Physics 19 Mechanical Waves (13 of 21) The Interference of Waves (Opposite Directions) (2023)

Introduction

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In this video I will show you how to develop the interference of waves equation traveling in the opposite directions.

Content

Okay, continuing on the path of working with interference of waves or wave passing by one another.

In this case we have a situation where the waves are actually travelling in an opposite direction.

Here we have the two wave equations notice.

They still have the same.

Amplitude still have the same wave length same velocity, except one is in the positive direction.

The other one is in the negative direction, so we have wave number one moving to the right wave number two moving to the to the left.

The magnitude of the velocities are the same, but what will we end up with when we do that when two waves interfere with one with each other? In that particular respect again, we're going to need to need need to know or use this particular trigonometric identity.

So let's go ahead and do that notice that when we add the two together we have Y 1 plus y 2 is equal to a times the sine of KX minus Omega, T, plus a times the sine of KX plus Omega T.

The plus simply means it moves to the left and set it to the right.

We can factor out an a so this is equal to a times the sine of KX minus Omega T and that would be plus the sine of KX plus Omega T.

All right now we're going to go ahead and use that identity notice that this here will be our angle.

A this here will be our angle, B see what we get so it's two times.

So this is a times two times the sine of half the sum of the two angles that would be KX: minus Omega, T, plus KX, plus Omega T, all right now times the cosine of one half.

That would be the first angle: KX minus Omega T.

Now it's minus the second angle, so it's minus KX and a minus times a minus minus times a plus.

It's also minus Omega T.

All right good thing, I check that.

Ok, now simplifying things a little bit, we can go ahead and multiply the eight times the 2.

So this is equal to 2 times a times the sine of now we have KX plus KX.

That gives us 2 KX but times 1/2.

That gives us a single KX and a minus Omega T, plus Omega T.

They cancel out and so simply left with the sine of KX.

Now for the cosine, we have 1/2 times the sum of these, so we have KX minus KX.

All that disappears that cancels out and a minus Omega t minus Omega T is minus 2, Omega T, but times 1/2 is minus a single Omega T.

Now here we can use the identity where we have the cosine of a negative angle, which is exactly the same as the cosine of a positive angle.

So this is equal to 2ei times the sine of KX times the cosine of Omega T.

Now, let's find out what we're going to end up with when we have two ways pass filter by each other.

By like that and notice, it's going to be really really interesting.

First of all, we have a sine function.

That only depends on position.

Then we have a cosine function.

That only depends on time not on position all right.

Now, let's draw this right here: let's draw the sine of KX, okay, the sine of KX notice.

That would be the same as what we had over here notice.

The amplitude will be twice as large, so we'll draw much larger amplitude, but it looks like the same wavelength because nothing has changed here.

We still have the same K and K is, of course, the wave number K is 2 PI over lambda, so same wavelength, same frequency, because Omega is still Omega, but of course remember that the sine is not dependent on the frequency is only positive, dependent on position here all right now.

What about the cosine of Omega, T? Well notice? Omega is whatever it is, but T varies, which means that this angle right here will cycle through anywhere from zero to PI, to 2 pi, so whatever's.

This value right here will start at 1.

If T is zero that end up at zero, then I left minus one, then I love zero, Ellen up, 1, 0, minus 1, 0 and so forth.

In other words, this value will bounce or cycle or fluctuate between positive 1 and negative 1 up and down up and down like a co sub function should, but it's that number multiplied times the magnitude of the of the function right here.

So this is a number that's multiplied times, 2 a so sometimes it's 1 we get to a sometimes it's 0 we get 0.

Sometimes it's minus 1.

We get minus 2 a in other words.

What that means is that these amplitudes will be multiplied by a function that varies over time between 1 and negative 1, which means that these will become smaller, go to 0, become negative 1 go to 0, 0 0, negative 1.

In other words, the wave amplitude will fluctuate back and forth like that, but will not change in the horizontal direction, because it's locked in by the function, the sine of KX and that is K- is a constant, and so simply it will stay in place.

The words there will be places like this, where the wave will not change in amplitude and then the rest of it will get larger and smaller and the wave will fluctuate up and down, and essentially what you'll end up with then is something that looks like this.

Where the waves will get smaller will get smaller will get smaller, smaller go to 0 then becomes larger in the negative direction and all the way up to maximum amplitude and small, so back and forth, and back and forth in all these places right here, so the wave will simply so as this one goes up.

This one goes down so back and forward back and forth like that, so the wave will do something like that and of course these will go up.

At the same time, these will go up at the same time.

This will go up at different times like that, so we have what we call a standing wave.

This is the hallmark of what we call a standing wave, a wave that no longer travels either to the left or to the right and where the amplitude of the individual sections will vary from positive to negative maximum amplitude, which in this case will be twice the amplitude of the original waves.

So that's a very interesting result.

So what we end up with when we have two waves that have the same velocity but in opposite directions: the same wavelength, the same frequency and the same amplitude.

We end up with something called a standing wave where the lobes are to speak of the wave, just oscillate back and forth back and forth back and forth like this.

But they don't move in the in the x-direction either to the left or to the right very interesting result and that's how you do.