1 00:00:00,203 --> 00:00:01,801 - [Narrator] So, let's say you had a string here, 2 00:00:01,801 --> 00:00:04,458 and you give that string a little yoink up and down. 3 00:00:04,458 --> 00:00:07,860 You'd see a little wave pulse travel down that string. 4 00:00:07,860 --> 00:00:08,917 But here's a question, 5 00:00:08,917 --> 00:00:11,413 if someone else on the other end of this string 6 00:00:11,413 --> 00:00:15,749 also sent a wave pulse down the line toward the first wave, 7 00:00:15,749 --> 00:00:17,501 what would happen when they overlap? 8 00:00:17,501 --> 00:00:18,460 So, let's try to figure this out. 9 00:00:18,460 --> 00:00:20,317 Let's say you had a wave coming in this way, 10 00:00:20,317 --> 00:00:21,971 and yes, this is square. 11 00:00:21,971 --> 00:00:23,510 Kinda weird. It'd be hard. 12 00:00:23,510 --> 00:00:26,150 You have to be pretty talented to do this on a string, 13 00:00:26,150 --> 00:00:27,532 but this doesn't have to be a string. 14 00:00:27,532 --> 00:00:29,175 Let's say it could be a sound wave, 15 00:00:29,175 --> 00:00:31,386 an electromagnetic wave, any wave at all. 16 00:00:31,386 --> 00:00:33,442 The fact that it's a square is just gonna make it easier 17 00:00:33,442 --> 00:00:34,704 for us to analyze. 18 00:00:34,704 --> 00:00:36,019 So, you got this wave coming in this way 19 00:00:36,019 --> 00:00:38,346 and then another wave coming in this way. 20 00:00:38,346 --> 00:00:40,413 So, to be clear, this is the same string. 21 00:00:40,413 --> 00:00:41,731 There aren't like, two strings here. 22 00:00:41,731 --> 00:00:43,954 There is one string, two waves coming at each other. 23 00:00:43,954 --> 00:00:47,110 So, to be real, I mean, honestly, there's only one string, 24 00:00:47,110 --> 00:00:48,406 so this should be string coming here, 25 00:00:48,406 --> 00:00:49,992 and then there's a pulse up this way, 26 00:00:49,992 --> 00:00:51,512 so this string shouldn't be here. 27 00:00:51,512 --> 00:00:53,337 This string moved up to that point. 28 00:00:53,337 --> 00:00:55,908 It got disturbed, then it comes back down to zero, 29 00:00:55,908 --> 00:00:57,465 and then there shouldn't be two strings here. 30 00:00:57,465 --> 00:00:59,664 You don't have two strings in the same spot, 31 00:00:59,664 --> 00:01:02,001 so this wouldn't be, the string up there, 32 00:01:02,001 --> 00:01:03,251 and then it comes back down, 33 00:01:03,251 --> 00:01:04,247 but in these examples 34 00:01:04,247 --> 00:01:05,778 I don't wanna have to erase these all the time. 35 00:01:05,778 --> 00:01:07,344 It'd make the video really long, so let's just 36 00:01:07,344 --> 00:01:10,018 any time there's a string underneath a pulse 37 00:01:10,018 --> 00:01:12,374 we're just gonna pretend like there's no string in there. 38 00:01:12,374 --> 00:01:13,207 So, what would happen? 39 00:01:13,207 --> 00:01:15,094 What would happen when these pulses overlap? 40 00:01:15,094 --> 00:01:16,574 Well, let's just find out if I take one, 41 00:01:16,574 --> 00:01:20,173 and I move this here, and then another wave's gonna move 42 00:01:20,173 --> 00:01:21,685 over the top of that one. 43 00:01:21,685 --> 00:01:23,283 I'm gonna get wave interference. 44 00:01:23,283 --> 00:01:25,232 This is the term, wave interference, 45 00:01:25,232 --> 00:01:28,884 for when two or more waves overlap in the same region. 46 00:01:28,884 --> 00:01:29,883 So, what's gonna happen? 47 00:01:29,883 --> 00:01:32,191 Well, the string can't be in two places at once. 48 00:01:32,191 --> 00:01:35,487 There can only be one string and one shape of that string. 49 00:01:35,487 --> 00:01:37,352 And the way you find out what the total wave 50 00:01:37,352 --> 00:01:40,566 is gonna look like is simply by adding up the contributions 51 00:01:40,566 --> 00:01:42,940 of the two waves that are overlapping. 52 00:01:42,940 --> 00:01:45,010 So, in other words, if I wanna know the height 53 00:01:45,010 --> 00:01:48,159 of the total wave, I'm gonna call that height YT, 54 00:01:48,159 --> 00:01:49,291 T for total. 55 00:01:49,291 --> 00:01:51,752 That's just gonna equal the height of the first wave, 56 00:01:51,752 --> 00:01:54,713 I'll call that Y1, plus the height of the second wave, 57 00:01:54,713 --> 00:01:56,316 and I'll call that Y2. 58 00:01:56,316 --> 00:01:58,719 So, if you're familiar with the wave equations 59 00:01:58,719 --> 00:02:00,916 you could just plug in those two wave equations here, 60 00:02:00,916 --> 00:02:03,068 add 'em up, and you get a total wave equation. 61 00:02:03,068 --> 00:02:04,541 But a lotta times you don't have to resort 62 00:02:04,541 --> 00:02:07,546 to the full blown mathematics of the wave equation. 63 00:02:07,546 --> 00:02:09,080 You can kinda just look at the picture 64 00:02:09,080 --> 00:02:10,711 and figure out what the total wave would look like. 65 00:02:10,711 --> 00:02:11,855 So, let's do that. 66 00:02:11,855 --> 00:02:13,354 Let's put a little backdrop here, 67 00:02:13,354 --> 00:02:14,558 so we can add these up. 68 00:02:14,558 --> 00:02:16,052 So, we'll call this one unit high, 69 00:02:16,052 --> 00:02:17,207 and this is gonna be two units high, 70 00:02:17,207 --> 00:02:18,606 and this is gonna be three units high. 71 00:02:18,606 --> 00:02:21,032 It could be meters or centimeters, but it doesn't matter. 72 00:02:21,032 --> 00:02:22,995 We'll just say one unit, two unit, three unit. 73 00:02:22,995 --> 00:02:25,421 And now to figure out what the total wave's gonna look like 74 00:02:25,421 --> 00:02:28,301 I just add up the contributions from each individual wave. 75 00:02:28,301 --> 00:02:30,754 So, both waves are zero over here, so that's easy. 76 00:02:30,754 --> 00:02:32,256 Zero plus zero is zero. 77 00:02:32,256 --> 00:02:33,281 And then it gets to here. 78 00:02:33,281 --> 00:02:35,382 The blue wave, we'll call that wave one, 79 00:02:35,382 --> 00:02:38,239 has a value of one unit high. 80 00:02:38,239 --> 00:02:40,304 The pink wave, we'll call that wave two, 81 00:02:40,304 --> 00:02:42,039 has a value of two units high. 82 00:02:42,039 --> 00:02:43,327 They're going in different directions. 83 00:02:43,327 --> 00:02:44,160 Doesn't matter. 84 00:02:44,160 --> 00:02:46,487 Right now they're overlapping, so one unit high 85 00:02:46,487 --> 00:02:50,554 plus two units high is gonna be equal to three units high. 86 00:02:50,554 --> 00:02:52,745 My total wave would look something like this. 87 00:02:52,745 --> 00:02:55,561 So, if I were to ask what would the wave actually look like? 88 00:02:55,561 --> 00:02:57,292 The string, if this were a string, 89 00:02:57,292 --> 00:02:58,701 would actually look like this. 90 00:02:58,701 --> 00:03:01,711 It would just be one big three unit high wave, 91 00:03:01,711 --> 00:03:03,652 while those two waves are overlapping. 92 00:03:03,652 --> 00:03:04,616 So, that one was kinda easy. 93 00:03:04,616 --> 00:03:05,642 How does this get harder? 94 00:03:05,642 --> 00:03:06,729 Well, let's say we ask the question 95 00:03:06,729 --> 00:03:08,343 what do these two waves look like 96 00:03:08,343 --> 00:03:10,253 when they're only partially overlapping? 97 00:03:10,253 --> 00:03:11,927 So, maybe when they get to this point, 98 00:03:11,927 --> 00:03:13,776 where they're only halfway overlapped, 99 00:03:13,776 --> 00:03:15,569 what's that total wave gonna look like? 100 00:03:15,569 --> 00:03:16,749 We're still gonna use this rule. 101 00:03:16,749 --> 00:03:19,152 We're gonna add up both contributions to get the total. 102 00:03:19,152 --> 00:03:22,838 So, over here we have zero, and zero plus zero is zero. 103 00:03:22,838 --> 00:03:24,130 Until you get to here. 104 00:03:24,130 --> 00:03:28,287 Now the blue wave, wave one, has a value of one unit high. 105 00:03:28,287 --> 00:03:29,820 The height of this wave is one unit. 106 00:03:29,820 --> 00:03:32,905 The height of the pink wave, wave two, is zero units. 107 00:03:32,905 --> 00:03:36,134 One plus zero is one, so my total wave would look like this 108 00:03:36,134 --> 00:03:37,234 in that region, 109 00:03:37,234 --> 00:03:40,030 and now in this region the blue wave is one unit high, 110 00:03:40,030 --> 00:03:42,201 the pink wave is two units high, 111 00:03:42,201 --> 00:03:44,623 one plus two is three, so it would look like this 112 00:03:44,623 --> 00:03:45,540 in this region. 113 00:03:45,540 --> 00:03:47,717 Now over here, since the blue wave dropped down 114 00:03:47,717 --> 00:03:49,128 we have to figure out a new value, 115 00:03:49,128 --> 00:03:51,263 so the blue wave has a value of zero. 116 00:03:51,263 --> 00:03:53,122 The pink wave has a value of two. 117 00:03:53,122 --> 00:03:55,057 Two plus zero is two, 118 00:03:55,057 --> 00:03:57,063 and so my total wave's gonna look like this. 119 00:03:57,063 --> 00:03:59,298 So, the total string when those are overlapping halfway 120 00:03:59,298 --> 00:04:01,046 would look something like this, 121 00:04:01,046 --> 00:04:03,072 which if it was a string would be really hard to do, 122 00:04:03,072 --> 00:04:04,860 'cause it's hard to get an exactly square wave, 123 00:04:04,860 --> 00:04:07,095 but electronic signals can have square waves, 124 00:04:07,095 --> 00:04:08,498 and this is what they would look like 125 00:04:08,498 --> 00:04:09,922 if they were overlapping. 126 00:04:09,922 --> 00:04:11,054 Now I wanna warn you about one thing. 127 00:04:11,054 --> 00:04:13,751 This idea of wave interference is a cool idea, 128 00:04:13,751 --> 00:04:14,709 but you gotta be careful. 129 00:04:14,709 --> 00:04:16,103 The term interference is a little misleading. 130 00:04:16,103 --> 00:04:18,799 Yes, while these waves are overlapping 131 00:04:18,800 --> 00:04:20,289 they create a different wave. 132 00:04:20,289 --> 00:04:22,041 They get distorted because the total wave 133 00:04:22,041 --> 00:04:23,989 will be the sum of the two waves. 134 00:04:23,989 --> 00:04:25,914 But these waves pass right through each other, 135 00:04:25,914 --> 00:04:28,178 which is great, 'cause when our phones send out 136 00:04:28,178 --> 00:04:30,697 a text message or a call to someone else, 137 00:04:30,697 --> 00:04:33,283 everyone else's phone is also sending out a message 138 00:04:33,283 --> 00:04:34,670 in that same air. 139 00:04:34,670 --> 00:04:36,536 Those electromagnetic waves are traveling 140 00:04:36,536 --> 00:04:37,369 right through each other. 141 00:04:37,369 --> 00:04:39,028 If they bounced off of each other, 142 00:04:39,028 --> 00:04:41,556 right, if these waves like bounced or got corrupted, 143 00:04:41,556 --> 00:04:43,024 and the information got changed 144 00:04:43,024 --> 00:04:45,445 so that the shape isn't the same after 145 00:04:45,445 --> 00:04:48,340 as it was before, it'd be really hard to make phone calls 146 00:04:48,340 --> 00:04:49,755 and send text messages, 147 00:04:49,755 --> 00:04:51,605 but the wave interference is only happening 148 00:04:51,605 --> 00:04:52,875 while they're overlapping. 149 00:04:52,875 --> 00:04:54,986 The waves make it through unaffected. 150 00:04:54,986 --> 00:04:56,361 So, the interference is only happening 151 00:04:56,361 --> 00:04:58,016 while they're overlapping, 152 00:04:58,016 --> 00:05:00,971 otherwise they pass right through each other unaffected, 153 00:05:00,971 --> 00:05:02,041 which is good. 154 00:05:02,041 --> 00:05:03,281 So, let's look at one more example 155 00:05:03,281 --> 00:05:04,728 that's a little more challenging. 156 00:05:04,728 --> 00:05:06,120 So, I'm gonna get rid of these, 157 00:05:06,120 --> 00:05:08,569 and let's say you had these two wave pulses, 158 00:05:08,569 --> 00:05:09,904 the same square pulse, 159 00:05:09,904 --> 00:05:12,853 and then you got this weird triangular pulse coming in, 160 00:05:12,853 --> 00:05:14,107 and they're gonna overlap. 161 00:05:14,107 --> 00:05:16,122 So, this wave pulse makes to here. 162 00:05:16,122 --> 00:05:18,345 This triangular pulse makes it to here. 163 00:05:18,345 --> 00:05:20,499 Oh, and you get the state of Nevada. 164 00:05:20,499 --> 00:05:23,553 So, the string is not gonna take the shape of Nevada. 165 00:05:23,553 --> 00:05:25,648 The string can't be in two places at once, 166 00:05:25,648 --> 00:05:26,994 so what's our total wave gonna look like? 167 00:05:26,994 --> 00:05:29,032 We're gonna use the same rules that we did before 168 00:05:29,032 --> 00:05:30,071 for wave interference. 169 00:05:30,071 --> 00:05:32,548 We're gonna add up the values of each wave 170 00:05:32,548 --> 00:05:33,642 at a particular point 171 00:05:33,642 --> 00:05:35,920 to get the value of the total wave at that point. 172 00:05:35,920 --> 00:05:36,753 So, what do we get? 173 00:05:36,753 --> 00:05:38,285 We've got zero plus zero over here, 174 00:05:38,285 --> 00:05:39,118 so that's easy, 175 00:05:39,118 --> 00:05:41,942 and now at this moment the value of the blue wave is one. 176 00:05:41,942 --> 00:05:43,951 The value of the pink wave is zero, 177 00:05:43,951 --> 00:05:46,191 so zero plus one is one. 178 00:05:46,191 --> 00:05:47,440 And then here it's a little weird, 179 00:05:47,440 --> 00:05:48,850 like you've got this pink wave changing, 180 00:05:48,850 --> 00:05:50,034 but over here it's easy, 181 00:05:50,034 --> 00:05:52,565 'cause the blue wave has a value of one. 182 00:05:52,565 --> 00:05:55,416 The pink wave let's assume this drops down one as well, 183 00:05:55,416 --> 00:05:56,803 so this is negative one unit. 184 00:05:56,803 --> 00:05:59,463 Blue wave is one, pink wave is negative one. 185 00:05:59,463 --> 00:06:00,582 That's gonna be zero. 186 00:06:00,582 --> 00:06:02,354 One plus negative one is zero. 187 00:06:02,354 --> 00:06:03,718 And after there it's gonna be zero, 188 00:06:03,718 --> 00:06:04,952 but what is it in between? 189 00:06:04,952 --> 00:06:06,515 Well, the simplest answer is actually 190 00:06:06,515 --> 00:06:07,657 the correct answer here. 191 00:06:07,657 --> 00:06:09,476 It just drops down like this. 192 00:06:09,476 --> 00:06:10,653 Why does it do that? 193 00:06:10,653 --> 00:06:12,702 Well, let's consider a point in the middle. 194 00:06:12,702 --> 00:06:13,765 This point in the middle, 195 00:06:13,765 --> 00:06:15,538 this blue wave has a value of one. 196 00:06:15,538 --> 00:06:18,861 The pink wave has a value of negative 1/2, 197 00:06:18,861 --> 00:06:22,113 so one plus negative 1/2 is positive 1/2. 198 00:06:22,113 --> 00:06:23,806 Or, consider a point over here. 199 00:06:23,806 --> 00:06:26,234 The value of the wave right here for the blue wave is one. 200 00:06:26,234 --> 00:06:29,635 The value of the pink wave is like, negative 3/4, 201 00:06:29,635 --> 00:06:32,213 so the value of the total wave would be positive 1/4. 202 00:06:32,213 --> 00:06:35,739 That's why this drops down linearly if this is linear. 203 00:06:35,739 --> 00:06:38,149 This pink wave just keeps taking a bigger and bigger bite 204 00:06:38,149 --> 00:06:39,718 out of this blue wave, 205 00:06:39,718 --> 00:06:41,107 and the total wave would look like this. 206 00:06:41,107 --> 00:06:42,215 So, if we get rid of these 207 00:06:42,215 --> 00:06:44,105 this would be what the total wave looks like 208 00:06:44,105 --> 00:06:46,360 when these two waves overlap. 209 00:06:46,360 --> 00:06:48,792 So, I should say that this technique of just adding up 210 00:06:48,792 --> 00:06:51,381 the values of each wave at that point is called 211 00:06:51,381 --> 00:06:53,419 the superposition principle. 212 00:06:53,419 --> 00:06:57,059 It's a very lofty, intimidating name for something 213 00:06:57,059 --> 00:06:58,231 that's actually pretty simple. 214 00:06:58,231 --> 00:07:00,635 To find the total wave, you just add up the values 215 00:07:00,635 --> 00:07:02,447 of the individual waves. 216 00:07:02,447 --> 00:07:05,209 So, recapping, wave interference is the term we use 217 00:07:05,209 --> 00:07:08,013 to refer to the situation where two or more waves 218 00:07:08,013 --> 00:07:09,627 are overlapping in the same region. 219 00:07:09,627 --> 00:07:11,647 And, to find the value of the total wave 220 00:07:11,647 --> 00:07:12,905 while they're overlapping 221 00:07:12,905 --> 00:07:14,601 you can use the superposition principle, 222 00:07:14,601 --> 00:07:18,511 which just says to add up the values of the individual waves 223 00:07:18,511 --> 00:07:19,587 at a given point 224 00:07:19,587 --> 00:00:00,000 to find the value of the total wave at that point.