1 00:00:00,334 --> 00:00:02,561 - [Narrator] I want to show you the equation of a wave 2 00:00:02,561 --> 00:00:04,840 and explain to you how to use it, 3 00:00:04,840 --> 00:00:07,711 but before I do that, I should explain what do we even mean 4 00:00:07,711 --> 00:00:09,417 to have a wave equation? 5 00:00:09,417 --> 00:00:12,309 What does it mean that a wave can have an equation? 6 00:00:12,309 --> 00:00:13,279 And here's what it means. 7 00:00:13,279 --> 00:00:15,909 So imagine you've got a water wave and it looks like this. 8 00:00:15,909 --> 00:00:18,173 And we graph the vertical height of the water wave 9 00:00:18,173 --> 00:00:19,693 as a function of the position. 10 00:00:19,693 --> 00:00:22,143 So for instance, say you go walk out on the pier 11 00:00:22,143 --> 00:00:25,348 and you go look at a water wave heading towards the shore, 12 00:00:25,348 --> 00:00:26,938 so the wave might move like this. 13 00:00:26,938 --> 00:00:29,243 You'll see this wave moving towards the shore. 14 00:00:29,243 --> 00:00:31,871 Now, realistic water waves on an ocean 15 00:00:31,871 --> 00:00:33,364 don't really look like this, 16 00:00:33,364 --> 00:00:36,763 but this is the mathematically simplest wave 17 00:00:36,763 --> 00:00:37,596 you could describe, 18 00:00:37,596 --> 00:00:39,413 so we're gonna start with this simple one 19 00:00:39,413 --> 00:00:40,365 as a starting point. 20 00:00:40,365 --> 00:00:41,422 So let's say this is your wave, 21 00:00:41,422 --> 00:00:42,798 you go walk out on the pier, 22 00:00:42,798 --> 00:00:44,365 and you go stand at this point 23 00:00:44,365 --> 00:00:46,040 and the point right in front of you, 24 00:00:46,040 --> 00:00:48,759 you see that the water height is high 25 00:00:48,759 --> 00:00:50,290 and then one meter to the right of you, 26 00:00:50,290 --> 00:00:52,504 the water level is zero, 27 00:00:52,504 --> 00:00:54,344 and then two meters to the right of you, 28 00:00:54,344 --> 00:00:57,174 the water height, the water level is negative three. 29 00:00:57,174 --> 00:00:58,007 What does that mean? 30 00:00:58,007 --> 00:01:00,288 It means that if it was a nice day out, right, 31 00:01:00,288 --> 00:01:02,097 there was no waves whatsoever, 32 00:01:02,097 --> 00:01:04,025 there'd just be a flat ocean or lake 33 00:01:04,025 --> 00:01:05,319 or wherever you're standing. 34 00:01:05,319 --> 00:01:07,864 But if there's waves, that water level can be higher 35 00:01:07,864 --> 00:01:11,172 than that position or lower than that water level position. 36 00:01:11,172 --> 00:01:13,982 We'll just call this water level position zero 37 00:01:13,982 --> 00:01:16,710 where the water would normally be if there were no waves. 38 00:01:16,710 --> 00:01:18,701 So you graph this thing and you get this graph like this, 39 00:01:18,701 --> 00:01:20,873 which is really just a snapshot. 40 00:01:20,873 --> 00:01:24,226 Because this is vertical height versus horizontal position, 41 00:01:24,226 --> 00:01:25,638 it's really just a picture. 42 00:01:25,638 --> 00:01:27,908 So in other words, I could just fill this in with water, 43 00:01:27,908 --> 00:01:29,792 and I'd be like, "Oh yeah, that's what the wave looks like 44 00:01:29,792 --> 00:01:31,436 "at that moment in time." 45 00:01:31,436 --> 00:01:33,632 And if I were to show what the wave does, 46 00:01:33,632 --> 00:01:35,849 it travels toward the shore like this 47 00:01:35,849 --> 00:01:36,933 and you'd see it move, 48 00:01:36,933 --> 00:01:39,180 so that's what this graph really is. 49 00:01:39,180 --> 00:01:41,571 If you've got a height versus position, 50 00:01:41,571 --> 00:01:43,622 you've really got a picture or a snapshot 51 00:01:43,622 --> 00:01:46,507 of what the wave looks like at all horizontal positions 52 00:01:46,507 --> 00:01:48,900 at one particular moment in time. 53 00:01:48,900 --> 00:01:50,726 And so what should our equation be? 54 00:01:50,726 --> 00:01:53,091 It should be an equation for the vertical height 55 00:01:53,091 --> 00:01:57,269 of the wave that's at least a function of the positions, 56 00:01:57,269 --> 00:01:58,861 so this is function of. 57 00:01:58,861 --> 00:02:00,137 This isn't multiplied by, 58 00:02:00,137 --> 00:02:03,361 but this y should at least be a function of the position 59 00:02:03,361 --> 00:02:06,141 so that I get a function where I can plug in 60 00:02:06,141 --> 00:02:07,735 any position I want. 61 00:02:07,735 --> 00:02:09,136 Let's say x equals zero. 62 00:02:09,136 --> 00:02:11,480 And it should tell me, oh yeah, that's at three. 63 00:02:11,480 --> 00:02:14,024 So this wave equation should spit out three 64 00:02:14,024 --> 00:02:15,847 when I plug in x equals zero. 65 00:02:15,847 --> 00:02:17,698 When I plug in x equals one, 66 00:02:17,698 --> 00:02:20,079 it should spit out, oh, that's at zero height, 67 00:02:20,079 --> 00:02:22,372 so it should give me a y value of zero, 68 00:02:22,372 --> 00:02:25,373 and if I were to plug in an x value of 6 meters, 69 00:02:25,373 --> 00:02:29,082 it should tell me, oh yeah, that y value is negative three. 70 00:02:29,082 --> 00:02:31,967 So no matter what x I plug in here, say seven, 71 00:02:31,967 --> 00:02:34,353 it should tell me what the value of the height 72 00:02:34,353 --> 00:02:37,573 of the wave is at that horizontal position. 73 00:02:37,573 --> 00:02:39,186 So what would this equation look like? 74 00:02:39,186 --> 00:02:40,309 Well, let's just try to figure it out. 75 00:02:40,309 --> 00:02:42,608 Y should equal as a function of x, 76 00:02:42,608 --> 00:02:45,754 it should be no greater than three or negative three 77 00:02:45,754 --> 00:02:47,470 and this is called the amplitude. 78 00:02:47,470 --> 00:02:50,906 So if we call this here the amplitude A, 79 00:02:50,906 --> 00:02:53,225 it's gonna be no bigger than that amplitude, 80 00:02:53,225 --> 00:02:55,411 so in this case the amplitude would be three, 81 00:02:55,411 --> 00:02:57,335 but I'm just gonna write amplitude, so this is 82 00:02:57,335 --> 00:03:00,203 a general equation that you could apply to any wave. 83 00:03:00,203 --> 00:03:01,480 And then look at the shape of this. 84 00:03:01,480 --> 00:03:03,649 This is like a sine or a cosine graph. 85 00:03:03,649 --> 00:03:04,857 Which one is this? 86 00:03:04,857 --> 00:03:06,641 Well, because at x equals zero, 87 00:03:06,641 --> 00:03:08,997 it starts at a maximum, I'm gonna say 88 00:03:08,997 --> 00:03:11,102 this is most like a cosine graph 89 00:03:11,102 --> 00:03:14,662 because cosine of zero starts at a maximum value, 90 00:03:14,662 --> 00:03:15,533 so I'm gonna say that this is 91 00:03:15,533 --> 00:03:18,378 like cosine of some stuff in here. 92 00:03:18,378 --> 00:03:20,489 Now you might be tempted to just write x. 93 00:03:20,489 --> 00:03:21,489 But that's not gonna work. 94 00:03:21,489 --> 00:03:23,288 If I just wrote x in here, 95 00:03:23,288 --> 00:03:26,081 this wouldn't be general enough to describe any wave. 96 00:03:26,081 --> 00:03:26,952 Because think about it, 97 00:03:26,952 --> 00:03:31,114 if I've just got x, cosine of x will reset every time 98 00:03:31,114 --> 00:03:32,589 x gets to two pi. 99 00:03:32,589 --> 00:03:35,508 So every time the total inside here gets to two pi, 100 00:03:35,508 --> 00:03:36,702 cosine resets. 101 00:03:36,702 --> 00:03:38,208 But look at this cosine. 102 00:03:38,208 --> 00:03:40,881 It resets after four meters. 103 00:03:40,881 --> 00:03:43,589 And some other wave might reset after eight meters, 104 00:03:43,589 --> 00:03:44,812 and some other wave might reset 105 00:03:44,812 --> 00:03:46,432 after a different distance. 106 00:03:46,432 --> 00:03:48,823 I need a way to specify in here 107 00:03:48,823 --> 00:03:51,432 how far you have to travel in the x direction 108 00:03:51,432 --> 00:03:53,168 for the wave to reset. 109 00:03:53,168 --> 00:03:54,839 So x alone isn't gonna do it, 110 00:03:54,839 --> 00:03:56,033 because if you've just got x, 111 00:03:56,033 --> 00:03:57,920 it always resets after two pi. 112 00:03:57,920 --> 00:03:59,006 So what do I do? 113 00:03:59,006 --> 00:04:01,004 I play the same game that we played 114 00:04:01,004 --> 00:04:02,709 for simple harmonic oscillators. 115 00:04:02,709 --> 00:04:04,702 And I say that this is two pi, 116 00:04:04,702 --> 00:04:07,725 and I divide by not the period this time. 117 00:04:07,725 --> 00:04:09,311 This is not a function of time, 118 00:04:09,311 --> 00:04:10,356 at least not yet. 119 00:04:10,356 --> 00:04:11,198 It's not a function of time. 120 00:04:11,198 --> 00:04:12,629 This is just of x. 121 00:04:12,629 --> 00:04:13,711 So this wouldn't be the period. 122 00:04:13,711 --> 00:04:15,439 This would not be the time it takes 123 00:04:15,439 --> 00:04:17,019 for this function to reset. 124 00:04:17,019 --> 00:04:19,521 It would actually be the distance that it takes 125 00:04:19,522 --> 00:04:20,788 for this function to reset. 126 00:04:20,788 --> 00:04:22,790 In other words, what we call the wavelength. 127 00:04:22,790 --> 00:04:26,662 So the distance between two peaks is called the wavelength. 128 00:04:26,662 --> 00:04:28,891 And we represent it with this Greek letter lambda. 129 00:04:28,891 --> 00:04:32,194 So the distance it takes a wave to reset in space 130 00:04:32,194 --> 00:04:33,145 is the wavelength. 131 00:04:33,145 --> 00:04:34,627 That's what we would divide by, 132 00:04:34,627 --> 00:04:36,444 because that has units of meters. 133 00:04:36,444 --> 00:04:39,091 And then finally, we would multiply by x in here. 134 00:04:39,091 --> 00:04:41,754 That way, if I start at x equals zero, 135 00:04:41,754 --> 00:04:43,311 cosine starts at a maximum, 136 00:04:43,311 --> 00:04:44,853 I would get three. 137 00:04:44,853 --> 00:04:48,400 If I say that my x has gone all the way to one wavelength, 138 00:04:48,400 --> 00:04:49,888 and in this case it's four meters. 139 00:04:49,888 --> 00:04:52,164 If I go all the way at four meters or one wavelength, 140 00:04:52,164 --> 00:04:56,068 once I plug in wavelength for x, that wavelength 141 00:04:56,068 --> 00:04:57,365 would cancel this wavelength. 142 00:04:57,365 --> 00:04:59,983 We'd get two pi and this cosine would reset, 143 00:04:59,983 --> 00:05:03,203 because once the total inside becomes two pi, 144 00:05:03,203 --> 00:05:04,604 the cosine will reset. 145 00:05:04,604 --> 00:05:05,682 And that's what happens for this wave. 146 00:05:05,682 --> 00:05:08,346 It should reset after every wavelength. 147 00:05:08,346 --> 00:05:09,999 You go another wavelength, it resets. 148 00:05:09,999 --> 00:05:11,370 Another wavelength, it resets. 149 00:05:11,370 --> 00:05:12,742 And that's what would happen in here. 150 00:05:12,742 --> 00:05:14,902 So how would we apply this wave equation 151 00:05:14,902 --> 00:05:16,422 to this particular wave? 152 00:05:16,422 --> 00:05:17,255 Well, let's take this. 153 00:05:17,255 --> 00:05:19,278 It's already got cosine, so that's cool 154 00:05:19,278 --> 00:05:20,492 because I've got this here. 155 00:05:20,492 --> 00:05:24,346 You could use sine if your wave started at this point 156 00:05:24,346 --> 00:05:25,775 and went up from there, 157 00:05:25,775 --> 00:05:28,390 but ours start at a maximum, so we'll use cosine. 158 00:05:28,390 --> 00:05:30,755 So we'll say that our amplitude, not just A, 159 00:05:30,755 --> 00:05:33,285 our amplitude happens to be three meters 160 00:05:33,285 --> 00:05:36,640 because our water gets as high as three meters 161 00:05:36,640 --> 00:05:38,453 above the equilibrium level. 162 00:05:38,453 --> 00:05:40,200 And we'll leave cosine in here. 163 00:05:40,200 --> 00:05:42,311 The two pi stays, but the lambda does not. 164 00:05:42,311 --> 00:05:45,570 Our wavelength is not just lambda. 165 00:05:45,570 --> 00:05:46,556 That's just too general. 166 00:05:46,556 --> 00:05:47,572 We gotta write what it is, 167 00:05:47,572 --> 00:05:49,839 and it's the distance from peak to peak, 168 00:05:49,839 --> 00:05:51,734 which is four meters, or you could measure it 169 00:05:51,734 --> 00:05:55,158 from trough to trough, or you could call these valleys. 170 00:05:55,158 --> 00:05:57,923 Valley to valley, that'd also be four meters. 171 00:05:57,923 --> 00:05:59,333 Regardless of how you measure it, 172 00:05:59,333 --> 00:06:02,044 the wavelength is four meters. 173 00:06:02,044 --> 00:06:03,826 And then what do I plug in for x? 174 00:06:03,826 --> 00:06:05,388 I don't, because I want a function. 175 00:06:05,388 --> 00:06:06,717 This is a function of x. 176 00:06:06,717 --> 00:06:08,923 I mean, I can plug in values of x. 177 00:06:08,923 --> 00:06:10,217 Actually, let's do it. 178 00:06:10,217 --> 00:06:11,661 Let's see if this function works. 179 00:06:11,661 --> 00:06:13,526 If I leave it as just x, it's a function 180 00:06:13,526 --> 00:06:16,188 that tells me the height of the wave at any point in x. 181 00:06:16,188 --> 00:06:17,053 But we should be able to test it. 182 00:06:17,053 --> 00:06:18,545 Let's test if it actually works. 183 00:06:18,545 --> 00:06:21,154 So let's take x and let's just plug in zero. 184 00:06:21,154 --> 00:06:22,932 So if I plug in zero for x, 185 00:06:22,932 --> 00:06:24,762 what does this function tell me? 186 00:06:24,762 --> 00:06:27,900 It tells me that the cosine of all of this would be zero. 187 00:06:27,900 --> 00:06:30,739 And I know cosine of zero is just one. 188 00:06:30,739 --> 00:06:33,266 So tell me that this whole function's gonna equal 189 00:06:33,266 --> 00:06:35,564 three meters, and that's true. 190 00:06:35,564 --> 00:06:37,787 The height of this wave at x equals zero, 191 00:06:37,787 --> 00:06:40,609 so at x equals zero, the height of the wave is three meters. 192 00:06:40,609 --> 00:06:41,531 So that one worked. 193 00:06:41,531 --> 00:06:42,453 Let's try another one. 194 00:06:42,453 --> 00:06:46,513 Let's say we plug in a horizontal position of two meters. 195 00:06:46,513 --> 00:06:48,893 If I plug in two meters over here, 196 00:06:48,893 --> 00:06:51,277 and then I plug in two meters over here, 197 00:06:51,277 --> 00:06:52,110 what do I get? 198 00:06:52,110 --> 00:06:56,033 This is gonna be three meters times cosine of, 199 00:06:56,033 --> 00:06:58,609 well, two times two is four, over four is one, 200 00:06:58,609 --> 00:07:01,517 times pi, it's gonna be cosine of just pi. 201 00:07:01,517 --> 00:07:03,916 And the cosine of pi is negative one. 202 00:07:03,916 --> 00:07:06,175 So I'm gonna get negative three out of this. 203 00:07:06,175 --> 00:07:07,881 Negative three meters, and that's true. 204 00:07:07,881 --> 00:07:10,012 The height of this wave at two meters 205 00:07:10,012 --> 00:07:11,675 is negative three meters. 206 00:07:11,675 --> 00:07:13,669 So this function's telling us the height of the wave 207 00:07:13,669 --> 00:07:17,082 at any horizontal position x, which is pretty cool. 208 00:07:17,082 --> 00:07:19,084 However, you might've spotted a problem. 209 00:07:19,084 --> 00:07:21,076 You might be like, "Wait a minute, that's fine and all, 210 00:07:21,076 --> 00:07:23,135 "but this is for one moment in time. 211 00:07:23,135 --> 00:07:25,597 "This wave's moving, remember?" 212 00:07:25,597 --> 00:07:27,850 This whole wave moves toward the shore. 213 00:07:27,850 --> 00:07:30,050 So at a particular moment in time, 214 00:07:30,050 --> 00:07:33,470 yeah, this equation might give you what the wave shape is 215 00:07:33,470 --> 00:07:35,043 for all values of x, 216 00:07:35,043 --> 00:07:36,770 but if I wait just a moment, boop, 217 00:07:36,770 --> 00:07:38,131 now everything's messed up. 218 00:07:38,131 --> 00:07:41,597 Now, at x equals two, the height is not negative three. 219 00:07:41,597 --> 00:07:44,714 And at x equals zero, the height is no longer three meters. 220 00:07:44,714 --> 00:07:46,188 It only goes up to here now. 221 00:07:46,188 --> 00:07:47,021 So what do we do? 222 00:07:47,021 --> 00:07:49,811 How do we describe a wave that's actually moving 223 00:07:49,811 --> 00:07:52,498 to the right in a single equation? 224 00:07:52,498 --> 00:07:53,907 Well, it's not as bad as you might think. 225 00:07:53,907 --> 00:07:54,740 Let me get rid of this 226 00:07:54,740 --> 00:07:55,656 Let's clean this up. 227 00:07:55,656 --> 00:07:57,589 We're really just gonna build off of this function 228 00:07:57,589 --> 00:07:58,422 over here. 229 00:07:58,422 --> 00:08:01,426 What I really need is a wave equation that's not only 230 00:08:01,426 --> 00:08:05,343 a function of x, but that's also a function of time. 231 00:08:05,343 --> 00:08:08,775 So this function up here has to not just be a function of x, 232 00:08:08,775 --> 00:08:10,552 it's got to also be a function of time 233 00:08:10,552 --> 00:08:13,298 so that I could plug in any time at any position, 234 00:08:13,298 --> 00:08:14,996 and it would tell me what the value 235 00:08:14,996 --> 00:08:16,947 of the height of the wave is. 236 00:08:16,947 --> 00:08:19,195 So how do I get the time dependence in here? 237 00:08:19,195 --> 00:08:20,412 Well, I'm gonna ask you to remember, 238 00:08:20,412 --> 00:08:23,392 if you add a phase constant in here. 239 00:08:23,392 --> 00:08:26,730 Remember, if you add a number inside the argument cosine, 240 00:08:26,730 --> 00:08:28,330 it shifts the wave. 241 00:08:28,330 --> 00:08:31,019 In fact, if you add a little bit of a constant, 242 00:08:31,019 --> 00:08:32,530 it's gonna take your wave, 243 00:08:32,530 --> 00:08:34,408 it actually shifts it to the left. 244 00:08:34,408 --> 00:08:35,374 So we're not gonna want to add. 245 00:08:35,374 --> 00:08:36,953 If we've got a wave going to the right, 246 00:08:36,953 --> 00:08:38,914 we're gonna want to subtract 247 00:08:38,914 --> 00:08:40,499 a certain amount of shift in here. 248 00:08:40,500 --> 00:08:42,995 But subtracting a certain amount, so that's cool, 249 00:08:42,995 --> 00:08:44,950 because subtracting a certain amount shifts the wave 250 00:08:44,950 --> 00:08:46,096 to the right. 251 00:08:46,096 --> 00:08:48,708 But if I just had a constant shift in here, 252 00:08:48,708 --> 00:08:49,550 that wouldn't do it. 253 00:08:49,550 --> 00:08:51,742 Like, the wave at the beach does not just move 254 00:08:51,742 --> 00:08:53,828 to the right and then boop it just stops. 255 00:08:53,828 --> 00:08:54,917 It just keeps moving. 256 00:08:54,917 --> 00:08:57,350 We need a wave that keeps on shifting. 257 00:08:57,350 --> 00:08:58,594 So you might realize if you're clever, 258 00:08:58,594 --> 00:08:59,993 you could be like, "Wait, why don't I just 259 00:08:59,993 --> 00:09:02,626 "make this phase shift depend on time? 260 00:09:02,626 --> 00:09:05,801 "That way, as time keeps increasing, 261 00:09:05,801 --> 00:09:08,340 the wave's gonna keep on shifting more and more." 262 00:09:08,340 --> 00:09:12,175 So if this wave shift term kept getting bigger 263 00:09:12,175 --> 00:09:13,707 as time got bigger, 264 00:09:13,707 --> 00:09:16,250 your wave would keep shifting to the right. 265 00:09:16,250 --> 00:09:17,973 You'd have an equation that describes a wave 266 00:09:17,973 --> 00:09:19,842 that's actually moving, 267 00:09:19,842 --> 00:09:21,420 so what would you put in here? 268 00:09:21,420 --> 00:09:22,845 It might seem daunting. 269 00:09:22,845 --> 00:09:24,394 You might be like, "Man, that's gonna be complicated. 270 00:09:24,394 --> 00:09:25,450 "How do we figure that out?" 271 00:09:25,450 --> 00:09:28,232 But it's not too bad, because just like the wavelength 272 00:09:28,232 --> 00:09:31,128 is the distance it takes for the wave to reset, 273 00:09:31,128 --> 00:09:33,183 there's also something called the period, 274 00:09:33,183 --> 00:09:34,825 and we represent that with a capital T. 275 00:09:34,825 --> 00:09:37,409 And the period is the time it takes 276 00:09:37,409 --> 00:09:39,195 for the wave to reset. 277 00:09:39,195 --> 00:09:40,840 So if I wait one whole period, 278 00:09:40,840 --> 00:09:42,987 this wave will have moved in such a way 279 00:09:42,987 --> 00:09:45,568 that it gets right back to where you couldn't really tell. 280 00:09:45,568 --> 00:09:48,137 It looks like the exact same wave, in other words. 281 00:09:48,137 --> 00:09:49,373 So we've showed that over here. 282 00:09:49,373 --> 00:09:51,103 Let's say you had your water wave up here. 283 00:09:51,103 --> 00:09:52,311 And I take this wave. 284 00:09:52,311 --> 00:09:56,042 If you wait one whole period, the wave will have shifted 285 00:09:56,042 --> 00:09:58,643 right back and it'll look like it did just before. 286 00:09:58,643 --> 00:10:00,240 So the whole wave is moving toward the beach. 287 00:10:00,240 --> 00:10:03,671 If you close your eyes, and then open them one period later, 288 00:10:03,671 --> 00:10:05,590 the wave looks exactly the same. 289 00:10:05,590 --> 00:10:07,635 So I'm gonna use that fact up here. 290 00:10:07,635 --> 00:10:11,618 We need this function to reset not just after a wavelength. 291 00:10:11,618 --> 00:10:14,274 We need it to reset after a period as well. 292 00:10:14,274 --> 00:10:15,583 So how do we represent that? 293 00:10:15,583 --> 00:10:16,941 We play the exact same game. 294 00:10:16,941 --> 00:10:20,366 We say that, all right, I can't just put time in here. 295 00:10:20,366 --> 00:10:22,311 What I'm gonna do is I'm gonna put two pi 296 00:10:22,311 --> 00:10:26,865 over the period, capital T, and then I multiply by the time. 297 00:10:26,865 --> 00:10:30,989 That way, just like every time x went through a wavelength, 298 00:10:30,989 --> 00:10:33,700 every time we walk one wavelength along the pier, 299 00:10:33,700 --> 00:10:36,771 we see the same height, because this becomes two pi. 300 00:10:36,771 --> 00:10:39,232 Every time we wait one whole period, 301 00:10:39,232 --> 00:10:40,515 this becomes two pi, 302 00:10:40,515 --> 00:10:42,466 and this whole thing is gonna reset again. 303 00:10:42,466 --> 00:10:43,892 So this is the wave equation, 304 00:10:43,892 --> 00:10:46,082 and I guess we could make it a little more general. 305 00:10:46,082 --> 00:10:48,323 This cosine could've been sine. 306 00:10:48,323 --> 00:10:50,510 So if you end up with a wave that's better described 307 00:10:50,510 --> 00:10:51,343 with a sine, 308 00:10:51,343 --> 00:10:52,525 maybe it starts here and goes up, 309 00:10:52,525 --> 00:10:54,143 you might want to use sine. 310 00:10:54,143 --> 00:10:57,001 And the negative, remember the negative caused this wave 311 00:10:57,001 --> 00:10:58,277 to shift to the right, 312 00:10:58,277 --> 00:10:59,879 you could use negative or positive 313 00:10:59,879 --> 00:11:02,160 because it could shift right with the negative, 314 00:11:02,160 --> 00:11:03,364 or if you use the positive, 315 00:11:03,364 --> 00:11:05,557 adding a phase shift term shifts it left. 316 00:11:05,557 --> 00:11:07,700 So a positive term up here would describe a wave 317 00:11:07,700 --> 00:11:09,557 moving to the left 318 00:11:09,557 --> 00:11:11,688 and technically speaking, you could make it 319 00:11:11,688 --> 00:11:13,335 just slightly more general 320 00:11:13,335 --> 00:11:16,093 by having one more constant phase shift term 321 00:11:16,093 --> 00:11:17,104 over here to the right. 322 00:11:17,104 --> 00:11:19,303 If we add this, then we could take into account 323 00:11:19,303 --> 00:11:22,694 cases that are weird where maybe the graph starts like here 324 00:11:22,694 --> 00:11:25,725 and neither starts as a sine or a cosine. 325 00:11:25,725 --> 00:11:28,453 You'd have to draw it shifted by just a little bit. 326 00:11:28,453 --> 00:11:29,713 But in our case right here, 327 00:11:29,713 --> 00:11:30,546 you don't have to worry about it 328 00:11:30,546 --> 00:11:32,462 because it started at a maximum, 329 00:11:32,462 --> 00:11:34,102 so you wouldn't have to have that phase shift. 330 00:11:34,102 --> 00:11:34,935 And this is it. 331 00:11:34,935 --> 00:11:35,768 This is the wave equation. 332 00:11:35,768 --> 00:11:36,758 This is what we wanted: 333 00:11:36,758 --> 00:11:39,167 a function of position in time 334 00:11:39,167 --> 00:11:40,810 that tells you the height of the wave 335 00:11:40,810 --> 00:11:43,370 at any position x, horizontal position x, 336 00:11:43,370 --> 00:11:44,786 and any time T. 337 00:11:44,786 --> 00:11:46,212 So let's try to apply this formula 338 00:11:46,212 --> 00:11:48,472 to this particular wave we've got right here. 339 00:11:48,472 --> 00:11:49,305 So I'm gonna get rid of this. 340 00:11:49,305 --> 00:11:52,402 This was just the expression for the wave 341 00:11:52,402 --> 00:11:54,057 at one moment in time. 342 00:11:54,057 --> 00:11:57,433 So maybe this picture that we took of the wave at the pier 343 00:11:57,433 --> 00:12:01,415 was at the moment, let's call it T equals zero seconds. 344 00:12:01,415 --> 00:12:03,795 So at T equals zero seconds, we took this picture. 345 00:12:03,795 --> 00:12:05,104 That's what the wave looks like, 346 00:12:05,104 --> 00:12:07,175 and this is the function that describes 347 00:12:07,175 --> 00:12:10,135 what the wave looks like at that moment in time, 348 00:12:10,135 --> 00:12:11,488 but we're gonna do better now. 349 00:12:11,488 --> 00:12:13,428 Now we're gonna describe what the wave looks like 350 00:12:13,428 --> 00:12:16,696 for any position x and any time T. 351 00:12:16,696 --> 00:12:17,529 So let's do this. 352 00:12:17,529 --> 00:12:18,664 What would the amplitude be? 353 00:12:18,664 --> 00:12:20,429 That's easy, it's still three. 354 00:12:20,429 --> 00:12:22,530 The wave never gets any higher than three, 355 00:12:22,530 --> 00:12:24,432 never gets any lower than negative three, 356 00:12:24,432 --> 00:12:26,954 so our amplitude is still three meters. 357 00:12:26,954 --> 00:12:31,296 And since at x equals zero and T equals zero, 358 00:12:31,296 --> 00:12:33,420 our graph starts at a maximum, 359 00:12:33,420 --> 00:12:35,063 we're still gonna want to use cosine. 360 00:12:35,063 --> 00:12:37,147 So we come in here, two pi x over lambda. 361 00:12:37,147 --> 00:12:38,773 Well, the lambda is still a lambda, 362 00:12:38,773 --> 00:12:41,193 so a lambda here is still four meters, 363 00:12:41,193 --> 00:12:44,175 because it took four meters for this graph to reset. 364 00:12:44,175 --> 00:12:46,522 You had to walk four meters along the pier 365 00:12:46,522 --> 00:12:47,950 to see this graph reset. 366 00:12:47,950 --> 00:12:49,348 That's a little misleading. 367 00:12:49,348 --> 00:12:50,527 I mean, you'd have to run really fast. 368 00:12:50,527 --> 00:12:52,471 The wave's gonna be moving as you're walking. 369 00:12:52,471 --> 00:12:54,295 So I should say, if you're standing at zero 370 00:12:54,295 --> 00:12:56,037 and a friend of yours is standing at four, 371 00:12:56,037 --> 00:12:58,039 you would both see the same height 372 00:12:58,039 --> 00:13:00,222 because the wave resets after four meters. 373 00:13:00,222 --> 00:13:02,172 Would we want positive or negative? 374 00:13:02,172 --> 00:13:03,943 Since this wave is moving to the right, 375 00:13:03,943 --> 00:13:05,417 we would want the negative. 376 00:13:05,417 --> 00:13:07,507 I wouldn't need a phase shift term 377 00:13:07,507 --> 00:13:09,828 because this starts as a perfect cosine. 378 00:13:09,828 --> 00:13:12,801 It doesn't start as some weird in-between function. 379 00:13:12,801 --> 00:13:15,042 The only question is what do I plug in for the period? 380 00:13:15,042 --> 00:13:17,097 So I would need one more piece of information. 381 00:13:17,097 --> 00:13:19,122 If I'm told the period, that'd be fine. 382 00:13:19,122 --> 00:13:21,148 But sometimes questions are trickier than that. 383 00:13:21,148 --> 00:13:24,249 Maybe they tell you this wave is traveling to the right 384 00:13:24,249 --> 00:13:26,332 at 0.5 meters per second. 385 00:13:27,306 --> 00:13:29,223 Let's say that's the wave speed, 386 00:13:29,223 --> 00:13:30,795 and you were asked, "Create an equation 387 00:13:30,795 --> 00:13:33,566 "that describes the wave as a function of space and time." 388 00:13:33,566 --> 00:13:35,277 So you'd do all of this, but then you'd be like, 389 00:13:35,277 --> 00:13:36,602 how do I find the period? 390 00:13:36,602 --> 00:13:38,506 We'd have to use the fact that, remember, 391 00:13:38,506 --> 00:13:40,404 the speed of a wave is either written as 392 00:13:40,404 --> 00:13:43,815 wavelength times frequency, or you can write it as 393 00:13:43,815 --> 00:13:45,516 wavelength over period. 394 00:13:45,516 --> 00:13:46,633 So I can solve for the period, 395 00:13:46,633 --> 00:13:48,506 and I can say that the period of this wave 396 00:13:48,506 --> 00:13:50,961 if I'm given the speed and the wavelength, 397 00:13:50,961 --> 00:13:52,757 I can find the wavelength on this graph. 398 00:13:52,757 --> 00:13:53,970 I'd say that the period of the wave 399 00:13:53,970 --> 00:13:56,650 would be the wavelength divided by the speed. 400 00:13:56,650 --> 00:13:59,361 So our wavelength was four meters, and our speed, 401 00:13:59,361 --> 00:14:00,698 let's say we were just told 402 00:14:00,698 --> 00:14:03,084 that it was 0.5 meters per second, 403 00:14:03,084 --> 00:14:05,174 would give us a period of eight seconds. 404 00:14:05,174 --> 00:14:06,937 So we'd have to plug in eight seconds over here 405 00:14:06,937 --> 00:14:07,770 for the period. 406 00:14:07,770 --> 00:14:08,603 And there it is. 407 00:14:08,603 --> 00:14:10,538 That's my equation for this wave. 408 00:14:10,538 --> 00:14:13,457 This describes, this little equation is amazing. 409 00:14:13,457 --> 00:14:15,489 It describes the height of this wave 410 00:14:15,489 --> 00:14:18,607 at any position x and any time T. 411 00:14:18,607 --> 00:14:21,649 So in other words, I could plug in three meters for x 412 00:14:21,649 --> 00:14:24,572 and 5.2 seconds for the time, 413 00:14:24,572 --> 00:14:27,465 and it would tell me, "What's the height of this wave 414 00:14:27,465 --> 00:14:30,588 "at three meters at the time 5.2 seconds?" 415 00:14:30,588 --> 00:14:31,674 Which is pretty amazing. 416 00:14:31,674 --> 00:14:33,752 So recapping, this is the wave equation 417 00:14:33,752 --> 00:14:35,308 that describes the height of the wave 418 00:14:35,308 --> 00:14:37,287 for any position x and time T. 419 00:14:37,287 --> 00:14:38,367 You would use the negative sign 420 00:14:38,367 --> 00:14:40,002 if the wave is moving to the right 421 00:14:40,002 --> 00:00:00,000 and the positive sign if the wave was moving to the left.