00:01
And i have a thin taut string, which is tied to ends, two ends.
00:06
So i'm going to draw those ends right here.
00:10
And these ends are fixed, so we're going to be dealing with standing waves.
00:13
It's oscillating in its third harmonic, which means there are three anti -notes.
00:26
The number of the harmonic is the number of anti -notes.
00:30
And so if we're going to draw what this looks like here, we have to sketch a wave with three anti -notes.
00:38
This is what it would look like.
00:40
So we have an anti -note here, anti -note here, anti -note here.
00:47
And then we have a node here, a note here, a note here, and a note here.
01:00
So the first part of this question here, it says, draw a sketch that shows the standing wave pattern.
01:06
We've essentially done that here.
01:08
So part a done.
01:10
The second question gets into doing some calculations.
01:13
So we're going to note something that we're given.
01:15
We're given the functional form of this wave here.
01:18
It says y of x comma t is 5 .6 centimeters times sine of 0 .034 radiance per centimeter times x times sign of 150 radiance per second times t.
01:45
So one thing we're going to note real quick.
01:47
Is that the general equation for a standing wave is y of x comma t is equal to a the standing wave amplitude as w times sine of k x times sign of omega t so from this we can immediately identify some things we can identify what omega is what k is and what the standing wave amplitude is and that'll help us for later problems the second part of the question b asks, find the amplitude of the two traveling waves that make up the standing wave.
02:25
Now, if you don't remember, a standing wave like this, is formed by the superposition of a traveling wave going to the right and a traveling wave going to the left.
02:37
And it's these two waves that superposed together that form a standing wave.
02:42
And from this, there's a relation that the standing wave amplitude is equal to twice the traveling wave amplitude here.
02:53
And so we can read off what the standing wave amplitude is.
02:57
A standing wave is equal to 5 .6 centimeters, and then deduce that the amplitude of the traveling waves that are forming it is then therefore equal to 2 .8 centimeters.
03:22
So part c asks, what is the length of the string? so based on this sketch, so in general, we know for standing waves, l is going to be n lambda over 2, which is the n is the harmonic number.
03:49
So for here it's three.
03:50
But we can also visually see that.
03:52
If this is half of a wavelength, so you remember here from this point to this point is a full wavelength.
04:02
That means this is a half of a wavelength, which means we have one to three halves of a wavelength in here.
04:12
So we can see how this is going to be 3 lambda over 2 here and then from there we need to calculate lambda so we can get lambda from this equation here lambda is 2 pi over k and we can identify what k is here from the functional form be 2 pi over 0 .034 radiance per centimeter and radiance isn't really a real unit and that is going to leave us with 184 .8 centimeters when we plug k in and then we can write l is going to be three halves times lambda which is going to be equal to 277 centimeters so that's what l is equal to so d asks, find the wavelength, the frequency, and the period, and the speed of the traveling waves.
05:38
So we got the wavelength here from this answer up here.
05:49
We know the wavelength is 184 .8 centimeters.
05:54
And then the, let's see, wavelength and frequency.
06:03
So now we have to find f, and we can use this equation.
06:08
F is going to be omega over 2 pi, which is going to be 150 radians per second over 2 pi, which is equal to 23 .87 hertz.
06:39
Fantastic.
06:40
So we got the frequency.
06:41
Now we want to find the period, so we can use this equation.
06:46
T is equal to 1 over f, the period was 1 over 23 .87 hertz which tells us the period is 0 .0 for 188 seconds and finally we want to find the speed...