it was that in spite of this desire no sound came from his throat. He continued to stare until Céline turned to him and swayed into his arms. In the moment of their triumph her strength was utterly gone and then the thing happened which brought the life back into him again with a shock. From far up the black finger of timber, where it bellied over the horizon of the plane, there floated down to them a chorus of sound. It was a human sound, the yapping, wolfish cry of an Eskimo horde closing in on man or beast. They had heard that same cry close on the heels of the fight and the clearing. Now it was made by many voices instead of two or three. It was accompanied almost instantly by the clear, sharp report of a rifle and a moment later the single shot was followed by a scattering fuselage. After that there was silence. Quickly Philip bundled Céline on the sledge and drove the dogs ahead, his eyes on a wide opening in the timber three or four hundred yards above the river. Five minutes later the sledge drew up in front of the cabin. In that time they heard no further outcry or sound of gunfire and from the cabin itself there came no sign of life unless the smoke meant life. Scarcely had the sledge stopped before Céline was on her feet and running to the door. It was locked and she beat against it excitedly with her little fists, calling a strange name. Standing close behind her Philip heard a shuffling movement beyond the log walls, the scraping of a bar and a man's voice so deep that it had in it the booming note of a drum. To it Céline replied with almost a shriek. The door swung inward and Philip saw a man's arms open and Céline run into them. He was an old man. His hair and beard were white. This much Philip observed before he turned with a sudden thrill toward the open in the forest. Only he had heard the cry that had come from that direction and now looking back he saw a figure running swiftly over the plane toward the cabin. Instantly he knew that it was a white man. With his revolver in his hand he advanced to meet him and in a brief space they stood face to face. The stranger was a giant of a man. His long reddish hair fell to his shoulders.  His eyes seemed to bulge out of their sockets as he stared at Philip, and Philip, almost dropping his revolver in his amazement, gasped incredulously, "'My God, is it you, Olaf Andersen?' End of Chapter 24 Recording by Roger Maline Chapter 25 Of the Golden Snare This LibriVox recording is in the public domain. Recording by Roger Maline The Golden Snare by James Oliver Kerwood Chapter 25 Following that first wild stare of uncertainty and disbelief in the Big Swede's eyes came a look of sudden and joyous recognition. He was clutching at Philip's hand like a drowning man before he made an effort to speak, still with his eyes on the other's face as if he was not quite sure they had not betrayed him. Then he grinned. There was only one man in the world who could grin like Olaf Andersen. In spite of blood and swollen features it transformed him. Men loved the red-headed swede because of that grin. Not a man in the service who knew him but swore that Olaf would die with the grin in his face. Because the tighter the hole he was in, the more surely would the grin be there. It was the grin that answered Philip's question. Just in time to the dot, said Olaf, still pumping Philip's hand and grinning hard. All dead but me. Colkins, Harris, and that little Dutchman, O'Flynn. Cold and stiff, Phil, every one of them. I knew an investigating patrol would be coming up pretty soon. Been looking for it every day. How many men you got? He looked beyond Philip to the cabin and the sledge. The grin slowly went out of his face and Philip heard the sudden catch in his breath. A swift glance revealed the amazing truth to Olaf. He dropped Philip's hand and stepped back, taking him in suddenly from head to foot. Alone! Yes, alone, nodded Philip, with the exception of Céli Armin, I brought her back to her father. A fellow named Blake is back there a little way with Yuppie's tribe. We beat them out, but I'm figuring it won't be long before they show up. The grin was fixed in Olaf's face again. Lord bless us, but it's funny, he grunted. They're coming on the next train, so to speak, and right over in that neck of woods is the other half of Yuppie's tribe chasing their short legs off to get me. And the comical part of it is you're alone. His eyes were fixed suddenly on the revolver. Ammunition, he demanded eagerly, and grub. Thirty or forty rounds of rifle, a dozen colt, and plenty of meat. Then into the cabin and the dogs with us almost shouted the squeed. From the edge of the forest came the report of a rifle, and over their heads went the humming drone of a bullet. They were back at the cabin in a dozen seconds tugging at the dogs. It cost an effort to get them through the door with a sledge after them. Half a dozen shots came from the forest. A bullet spattered against the log wall, found a crevice, and a metallic jingled inside. As Olaf swung the door shut and dropped the wooden bar in place, Philip turned for a moment toward Céli. She went to him, her eyes shining in the semi-gloom of the cabin, and put her arms up about his shoulders. The swede, looking on, stood transfixed, and the white-bearded armin stared incredulously. On her tiptoes Céli kissed Philip, and then, turning with her arms still about him, said something to the older man that brought an audible gasp from Olaf. In another moment she had slipped away from Philip and back to her father. The swede was flattening his face against a two-inch crevice between the logs when Philip went to his side. "'What did she say, Olaf?' he entreated.  "'Pretty lucky dog, I say, if it's true.  But it will never happen. If you don't believe it, fill your eyes with that out there.' Philip glued his eyes to the long crevice between the logs, and found the forest and the little finger of plain between straight and his vision. The edge of the timber was alive with men. There must have been half a hundred of them, and they were making no effort to conceal themselves. For the first time, Olaf began to give him an understanding of the situation. "'This is the fortieth day we've held them off,' he said, in their quick-cut business-like voice he might have used in rendering a report to a superior. "'Eighty cartridges to begin with, and a month's ration of grub for two. All but the three last cartridges went day before yesterday. Yesterday everything quiet. On the edge of starvation this morning when I went out on scout duty and to take a chance at game.' Surprised a couple of them carrying meat and had a tall fight. Others hove into action, and I had to use two of my cartridges. One left, and they're showing themselves because they know we don't dare to use ammunition at long range. "'My caliber is thirty-five. What's yours?' "'The same,' replied Philip quickly, his blood beginning to thrill with the anticipation of battle. "'I'll give you half. I'm on duty from Fort Churchill, off on a tangent of my own.' He did not take his eyes from the slit in the wall, as he told Anderson in a hundred words what had happened since his meeting with Bram Johnson. "'And with forty cartridges we'll give him a taste of hell,' he added. He caught his breath, and the last word half choked itself from his lips. He knew that Anderson was staring as hard as he. Up from the river and over the level sweep of plain between it and the timber came a sledge, followed by a second, a third, and a fourth. In the trail behind the sledges tried it a score and a half of fur-clad figures. "'It's Blake,' exclaimed Philip. Anderson drew himself away from the wall. In his eyes burned a curious greenish flame, and his face was set with the hardness of iron. In that iron was molded instinctively the terrible smile with which he always went into battle or fronted his man. Slowly he turned, pointing a long arm at each of the four walls of the cabin. "'That's the lay of the fight,' he said, making his words short and to the point. "'They can come at us on all sides, and so I've made a six-foot gun crevice in each wall. We can't count on Armin for anything but the use of a club if it comes to close quarters. The walls are built of saplings, and they've got guns out there to get through. Outside of that we've got one big advantage. The little devils are superstitious about fighting at night, and even Blake can't force them into it. Blake is the man I was after when I ran across Armin and his people. Gad!' There was an unpleasant snap in his voice as he peered through the gun-hole again. Philip looked across the room to Saley and her father as he divided the cartridges. They were both listening, yet he knew they did not understand what he and Olaf were saying. He dropped a half of the cartridges into the right-hand pocket of the Swede's service-coat and advanced then toward Armin with both his hands held out in greeting. Even in that tense moment he saw the sudden flash of pleasure in Saley's eyes. Her lips trembled, and she spoke softly and swiftly to her father, looking at Philip. Armin advanced a step and their hands met. At first Philip had taken him for an old man. Hair and beard were white, his shoulders were bent, his hands were long and thin, but his eyes, sunken deep in their sockets, had not aged with the rest of him. They were filled with the piercing scrutiny of a hawks as they looked into his own, measuring him in that moment so far as man can measure man. Then he spoke, and it was the light in Saley's eyes, her parted lips, and the flush that came swiftly into her face that gave him an understanding of what Armin was saying. From the end of the cabin Olaf's voice broke in. With it came the metallic working of his rifle as he filled the chamber with cartridges. He spoke first to Saley and Armin in their own language, then to Philip. It's a pretty safe gamble we'd better get ready for them, he said. They'll soon begin. Did you split even on the cartridges? Seventeen apiece. Philip examined his rifle and looked through the gun crevice toward the forest. He heard Olaf tugging at the dogs as he tied them to the bunk posts. He heard Armin say something in a strained voice and the Swedes' unintelligible reply, followed by a quick low-voiced interrogation from Saley. In the same moment his heart gave a sudden jump. In the fringe of the forest he saw a long, thin line of moving figures advancing. He did not call out a warning instantly. For a space in which he might have taken a long breath or two, his eyes and brain were centered on the moving figures and the significance of their drawn-out formation. Like a camera flash his eyes ran over the battleground. Halfway between the cabin and that fringe of forest four hundred yards away was a hogback in the snow, running a curving parallel with the plane. It formed scarcely more than a three- or four-foot rise in the surface, and he had given it no special significance until now. His lips formed words as the thrill of understanding leaped upon him. They're moving, he called to Olaf. They're going to make a rush for the little ridge between us and the timber. Good God, Anderson, there's an army of them. Not more than a hundred, replied the swede calmly, taking his place at the gun crevice. Take it easy, Phil. This will be good target practice. We've got to make an eighty percent kill as they come across the open. This is mighty comfortable compared with the trick they turned on us when they got Colkins, Harris, and O'Flynn. I got away in the night. The moving line had paused just within the last straggling growth of trees as if inviting the fire of the defenders. Olaf grunted as he looked along the barrel of his rifle. Strategy, he mumbled. They know we're shy of ammunition. In the moments of tense waiting, Philip found his first opportunity to question the man at his side. First he said, I guess maybe you understand, Olaf.  If we get out of this, she's going to be my wife. She promised me that, and yet I swear to heaven I don't know more than a dozen words of her language. What has happened? Who is she? Why was she with Bram Johnson? You know their language and have been with them. They're taking final orders," interrupted Olaf as if he had not heard. There's something more on foot than a rush to the ridge. It's blakes scheming. See those little groups forming? They're going to bring battering rams and make a second rush from the ridge. He drew in a deep breath and without a change in the even tone of his voice went on. Colkins, Harris, and O'Flynn went down in a good fight. Tell you about that later. Hit seven days west and run on the camp of Armin, his girl, and two white men, Russians, guided by two Kogmalaks from Coronation Gulf. You can guess some of the rest. The little devils had blakened his gang about us two days after I struck them. Bram Johnson and his wolves came along then from nowhere, going nowhere. The Kogmalaks think Bram is a great devil, and that each of his wolves is a devil. If it hadn't been for that, they would have murdered us in a hurry, and Blake would have taken the girl. They were queered by the way Bram would squat in his haunches and stare at her. The second day I saw him mumbling over something and looked sharp. He had one of Selie's long hairs, and when he saw me he snarled like an animal, as though he feared I would take it from him. I knew what was coming. I knew Blake was only waiting for Bram to get away from his Kogmalaks. So I told Selie to give Bram a strand of her hair. She did, with her own hands, and from that minute the madman watched her like a dog. I tried to talk with him, but couldn't. I didn't seem to be able to make him understand. And then the swede cut himself short. They're moving, Phil. Take the men with the battering rams and let them get halfway before your fire. You see, Bram and his wolves had to have meat. Blake attacked while he was gone. Russians killed, Armin and I cornered, fighting for the girl behind us when Bram came back like a burst of thunder. He didn't fight. He grabbed the girl and was off with her like the wind with his wolf team. Armin and I got into this cabin, and here forty days and nights. His voice stopped ominously. A fraction of a second later it was followed by the roar of his rifle, and at the first shot one of Blake's Kogmalaks crumpled up with a grunt halfway between the snow ridge and the forest. CHAPTER XXVI The Eskimos were advancing at a trot now over the open space. Philip was amazed at their number. There were at least a hundred and his heart choked with a feeling of despair even as he pulled the trigger for his first shot. He had seen the effect of Olaf's shot and following the Swedes' instructions aimed for his man in the nearest group behind the main line. He did not instantly see the result as a puff of smoke shut out his vision, but a moment later aiming again he saw a dark blotch left in the snow. From his end of the crevice Olaf had seen the man go down and he grunted his approbation. There were five of the group's bearing tree trunks for battering rams, and on one of these Philip concentrated the six shots in his rifle. Four of the tree-bearers went down and the two that were left dropped their burden and joined those ahead of them. Until Philip stepped back to reload his gun he had not noticed Seili. She was close at his side peering through the gun-hole at the tragedy out on the plane. Once before he had been astounded by the look in her face when they had been confronted by great danger and as his fingers worked swiftly in refilling the magazine of his rifle he saw it there again. It was not fear even now. It was a more wonderful thing than that. Her wide-open eyes glowed with a strange dark luster in the center of each of her cheeks was a vivid spot of color and her lips were parted slightly so that he cut the faintest gleam of her teeth.  And then he heard Anderson's voice. They're behind the ridge. We got eight of them. In half a dozen places Philip had seen where bullets had bored the way through the cabin and leaning his gun against the wall he sprang to say lee and almost carried her behind the bunk that was built against the logs. You must stay here, he cried.  Here! She nodded and smiled. It was a wonderful smile, a flash of tenderness telling him that she knew what he was saying and that she would obey him. She made no effort to detain him with her hands but in that moment, if life had been the forfeit, Philip would have stolen the precious time in which to take her in his arms. For a space he held her close to him. His lips crushed to hers and faced the wall again with the throb of her soft breast still beating against his heart. He noticed Armin standing near the door, his hand resting on a huge club which, in turn, rested on the floor. Calmly he was waiting for the final rush. Olaf was peering through the gun-hole again and then came what he had expected, a rattle of fire from the snow-ridge. The pit-pit-pit of bullets rained against the cabin in a dull tattoo. Through the door came a bullet sending a splinter close to Armin's face. Almost in the same instant a second followed it and a third came through the crevice so close to Philip that he felt the hissing breath of it in his face. One of the dogs emitted a wailing howl and flopped among its con-rads in uncanny convulsions. Olaf staggered back and faced Philip. There was no trace of the fighting grin in his face now. It was set like an iron mask. Get down! he shouted. Do you hear? Get down! he dropped on his knees crying out the warning to Armin in the other's language. They've got enough guns to make a sieve of this kennel if their ammunition holds out and the lower logs are heaviest. Flatten yourself out until they stop firing with your feet toward them like this and he stretched himself out on the floor parallel with the direction of fire. In place of following the Swedes example Philip ran to Selie. Halfway a bullet almost got him flipping the collar of his shirt. He dropped beside her and gathered her up completely in his arms with his own body between her and the fire. A moment later he thanked God for the protection of the bunk. He heard the ripping of a bullet through the saplings and cut distinctly the thud of it as the spent lead dropped to the floor. Selie's head was close on his breast. Her eyes were on his face. Her soft lips so near he could feel their breath. He kissed her, unbelieving even then that the end was near for her. It was monstrous, impossible. Lead was finding its way into the cabin like raindrops. He heard the Swedes voice again crying thickly from the floor. Hug below the lower log. You've got eight inches. If you rise above that they'll get you. He repeated the warning to Armin. As if to emphasize his words there came a howl of agony from another of the dogs. Still closer Philip held the girl to him. Her hands had crept convulsively to his neck. He crushed his face down against hers and waited. It came to him suddenly that Blake must be reckoning on this very protection which he was giving Selie. He was gambling on the chance that while the male defenders of the cabin would be wounded or killed Selie would be sheltered until the last moment from their fire. If that was so the firing would soon cease until Blake learned results. Scarcely had he made this guess when the fuselage ended. Instead of rifle fire there came a sudden strange howl of voices and Olaf sprang to his feet. Philip had risen when the Swedes voice came to him in a choking cry. Prepared for the rush he had expected Olaf was making an observation through the gun crevice. Suddenly without turning his head he yelled back at them, Good God! It's Bram! Bram Johnson! Even Selie realized the thrilling import of the Swedes' excited words. Bram Johnson! She was only a step behind Philip when he reached the wall. With him she looked out. Out of that finger of forest they were coming, Bram and his wolves. The pack was free, spreading out fan shape, coming like the wind. Behind them was Bram, a wild and monstrous figure against the whiteness of the plane bearing in his hand a giant club. His yell came to them. It rose above all other sound like the cry of a great beast. The wolves came faster and then the truth fell upon those in the cabin with a suddenness that stopped the beating of their hearts. Bram Johnson and his wolves were attacking the Eskimos. From the thrilling spectacle of the giant madman charging over the plane behind his ravenous beasts Philip shifted his amazed gaze to the Eskimos. They were no longer concealing themselves. Palsied by a strange terror, they were staring at the onrushing horde and the shrieking wolfman. In those first appalling moments of horror and stupefaction not a gun was raised or a shot fired. Then there rose from the ranks of the Kugmalaks a strange and terrible cry and in another moment the plane between the forest and the snow ridge was alive with fleeing creatures in whose heavy brains surged the monstrous thought that they were attacked not by man and beast but by devils. And in that same moment it seemed that Bram Johnson and his wolves were among them. From man to man the beasts leapt driven on by the shrieking voice of their master and now Philip saw the giant madman overtake one after another of the running figures and saw the crushing force of his club as it fell. Céli swayed back from the wall and stood with her hands to her face. The swede sprang past her, flung back the bar to the door and opened it. Philip was a step behind him. From the front of the cabin they began firing and man after man crumpled down under their shots. If Bram and his wolves sensed the shooting in the ferocity of their bloodlust they paid no more attention to it than to the cries for mercy that rose chokingly out of the throats of their enemies. In another sixty seconds the visible part of it was over. The last of the Kugmalaks disappeared into the edge of the forest. After them went the wolf-man and his pack. Philip faced his companion. His gun was hot and empty. The old grin was in Olaf's face. In spite of it he shuddered. We won't follow, he said. Bram and his wolves will attend to the trimmings and he'll come back when the job is finished. Meanwhile we'll get a little start for home, eh? I'm tired of this cabin. Forty days and nights.   If you have, let's see, where did I leave off in that story about Princess Selie and the Duke of Rugny? The—the what? Your tobacco-fill? In a dazed fashion Philip handed his tobacco-pouch to the Swede. You said Princess Selie, the Duke of Rugny? Olaf nodded as he stuffed his pipe-bowl. That's it! Armin is the Duke of Rugny, whatever Rugny is. He was chased off to Siberia a good many years ago when Selie was a kid that somebody else could get hold of the dukedom. Understand? Millions in it, I suppose. He says some of Rasputin's old friends were behind it and that for a long time he was kept in the dungeons of the fortress of St. Peter and St. Paul with the Neva River running over his head. The friends he had, most of them in exile or chased out of the country, thought he was dead and some of these friends were caring for Selie. Just after Rasputin was killed and before the Revolution broke out they learned Armin was alive and dying by inches somewhere up on the Siberian coast. Selie's mother was Danish, died almost before Selie could remember but some of her relatives and a bunch of Russian exiles in London framed up a scheme to get Armin back, chartered a ship, sailed with Selie on board and... Olaf paused to light his pipe. And they found the duke, he added. They escaped with him before they learned of the Revolution or Armin could have gone home with the rest of the Siberian exiles and claimed his rights. For a lot of reasons they put him aboard an American whaler and the whaler missed its plans by getting stuck in the ice for the winter up in Coronation Gulf. After that they started out with dogs and sledge and guides. There's a lot more, but that's the meat of it, Phil. I'm going to leave it to you to learn Selie's language and get the details firsthand from her. But she's a right-enough princess, old man, and her dad's a duke. It's up to you to Americanize him. Hey! What's that?  And you've got the job half done, Phil. The duke may go back and raise the devil with the people who put him in cold storage, but Lady Selie is going to like America. Yes, sir, she's going to like it better in any other place on the face of the earth. It was late that afternoon he was travelling slowly southward over the trail of the copper mine when they heard far behind them the wailing cry of Bram Johnson's wolves. The sound came only once, like the swelling surge of a sudden sweep of wind, yet when they camped at the beginning of darkness Philip was confident the madman and his pack were close behind them. Utter exhaustion blotted out the hours for Selie and himself buried in two heavy Eskimo coats he had foraged from the field of battle sat on guard through the night. Twice in the stillness of his long vigil he heard strange cries. Once it was the cry of a beast. The second time it was that of a man. The second day with dogs refreshed they travelled faster and it was this night that they camped in the edge of timber and built a huge fire. It was such a fire as illumined the space about them for fifty paces or more and it was into this light that Bram Johnson stocked so suddenly and so noiselessly that a sharp little cry sprang from Selie's lips and Olaf and Philip and the Duke of Rugny stared in wide-eyed amazement. In his right hand the Wolfman bore a strange object. It was an Eskimo coat tied into the form of a bag and in the bottom of this improvision was a lump half the size of a water pail. Bram seemed oblivious of all presence but that of Selie. His eyes were on her alone as he advanced and with a weird sound in his throat deposited the bundle at her feet. In another moment he was gone. The Swede rose slowly from where he was sitting and speaking casually to Selie took the Wolfman's gift up in his hands. Philip observed the strange look in his face as he turned his back to Selie in the firelight and opened the bag sufficiently to get a look inside. Then he walked out into the darkness and a moment later returned without the bundle and with a laugh apologized to Selie for his action. No need of telling her what it was, he said to Philip then. I explained that it was foul meat Bram had brought in as a present. As a matter of fact it was Blake's head. You know the Kugmaloks have a pretty habit of pleasing a friend by presenting him with the head of a dead enemy.  After all there were some very strange happenings in life and the adventurers of the Royal Northwest Mounted Police come upon their share. The case of Bram Johnson, the mad Wolfman of the Upper Country happened to be one of them and filed away in the archives of the department is a big envelope filled with official and personal documents signed and sworn to by various people. There is, for instance, the brief and straightforward deposition of Corporal Olaf Anderson of the Fort Churchill Division and there is the longer and more detailed testimony of Mr. and Mrs. Philip Reign and the Duke of Rugny. And attached to these depositions is a copy of an official decision, pardoning Bram Johnson to give him award of the great dominion instead of a criminal. He is no longer hunted. Let Bram Johnson alone is the word that had gone forth to the man-hunters of the service. It is a wise and human judgment. Bram's country is big and wild and he and his wolves still hunt there under the light of the moon and the stars. The End of Chapter 26 Recording by Roger Maline End of The Golden Snare by James Oliver Kerwood


So, good morning. In this class what we will do is to look at the computational assignment, it is actually problems at 3, but I have called it problems at 4 online. And what I will do is I will try to work through a couple of the problems and as I go along feel free to ask me any doubts that you have about the assignment and then later on about the course until this point of time we can discuss some of those things. Then I will talk about the new assignment I uploaded online yesterday. The computational one and explain how you can write a code to simulate that Swift-Henberg equation so that we can test some of the things that we have done in the theory. So, this is only work if you ask me your doubts so feel free to ask and you do not have to worry about the video going on that we can edit anyways. Alright, so the first question was the 1D Swift-Henberg problem, right so u is a function of x and t and x extends in this problem from minus infinity to plus infinity. And this equation is a simplified you could say averaged version of the Rayleigh-Benard equations. So, using the method of multiple scales what has been done here is essentially they looked at the set of Rayleigh-Benard and using a suitable averaging condensed everything into this simple equation. So, quantitatively this will not cannot be expected to give you the same results, but what has been done is all the physics of the Rayleigh-Benard problem have been captured with the simple equation. The idea being that by studying this equation we can understand a lot of basic features about pattern forming systems. And this has been derived in a book by Cross and Green Side which is the book that I have used to put together this assignment and the next one as well and that is a really good book for the course. So, I will give the reference a bit later. So, they derive it from a heuristic perspective and then from a rigorous mathematical point of view also. So, keeping that in mind this parameter r is similar to the Rayleigh parameter. So, we want to find the value of r for which the system is unstable and there is a fourth order term which will come from the coupling of I think temperature and velocity and then you have a second order term. So, everything fits together similar to the Rayleigh-Benard if you just look at it from a far. So, you can be interpreted as the vertical component of velocity. So, for example, if you say I mean say you wanted to compare this with some experimental data then if I have my Rayleigh-Benard cell and it extends to infinity in both directions then as you already know at some point I will have maybe some kind of role something will be happened and if it is stable there will be no convection at all. So, u is almost the cutout along the z direction as h by 2. So, the and it is the v z the z component of velocity that I have cut out through the centre of the channel. So, now it is only of and I have considered to be only a function of x. So, I have ignored the y direction variation say that cell is quite narrow. So, I am looking at the x direction. So, with this interpretation we will move on to the analysis which is actually quite simple because it is just one equation. So, the first step is of course to linearize the problem. So, by that we would write u as u steady state plus u let us call it pot up and u steady state is basically 0 here. So, homogeneous state and that makes sense with the fact that there is no flow in the flow. So, u steady state is 0 and if you were to substitute here and retain terms I will call multiply this by epsilon for magnitude retain terms about a epsilon then we would get the and the u cube which is the nonlinearity does not survive in the linearization. Now, the way this equation has translated the entire nonlinear term has simply been knocked off but this would not always happen such a kind of thing happens only if your base state is identically 0. If I had a nonzero USS then I would have a term like USS square times u per term and other terms as well. So, you need to be careful when you are linearizing only if the base state is 0 you can write it down by inspection, but on other cases is better to substitute always and make sure. So, this is the perturbed equation. Now, the next step is to take a form for the perturbation or the normal mode analysis. So, I will say a few things about that. So, typically what we have been doing throughout the course is at this stage writing up as some constant a which is the initial condition of the perturbation. We take an exponential form in time because it is the first order system and then this is the most important part where we write down something like e to the power ikx and this means we are considering the Fourier modes. Okay, so first of all I will just go through it because some people have asked me this question. So, the first thing is that this is a complex factor but then up has to be real. So how does it resolve? Well, it resolves with the Euler identity that relates I mean complex exponential terms to cos h and sin h. So, when we look at the Fourier series in this manner we actually need to consider k's from okay because we are looking at we are not looking at a system where k would be discrete right now. If we have the entire if x belongs to the real number line so if x is a continuous variable from minus infinity to plus infinity k should take all values from minus infinity to plus infinity. So, we are looking at both k's negative k and positive k even though the dispersion curve we always plot positively it is understood that we are also considering the negative k.  So, eventually this complex guy will manifest itself manifest itself via the Euler formula as something like I mean this will be the dependence finally you will get cos kx and or depending on the initial condition sin kx. So, if you want to go plot your variation you cannot plot this you can plot cos kx and sin kx. So, this coefficient a is also complex that has to be kept in mind. So, this a is not a real thing it is complex because this is also complex.  So, in other words I have a mode this will be one mode this mode will have to be added with a star where a star is the complex conjugate of a and that will be obvious when you look at the equation because this when you write instead of k if you take minus k it is basically looking at this same equation and replacing i by minus i which means that any unknowns will just be the conjugates of what you get in any unknowns here will be the conjugates of what you get here that is clear. So, this is how even though we are looking at the complex modes ultimately they are going to manifest in a real fashion and this will always happen in all these problems it has nothing to do with the physics and all this is just the way the Fourier modes work right. Sir. A is the combination of both real and imaginary part of a plus b i that one e power i k x can be written as cos k x plus i sin k x. So, there will be some when you multiply this is a complex term this is a complex term this is its complex conjugate right if I add it I want to get a real. Just the real part of those both. No this is some complex number I can write it as a plus i b this is the same complex number conjugated I will write it as a minus i b and then I will add it and I will get 2 a and the sin part of it I mean if you what you are saying you will add the cos parts the sin parts will also get added and ultimately give you real and if you are still have a doubt just go back to some of your old books on the Euler identity and how you solve the equation I mean this all of this business comes in this equation. So, you see how the solution here is done you here you can write it directly write sin x cos x sin y cos y solutions or you can proceed with complex exponentials and then see how the 2 are resolved. So, that should clear any doubts here. So, ultimately when you write this it means you are looking at both cos and sin and that which of them come depend on initial condition boundary condition that is one important point the other important point is that here k will k is real when this is the case. So if x goes from minus infinity to plus infinity k takes all the values on the real number line from minus infinity to plus infinity. But if x belongs to some interval like say minus pi to plus pi with periodic boundary conditions. So, when would this come up suppose I am doing a numerical simulation. So, if I want to simulate this numerically which is what we are going to do in the next problem I cannot very well have a infinite long domain I will be waiting forever for the problem to get solved. So, one way to do it is to take a very large domain. So, may be minus 100 to plus 100 where the characteristic wavelength is of order 1 then that might work. Another way to look at it is I will take a small domain like minus pi to plus pi and impose periodic boundary conditions to simulate what would happen in the infinite case because when it is infinite our argument is it has to be periodic if it is not periodic then it has to blow off infinity at one of the extents. So, realizing that it must be periodic we could take a periodic domain and of course taking a periodic domain restricts the modes that we would get. So, that is my point here if we take a domain like this minus pi to plus pi then k will belong to a discrete set and so on. So, here k takes the integral values. So, because minus pi to plus pi so it will be sin x sin 2 x sin 3 x. So, it has the wave has to fit properly in the periodic domain it cannot you know fit half of it. If you had minus l to plus l then k will belong to 2n pi by l actually sorry I think this you will have minus n pi by l yeah it is 2n pi by l 2n pi by 2l I guess because the 2l is the length of the domain. So, again the n takes values and will belong to the integers. So, the idea is that if the domain is infinite the k will vary over all the real numbers if the domain is finite and periodic then k takes discrete special values. So, this is the same thing that happens in Fourier series and Fourier integral. So, the Fourier series is how you split up a function on a finite domain. So, when you write the Fourier series you sum over the series and the series sums over integer values of n that is what is happening here when you do Fourier integral you do the Fourier integral for a function on an infinite domain then you have to integrate across all the k values because k is not discrete anymore it is the whole real number line. So, with this understanding we are writing e to the power ikx. So, if x is infinite k takes all real numbers and your dispersion curve is continuous if x is finite in a finite periodic domain k takes only discrete values along that dispersion curve. So, what it means is if I take a domain minus pi to plus pi my k can only be 1 and then 2 k cannot be 1.5 because 1.5 that mode will not fit in that minus pi to plus pi. So, by controlling the size of the domain you can exclude modes or include modes in the periodic case if you take the infinite case all the modes will come. So, that is something that you should remember and thoroughly understand before you try to simulate the problem because when you take a periodic domain you will actually be restricting the modes that are entering the problem and then that should that can lead to different results unless you are aware that was one point the other point is with someone asked me is now when we do this let us carry on and I will come back to it. So, we have got this form now. So, we will go back and substitute it here and get the equation for the dispersion curve you are agree those of you all have done it. So, this problem is very simple because you start off with an PDE but with only one spatial direction because we are knocking off time and space we will just get the algebraic equation which is the dispersion curve it gives us the dispersion curve immediately. Now, at this point some of you all asked me the question as you know what happened to the boundary conditions because we had a PDE and then we certainly I mean reached a solution and we did not seem to have said anything about the boundaries. But the factor is that once again when you are writing e to the power ikx it implies certain boundary conditions. So, what it implies is either that you have minus infinity to plus infinity in the domain is basically going to be periodic. If your thing was a finite domain then taking e to the power ikx again forces it to be periodic boundary conditions where this minus l to plus l will control which are which modes come. So, the length 2l of the domain controls which of the k modes are allowed and what are those periodic boundary conditions let me just I will just write. So, periodic boundary conditions are simply u at is u at plus l and all its derivatives. So, n is n is 0 means just the values are equal when n is 1 it means the first derivative. So, it is 0, 1, 2, 3. So, you equate the values and all the higher derivatives until you get 4 boundary conditions because I need 4 boundary conditions in the x direction that clear. So, it is with respect to these boundary conditions periodic boundary conditions that we are using the form e to the power ikx when x is on a finite domain minus l to plus l. And if it is on an infinite domain then the idea is like we are splitting it up into the Fourier modes. So, that is fine and now we can you will have the dispersion curve. So, when k is 0 what will happen I get r minus 1. So, let us look at the case of r equal to 0 because you already know that that is going to be the critical k. So, when r is 0 basically k will be when k is 0 and r is 0 sigma will be equal to minus 1. So, it will be somewhere here and then it does something like that. So, it becomes critical here at any higher value it will cross right. So, basically you have a region of unstable modes right this is for r equal to 0 this is for r greater than 0 and r less than 0 it would have all been negative. And this peak it remains the same throughout which you can find out what it is and I will call it I mean this portion is called k max. So, the k max gives me the maximum value of sigma for all values of r which means that the peak of the curve does not change with r. That is not always the case as you would know when you do the assignment in this problem it is true. So, what it tells me is that this mode corresponding to k max is the fastest growing mode but there is also a range of modes along with it all of which are unstable and the number of modes which are unstable keeps increasing as my supercriticality keeps increasing as r becomes larger and larger than the critical value 0. So, that is as far as the first few points of the assignment work any doubts about this. So, essentially what we get finally is that at r equal to 0 or for r close to 0 you have the critical k or the k max basically equal to I think you will find 1 because that is where sigma becomes 0 when r is 0. So, k max is this is basically 1. So, the idea now is if you try to interpret pattern formation what will happen is if you have a r just above 0 maybe 0.1 or 0.5 also then you will see a mode manifesting which will be the mode k equal to 1. If you do the non-linear simulation of the equations you will find that from the base state 0 you will get a new steady state which has the same wavelength as this k equal to 1 which means the solution will look very similar to sin kx with some magnitude like a sin kx something like that the new non-linear state will close to r equal to 0. For r much greater than 0 higher non-linearities come in and then the steady state can change, but close to this onset the kind of wavelength you will see in the pattern in this case it is just one dimension will be a wavelength of k equal to 1 means lambda will be 2 pi. So, if my domain is minus pi to plus pi for example if I look at the thing in minus pi to plus pi whether it extends to infinity or not if I just look at the thing minus pi to plus pi I will just get one sin way and this is basically sin I mean this could this is sin x I could also get cos x because that is also equally valid. So, then I could have got I am not very good at drawing this, but basically got the idea this sin is 0 at both ends you would get cos where the derivatives are 0 at both ends is just the 90 degree phase shift. Let us try to do this sin cuts whatever but we will get cos x. So, that is fine now the second part of the question was what happens when you go to the 2 dimensional Swift-Henberg. So, in the 2D Swift-Henberg basically you will have these partial derivatives, but now extended in x and y. So, if you go through the analysis what you will find ultimately is that we will get the same story except instead of k square I will have to replace it by kx square plus ky square and this should not be a surprise because we have done this and seen this in many problems now in the course. So, this kx I mean this comes because I take e to the power i kx x e to the power i ky y. So, that is the kx and that is the ky. So, the interesting thing now is that in the 2D version of the problem my critical kc or whatever is actually given by kx square plus ky square equal to 1 and that is very different from what I had before that just k is equal to 1 in the 1D case now I have kx or plus ky square equal to 1. This is actually k square equal to 1 and then I took the square root. So, the point now is that the first question that I ask is is the mode therefore that will manifest is that mode unique is the wavelength that you will see unique is the kind of pattern that you are going to see unique in this case it was I mean okay there was it could be sinx it could be cos x but in an infinite domain or a periodic domain those things do not matter it is basically the same physical pattern you are seeing it just phase shifted. But here I can actually get an infinity of possible modes which look differently in and just look different in space mathematically they have a different representation in kx and ky apart from the phase shifting. So, what does that mean you understand why that is so if I you tell me a value of kx say 0.5 I can always find a value of ky to go along with that. So, in that sense I can get what actually happens is in the plane of kx ky I have an entire contour where it is maximum. So, all these points in the plane of kx ky are critical. So, physically what does it mean if I have a part of the domain just a piece of the infinite plane and I look at some manifestation of the mode. So, I do the nonlinear simulation give it a perturbation close to the onset suppose I take kx equal to 0 and ky equal to 1 that is one possibility or let me do kx equal to 1 because that is this basically falls back to the 1D case. It is almost like you had a 1D problem and I added a y direction and said nothing will change in y that is this case and that is also equally unstable. So, what we will get here is that there is a periodic variation in the x direction of wavelength 2 pi and in the y direction nothing is changing. So, you will just get stripes where this thing is means it is greater than 0 and this means it is less than 0. So, if you take any cross section you will just get the sin x or the cos x and you will get this kind of stripe pattern that is clear how that came. Now, if I looked at the opposite case where kx is 0 and ky is 0 then it is just a rotation of this pattern you will just have horizontal stripe sorry ky is you have horizontal stripe this is also equally valid as that if I took the intermediate case which is also equally valid I would get and the perpendicular vector will be oriented along the 45 that is right. So, now what you see is that and I can go on with this which I want thankfully. So, what you will see is that you have a whole range of patterns all of which are equally possible in terms of the stripes the basic thing is a stripes. Now, this is where the idea of symmetry breaking which is more often discussed by physicists but which is relevant to all of us is particularly important. So, the reason why you have all these different possibilities is actually very simple and obvious once you think about and that is that suppose I were to say that I am only going to consider the one dimensional problem. So, I will only have kx equal to 1 and ky equal to 0 means I am only going to get this pattern okay. So, fine but now the question is if I look at the physical system what is there to tell me what the x direction is. So, if you look at this table and that is my experimental system which is the x direction. So, you do not know that you can choose the x direction as you please because the system is isotropic and we were discussing a system like that in infinite fluid where nothing is particularly special about the fluid or about the boundary conditions or about the governing equation in any particular direction x and y. So, that was totally I mean it was just our choice of the axis and coordinate system to allow us to proceed mathematically but when we come back to look at the physical system there is no x direction. So, because of that you would immediately realize that it has to be that all orientations of this should be equally unstable because I could choose any direction as the x direction. So, any rotation of this pattern is equally valid because I could have taken that as the x direction and this is simply saying the same thing in a mathematical way. So, all of these patterns are just rotations of this pattern and the fact that all the rotations are equally unstable tells you that the original problem had rotational symmetry which means that I could if the x y axis was there and I rotated it any way it does not change the problem or if I took the x y axis I rotated the coordinate system by theta and I made that substitution. So, I replaced x by x star y by y star the rotated axis I would get exactly the same equations the problem would not change at all. So, that invariance to a rotational transformation is what we mean by rotational symmetry of the not of the pattern rotational symmetry of the base equations and the base state. So, because of that what happens is that if I have a pattern like this I can confidently expect that all rotations of this pattern will be equally unstable or equally possible in my new steady state. So, that is an important thing. However, you must realize that this pattern is not rotationally symmetric because this is the pattern that has a clearly well defined orientation in the y direction and the x direction. If I rotate it and make it this these two patterns are not identical. So, there is the rotational symmetry is been broken when I went from the base state to the new state. So, the base state was flat homogeneous that is rotationally symmetric is just everything is just 0. But when it came to a new state the new state broke that symmetry. So, because this is no longer invariant to rotation all right. But because there is no specific directions in my problem because the original equations and base state were rotationally symmetric the new state all its rotation should be equally valid that is the idea. None of those states are rotationally symmetric but any rotations the whole family of them should be equally unstable. So, that is the idea of broken symmetry which I have written down in the solution and given some references you can read more about it. The same idea comes when we look at the phase shifts that is also a manifestation actually of broken symmetry. Because let me look at the 1D case I remember I said if you look at minus pi to plus pi in the infinite problem I could get a sine wave. But in truth because the problem is infinite I could get any phase shift of that sine wave. So, basically if sine x is a possible pattern sine x plus any phi will be possible where phi is a phase shift or basically the pattern of sine x translated by any amount is an equally valid pattern to emerge in the new study state. Why? Because again if you go back to the base problem there is no reason to there is no place which I can fix as the origin. I can take anything as the origin. So, if I shift my origin along the plane basically that is translating the sine x curve. So, in the 1D problem I did not have this question of rotation what I had was translational symmetry because I can translate the origin on the x axis and it is not going to change the problem. So, because the base state and the base equations were translationally symmetric when I got the new pattern all translations of the new pattern were equally valid as a new pattern as an unstable mode. Again once again you will see that this sine mode which is the unstable mode is not translationally symmetric. Translations will give you a different pattern but all those different patterns are equally possible because the base state was translationally invariant because the origin can be placed anywhere on the x axis. So, there is a very good book a popular account of these questions in physics, chemistry, biology, crystallography and so on. It is called fearful symmetry by Gulubitski and Stuart. Stuart is a very good writer and Gulubitski has done a lot of work in this area for many years and that it is a popular science book. So, you can just pick it up and read it and it tells you a lot of different patterns in nature how symmetry plays a role and you will be able to understand those things very easily now that we have done some of the math in the course. Along with this I would recommend for not only this problem but in fact the entire assignment is based on that book and the next assignment also and it explains many of the ideas and maths really well in that book. Just get the reference. I have given it in the solution. So, you can look at the full thing. It is pattern formation and dynamics in non-equivalent. It is by Hengie Cross and Green side, Michael Cross and Hengie Green side, Cambridge. That is a very good book for all the ideas that I have spoken about. So, now when you look at the non-linear simulations of this problem or an experiment, the idea is you should be ready to expect all these different orientations of patterns. That brings us quickly to the final point I want to make about the patterns is that not only are all orientations of these stripes state possible but even superpositions of different stripes can come up and there is nothing to stop that. So, if I tell you that all the modes on this circle are equally possible then either any one of them can emerge individually or maybe three or four of them can or six of them can superpose together and lead to some pattern or they can all grow together and once they grow initially the non-linear terms will come in and modify the solution it is called non-linear saturation and give you a new steady state. So, that new steady state could bear resemblance to any orientation of this stripe set. It could also be given by some combination of those stripes. So, just as an exercise if you look at these stripes and they are 90 degree rotation what you can do is you can go to MATLAB and plot the contours of cos kxx into cos kyy. Take that function plot its contours when you take kx equal to 1, ky equal to 0 you will get these stripes black white black white then you plot the other case then you plot their combination and see what you get. So, in some portions the say if I look at this and this is a in this portion everything is positive right in this portion everything will be negative here things will be sort of cancelled out. So, you will have almost a square checkerboard kind of pattern emerging simply out of these two combinations and those types of patterns are known to be seen like a checkerboard new steady state has known to be seen in these kinds of non-equilibrium systems after the symmetry breaks after the instability. A similar combination also gives rise to other lattices like the hexagonal lattice or the Rayleigh-Bennard cells. So, that is how you get so the hexagon seem to be nowhere in the whole problem either in the 1D problem of Rayleigh-Bennard nor in the 2D problem you found hexagons you just had these stripes basically. But it is these combinations of those stripes that lead you to those lattice states that you see in many non-equilibrium systems that you will get hexagons and different kinds of cells. And the question as to which of these patterns will come how do I know the hexagons will come or how do I know that these stripes will come that depends on very often on the boundary conditions which are not in this thing right now it depends on the initial condition also. But very strongly it also depends on the non-linear terms in the equation which we have totally neglected from the linear stability. So, in that sense it is similar to the case where you get you know the grow the eigenvalue to be 0 and then you cannot say anything about stability you need to go to the non-linear terms. So, to understand how it is going to grow. So, in a similar way here all these modes have zero growth rate in fact zero eigenvalue. So, now the question you are asking me is which of them will come actually is asking me which of them is going to grow and dominate. So, to understand that we need to look at these different modes the different stripes and look at the effect of non-linear terms on their growth. So, that is called weekly non-linear theory and that will tell you which of these patterns will come. So, those of you are interested in that can again look at this book and he has a discussion about those things and that those are the more advanced topics in the area of you know non-equilibrium dynamics pattern formation. So, now you can go back to your other problems that you have done the Rayleigh Taylor problem the Rayleigh jet Rayleigh plateau jet instability capillary instability and look at the same thing over there look at the base state identified symmetries and then see that in fact your new steady state or that the modes that you get should all obey this idea of broken symmetry that all they are the whole family of translations rotations and so on should be equally unstable and then it will not be surprising to you anymore. Do you all have any other doubts in the assignment I can address that now and then I want to talk a bit about the computation assignment I put up. So, if you will have any if you do not have any doubts in the assignment I will directly do that otherwise if you will have a doubt I will continue with the assignment. So, when you solve the non-linear equation you will get a graph for u1 and u2 like in this case why does it go like that and then we can. It is oscillating right. Yeah. Why I mean that is the system is like that you know the dynamics are oscillatory so it is a 2D system so it can oscillate and understand like in the sense that there is a there is a base steady state you solve the non-linear equations. So, now the base steady state was actually unstable possibly at that case the eigenvalues were had a complex part. So, it was complex positive and then you had an oscillatory kind of behavior if it is stable it will oscillate and die out to 0 if it is unstable it will oscillate and you know reach a new steady dynamic maybe it be a limit cycle maybe it will be an oscillation I mean it will be a steady state. The main idea is that it all of the things should have died out the steady state which we are looking at should have been stable under all conditions because the eigenvalues were minus epsilon and minus 2 epsilon but that does not happen because of non-normal growth and I have given in the solution I worked out a bit and given some references where you can read more about or any problems with that reaction diffusion problem. So, that is clear in that problem you need to look at the rate r of c I have not specified any form you should continue with the unspecified form. So, that means where I have written so, when you linearize it about USS you do not know what is r of USS plus epsilon you put up I mean you do not know it in the algebraic expression form but that is the whole idea is you anyway doing a Taylor series. So, if you want to approximate the value of this function about USS for small epsilon this will just be r of USS plus r prime of USS into epsilon UP. So, this is a constant depending on the USS. So, you do not know what USS is it does not matter. So, this is some constant number you can replace it by a if you want and then the rest is just epsilon UP. So, you will get a UP basically from this term into RSS basically a UP you will get from this term in the linear equation and then you will find that if r prime of USS is positive the system will be unstable and if r prime or negative the system would be stable. So, diffusion also if you look at it diffusion would already be stabilizing. So, this value of sigma will be given by simply r prime of USS itself and the diffusion just stabilizes it because sigma will be r prime of USS minus k square or something like that. So, in this problem diffusion is stabilizing. We might look later at a situation where diffusion is actually destabilizing and that happens when you have two chemical species and gives rise to what are called Turing patterns but I will not talk about them now because I think we will do it in the course. If you have any other doubts I have written the solution set down. So, you can go through that and then we can discuss them later especially about that classification of the different states you can just read that part. All right. So, now the next thing that we need to do is will be on a computers. So, again we are going to look at the 1D Swift to Hamburg and the idea is to now simulate the non-linear equation all right and then see whether in fact if I perturb the system for r greater than 0 whether that I will get a new steady state will the new steady states pattern be similar to sin x or cos x then how the effect of boundary conditions will change the instability and so on. So, all these questions that we are trying to answer analytically in this 1D problem it is quite simple to go and see for yourself numerically. So, second numerical experiment you can see whether the theory matches the computation. So, the equation is this similar thing I have put it up the assignment online with the list of questions and exercises and there I have given a short description of how you can write a numerical code to simulate it. So, I will just discuss some of those things here. So, that might make it easier all right. So, that is the governing non-linear equation. Now there are the next now here the important question comes of boundary conditions because I cannot simulate an infinite domain.  So, u the values of u and all the derivatives will be equal at both ends of the domain okay. Now this is where the important thing about that mathematics business comes in which I was talking to you about. So, if your domain is this minus r to plus l you have a length of 2l. So, the possible case that I would get under this periodic boundary conditions there will be 2 pi by 2l n okay where n will take values of and the negative values. If I look at only sine if I look e to the pi kx will be plus and minus otherwise you will have cos and sine with only the positive. But the idea is that the case that you will get on the dispersion curve won't be all the case the case will only be these discrete values all right. So, now because we know from theory that the most unstable mode is k equal to 1 right. So, I will want to allow that to manifest otherwise I cannot compare the thing with the theory. So, therefore if I want to allow the k critical equal to 1 to manifest it should be one of these possible modes one of these discrete set. So, therefore I should take l equal to pi or in general l equal to n pi. So, if I go from minus n pi to plus n pi any integral multiple pi then I can be sure that that mode this kc equal to 1 which came from the linear stability theory that that mode will be allowed to come in if my l is say you can go minus pi to plus pi okay. If my l is less than this then that mode will not be allowed only k equal to 0.5 or something will be will come in this k equal to 1 will not because that domain will be too small to allow sin x to come in because it needs a wavelength of 2 pi if you are not going to give it a wavelength of 2 pi how can it come. So, you are not allowing it only. So, that is an important thing you should keep in mind that the numerical simulation at some point will start failing if you do not observe that thing. This is more important when you do not have periodic boundary conditions but even in this case you can try it out and see. So, the important thing is you can keep this in mind and then keep it at domains of n pi. So, if you take n very large ultimately you will reach minus infinity to plus infinity then it will l will not matter anymore if you take it large enough but then you want to avoid that much of computation. So, that is the one point then about the numerical simulation one method of doing it is to do what is called method of lines. So, in a method of lines or mol the idea is to discretize the spatial variables and leave the partial derivatives and time as total derivatives. And then what you can do is instead of a set of PDEs you will get a set of ODE's in time. And using Matlab or Mathematica you can integrate those ODE's very easily using the inbuilt functions with robust numerical methods. So, you do not have to worry about the time stepping you just need to take care of the spatial dependence. So, I will do a simple example suppose I just had dou u by dou t equal to dou square u by dou x square diffusion equation. So, that is some like this is a more complex version let us look at this. So, I just had this simple diffusion equation then what I would do is discretize in space which means that if I have my x domain I will break it up into nodes. So, node 0, node 1, node 2 this is node n, node n minus 1, n minus 2 normal spatial finite difference discretization. And what I will do is I will replace the derivative in x by its central difference finite I mean central finite difference. So, I will get du i by dt where u i is the value at the node i. So, du i by dt because I have discretized in space and the second derivative at i is basically u i plus 1 plus u i minus 1 minus 2 u i upon delta x where delta x is this difference between 0 and 1. So, this is the central finite difference of second order. So, in the same way you can find central difference formulas for the fourth order derivative also and substitute here and u cube will just be u i cube. So, now what we have is we have a bunch of ODs. So, if I runs from the other question now you have this now where do I apply this equation I apply this equation for all the interior nodes. So, in this problem I would have 2 boundary conditions 1 boundary condition would be u 0 equal to u n let me write it here boundary conditions would give me u 0 equal to u n that would be the I mean equality of the values and the derivatives would be the derivative of u equal to derivative of n because I have a domain like this I can use a backward difference here and a forward difference from the other side because delta x is the same I will get u 1 minus u 0 should be equal to u n minus u n minus 1. So, actually here I have used a forward difference and there I have used a backward difference. So, there are other ways of doing this but this is a simple way you could also use central difference here and then you will have a dummy node and whatnot. But if you just start this is a simple thing you can do. So, use appropriate differences forward here and backward from the other side. So, then you have got a boundary conditions and this problem you will have to repeat that for the second derivative and the third derivative using again forward difference backward difference of the appropriate formulas. So, what you will see is that these equations allow you to relate the values of the exterior nodes u 0 and u n to the interior nodes. So, you can do that. So, basically ultimately what you need to do is you need to apply this equation for i going from 1 to n minus 1 okay. You apply this equation for 1 to n minus 1 those are the interior nodes you do not have equations for 0 and n but that gets provided by the 2 boundary conditions. So, you will have n minus 2 ods plus 2 boundary conditions which are basically algebraic equations and you can use them to describe the evolution of these n nodes. So, more on this is there in the assignment. So, with this basics you should be able to simulate it. So, we will discuss more later. Thanks.


Purdue's minority engineering program brought Don and Liz Thompson together and launched them to success. Don as CEO of McDonald's and now head of a venture capital firm, and Liz as a leader in education and philanthropy. My name is Don Thompson and I'm currently the CEO and founder of Cleveland Avenue. My name is Elizabeth Thompson, people call me Liz, and I am currently the president of the Cleveland Avenue Foundation for Education. The significance of the phrase Cleveland Avenue is that it's the street where we both grew up on here in Chicago. We lived four blocks apart and didn't realize it until the first day on campus at Purdue back in August of 1980. I thought she was kind of cute. He was I. So while we were at Purdue, the influence of the minority engineering program was so great in our lives. It helped us make a giant leap from high school kids from, in my case, Chicago, really nervous never having been in a supremely diverse environment as Purdue represented. The minority engineering program helped us make that giant leap through giving us early introductions to the engineering courses that we were going to take by introducing us to professors on campus. It helped us in ways that we probably didn't realize at the time, but we now know without a doubt helped us transition into the adults that we were to become. I guess if I look back and I had to characterize my career thus far, I would say it's been a career of one opportunity and two, I would call it risk taking. You know, started out as an engineer at Northrop Grumman. Wonderful career, loved it, and then an opportunity to go to a company that was so focused on people and operations and systems, but it was just a different widget. It was hamburgers and fries and drinks and service at McDonald's Corporation, which I first thought was McDonald's Douglas when they called, but ultimately, yeah, we came to McDonald's. But I think it, and even now into what we do in the venture capital space, what Purdue taught us was that every opportunity was also a problem to be solved. And that is the thing that we talk to students a lot about. Every opportunity is simply a problem to be solved. Thanks to Purdue, there have been doors of opportunities open to me and to our family because education is that exposures, that opportunity to open up those doors. And we've been blessed to be able to walk through those doors because we have a confidence and we have a confidence that a lot of which was gained through our upbringing and it was gained through being in school at Purdue and the learnings that we have from the university. Thanks to Purdue, I now have a platform, we have a platform to help countless young people who are on the same journey that we were on, perhaps from neighborhoods where there were not a lot of resources. We are now able to say, we can help you. We can help you start this journey and hopefully through programs like the Minority Engineering Program and others at Purdue, we can help see you to the end. And the other thing I'd like to say is thanks to Purdue, I met the love of my life, who I now call my husband, who's the most wonderful man I've ever met. So thank you Purdue. That was Don and Liz Thompson. Look for more stories at Purdue.edu slash footprints.