# Standing Waves

by Jared Rovny

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00:01 Now we're going to discuss what might be the most complicated or the most tricky aspect of sound.

00:06 And so these next few slides as we go through this idea, really pay close attention or at least give yourself some time to go over a few more times because it might be confusing.

00:16 What we're going to do is this, we're going to talk about what's called a standing wave and to represent a wave since we're going to have to talk about these pressure waves, these sounds as it moves around.

00:27 We're going to plot it as you see here.

00:30 First, in the blue we have a, showing a density of the sound, the density of the air rather as the sound moves through it.

00:37 We have less dense and more dense and less dense and more dense, etc.

00:41 What we'll do is we'll plot the pressure at different points along this actual physical density changing wave as we go along the wave.

00:50 So I'm plotting here the pressure of this wave.

00:53 So in the lighter blue, we have low pressure and then in the darker blue, we have high pressure and then so on low pressure, high pressure, low pressure.

01:00 And so when we're plotting these curves, these green curves like what I've shown here, be careful to understand that we're not just talking about something moving up and down.

01:11 In fact we're not talking about that at all, we're actually plotting the pressure of the sound wave and so we're just representing that using a graph but we're still talking about a pressure wave, a wave of sound.

01:23 A standing wave occurs when you have some object in which sound is occurring so this could be a pipe, it could be an organ pipe for example, it could be a flute, it could be anything that has a sound.

01:34 These pressure waves going through it that are what we again call a standing wave and a standing wave what you have is the oscillations, the parts that are oscillating the most with the highest to lowest pressures and the parts between those, maximum and minimum pressures.

01:50 These zero points that you can see are crossing the green access are not changing.

01:53 So in other words if you look at this particular pressure wave that I've drawn here.

01:58 The part where the blue line is crossing the green access, that part is what we call zero or atmosphere pressure so we're saying that there is no extra pressure above the typical atmosphere pressure that's in your pipe.

02:10 In the standing wave those points where the sound wave is crossing zero, in other words where to pin that in atmosphere pressure are not changing, they're staying exactly where they are.

02:19 In similarly, the maximum points the highest points which achieve the highest pressure at the end of the pipe and the minimum points where there's the lowest pressure in the pipe which is in fact less than atmosphere pressure that's the, where a faction is less densifying.

02:34 These highest and lowest points will also stay where they are.

02:37 The highest point will oscillate between high and low and high and low but there will always be the maximum points on your wave.

02:44 The zero points that cross the zero access will stay as zero points and this defined for us a standing wave.

02:51 For standing waves, we could talk about a few different types of object.

02:55 So for example, this pipe here, in this pipe right at the end of the pipe there's a wall.

03:01 So we're saying that this pipe is closed on one end and open on another end.

03:05 What we do is we define some conditions that the sound wave has to follow always at this particular kinds of junctures either at the wall or at the open point.

03:15 At the wall like we have in this pipe, there's just a physical actual wall so the air can't move pass it.

03:21 You will get maximum pressure because as the air comes in and pushes against the air next to the wall.

03:26 The air next to the wall doesn't have anywhere to go.

03:29 So it's really stuck there and that causes it to reach a maximum pressure, a highest pressure.

03:34 And similarly as the air moves away from that wall, no air can get in to compensate and so what we'll then go to a lowest pressure.

03:41 And so for a pipe with a closed physical end, that will always be the maximum pressure point inside that pipe.

03:48 At the open end of the pipe you have the exact opposite sort of condition that you have to maintain.

03:53 At the open end the pipe, the air in the atmosphere is so abundant that there's no way to really build up any pressure rather than open end because as soon as you have even the slightest pressure that's not at zero, air can rush in from the outside, where air can flow out to the outside to compensate and make sure you don't maintain any non-zero pressure at the open end of a pipe.

04:13 And so we have this, what we call boundary conditions on a pipe.

04:15 Again, you have a maximum pressure at the close end of a pipe and you have a minimum or atmosphere at zero pressure at the open end of a pipe.

04:25 Then, if this was true, if we in fact do have what we call this boundary conditions that for a standing wave and a pipe which again has these zero points staying where they are and the maximum points staying maximum, either getting most positive or most negative.

04:38 We can define for ourselves the lowest possible frequency or the largest, longest wavelength of sound that can possibly fit inside of a pipe.

04:48 So for example in this pipe which has a close end and an open end.

04:51 It is a requirement as we just said at the close end to be at a maximum pressure and the open end be at a minimum or zero temperature rather.

05:00 If this was true we can't have a wavelength any longer in the pipe in the one that I've shown here because if I try to make the wavelength any longer there's no way I can still have a maximum at the close end and the minimum or a zero right at the open end.

05:13 So this particular wave if I drew the entire thing out.

05:16 We could see what the entire wave look like and we can measure the wavelength of this wave, the frequency of this wave, etc.

05:24 So for example, so in this example of for any wave, where we're talking about the smallest wavelength or the largest wavelength or the smallest frequency or the highest frequency.

05:34 We call the smallest frequency the lowest most lowest sound that this pipe can create.

05:39 The fundamental frequency for this pipe and again we could describe these points that I've just discussed the maximum points and the minimum points.

05:50 The maximum points, we call the antinodes and the minimum where the zero points which again can't move, they have to stay where they are, are called the nodes.

05:58 And so we have nodes where the pressure can't build up or be decreased and we have the antinodes which are just the opposite or the negative, the antinodes and those are always the maxima or the minima.

06:08 For example, where we could see how these place out in the few physical scenarios.

06:13 If I did have these example the close end and an open end.

06:17 I could draw the entire wave as I discussed so let's see what some of the properties are.

06:22 If I draw this entire wave, we can see where it's wavelength is, it's a very long wavelength much longer than the length of the pipe.

06:28 So the pipe has some length then we can give it a name, L.

06:31 So some length of pipe and then some wave length of your sound wave which again is much longer than the length L of the pipe.

06:39 We can then see how these two are related though since we know that we have an antinode or maximum point right at the end of the pipe and a node or an atmosphere pressure at zero point at the open end the pipe.

06:52 We can draw the rest of the wave and see that we have 4L's, 4 lengths of pipe in the entirety of our wave.

06:59 By this we can solve for what the length of the pipe is as one quarter of the total wavelength of this fundamental frequency of this fundamental sound wave.

07:09 Just by re-arranging, we can know for a given pipe what the wavelength of the lowest frequency is that can fit in that pipe.

07:16 So this is the important point because remember that a real physical situation you would have access to a pipe.

07:23 You would have some physical pipe in front of you and you can measure its length just by looking at the length of this pipe with a close end and an open end.

07:30 You can solve for and re-arrange the equation and figure out what wavelength would fit in that pipe.

07:35 The longest wavelength being the lowest frequency so you can figure out what the fundamental frequency is of a wave and a pipe like this one.

07:43 This isn't the only frequency that can fit in the pipe however.

07:47 We said that this is the lowest frequency or the longest wavelength that can fit in the pipe but we could also decrease that wavelength.

07:55 So we could sort of smash the wave in and still meet our requirement.

07:58 So for example I've done that here, we could move the wave in until the next point, the next zero point in our compression has also crossed the zero axis at the one end of the pipe which is our requirement and so this wavelength which is much shorter than one we saw before also matches the requirements that we have for the ends of our pipe.

08:18 In this case, we could do the exact same sort of analysis and compare the length of the pipe with the wavelength of this wave.

08:27 This is no longer the fundamental wavelength.

08:29 This is no longer the fundamental frequency.

08:31 This is because instead of the longest wavelength we can possibly have, we've now shortened it.

08:36 So in this case, instead of the length being one quarter the overall wavelength.

08:40 We now have something different which is that the length is now three quarters of the overall wavelength and we can again re-arrange and find the wavelength that fits for this particular frequency.

08:51 If we kept doing this, kept compressing our wave to be smaller and smaller and smaller.

08:56 We could keep finding all the different wavelengths that could fit inside of our pipe.

09:00 So we could have a wavelength of 4 times the length of our pipe which is our fundamental. As well as four thirds and then four fifths and we would see this pattern repeating.

09:09 Always this number 4 over a sequence of different odd numbers, 1 and then 3 and then 5 as we compress our wavelength more and more and more.

09:20 There's a different way to re-write this.

09:21 So let's just look at and in fact the wavelength equation, we have on the bottom here.

09:26 We said that the number 4 was always in the numerator of all these numbers we wrote and we said that the denominator was a sequence of odd numbers, it was 1 and then 3 and then 5, etc.

09:38 The way that this is usually talked about and this might be one of the more confusing points so be a little careful here and maybe go over a few times to make sure it makes sense.

09:46 The way this is often talked about is to re-write that denominator in terms of some integer n.

09:52 So this n number that we have down here which this 2 times n plus one.

09:56 This n is always an integer.

09:58 So it can be a zero or it can be 1 or it can be 2 or it can be 3, etc.

10:03 But it can any integer, not just the odd or just the evens.

10:06 The reason we write 2 and plus 1 here is to ensure we have an odd number.

10:11 So 2 time an integer is always by definition even because an even number is the number you can divide by two and I can always divide 2 times n by 2, just by the way we constructed it.

10:24 If I divide 2 n by 2, I end up with an integer which again is the requirement for an even number.

10:30 So by adding 1 to 2 n, we always get an odd number.

10:35 So this is sort of a clever trick for always finding an odd number.

10:39 You take any integer multiplied it by 2 to get an even number and then add 1 that shift it away from being an even number.

10:46 So we know it's an odd number.

10:47 So for example, let's start plugging in some numbers here.

10:49 Supposed I put an n equals zero, then I have my lambda, my wavelength is equal to 4L over 1 which is our first frequency.

10:57 Plugging now the next number n equals one then I have 4L over 2 plus 1 or 4L over 3 which we saw was our next frequency.

11:07 The next wavelength that can fit in our pipe.

11:09 And see you see how this works, we can rewrite a sequence of numbers like the ones that we had, in terms of this integer n which is a way for us to count all the different wavelengths that could fit in the given pipe.

11:21 The frequency is also something we could find from this wavelength.

11:25 We've talked a lot about the fundamental frequency but just by rearranging the equation using our velocity our wave velocity equation.

11:32 We can solve for the frequency which will be the velocity of your wave and again this will be the velocity of sound if we're talking about sound waves divided by the wavelength that we just found and again this comes from our velocity equation for waves.

11:45 And now we have an equation for all the different possible frequencies of sound that could fit in our pipe and it looks like this.

11:52 Velocity divided by 4 times the length of the pipe and then times this 2n, plus 1 factor.

11:56 Where again, this will be counting all the different frequencies I can fit in your pipe and with the fundamental frequency would be if we use n equals zero, that would be the first frequency that would fit in your pipe and the lowest frequency.

The lecture Standing Waves by Jared Rovny is from the course Sound.

### Included Quiz Questions

1. The open end must be fixed at atmospheric pressure but the pressure at the closed end can be at maximum or minimum amplitude
2. Both the open and closed end are fixed at atmospheric pressure
3. Both the open and closed end can be at maximum or minimum pressure
4. The open end is at minimum pressure but the closed end is at maximum pressure
5. The pressure at the open end can be at maximum or minimum pressure as there is flexibility for oscillation but the closed end is fixed at atmospheric pressure as there is no flexibility for change
1. Air cannot move past the closed end and equilibrate the pressure whereas in the open end this is possible
2. The wall pushes air molecules to the highest possible pressure but in the open end this is not possible
3. Air molecules bounce against the wall and cause more pressure
4. The wall produces a beat frequency with higher pressure
5. The closed end is in opposition to the open end, where the pressure is produced
1. The boundary conditions at the ends of the pipe cannot be matched by sound waves that have longer wavelengths
2. The pipe cannot vibrate at lower frequencies, because of its limited size
3. Lower frequencies have very short wavelengths, which are too short to meet the end conditions of the pipe
4. Longer wavelengths will not be able to maximize or minimize the pressure at the closed or open ends of a pipe
5. Sound waves move too quickly to allow lower frequencies to stay in a pipe of finite length
1. n = 0, 1, 2, 3, …
2. n = 0, L, 2L, 3L, …
3. n = 1, 2, 3, ...
4. n = 1, 3, 5, ...
5. n = 0, 2, 4, ...
1. f = v(2n+1)/(4L)
2. f = 4Lv/(2n + 1)
3. f = 4L/(2n + 1)
4. (2n+1)/(4Lv)
5. f = 2πv(2n+1)/(4L)

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