## 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.

## FAQs

### What wave pattern forms when 2 equal sized waves travel in opposite directions and interfere with each other? ›

**Standing Waves**

A standing wave is a wave pattern that forms when two equal-sized waves travel in opposite directions and continuously interfere with each other.

**What is the interference of mechanical waves? ›**

Wave interference is **the phenomenon that occurs when two waves meet while traveling along the same medium**. The interference of waves causes the medium to take on a shape that results from the net effect of the two individual waves upon the particles of the medium.

**What is the formula for wave interference? ›**

The general formula for destructive interference due to a path difference is given by **δ = (m + 1/2) λ / n** where n is the index of refraction of the medium in which the wave is traveling, λ is the wavelength, δ is the path difference and m = 0, 1, 2, 3 ....

**What is it called when 2 waves meet in opposite directions and completely cancel each other? ›**

**Destructive interference** occurs when waves come together in such a way that they completely cancel each other out. When two waves interfere destructively, they must have the same amplitude in opposite directions.

**What happens when two waves travel in opposite directions? ›**

Two waves travelling in opposite directions **produce a standing wave**.

**What are the 2 types of wave interference called? ›**

**Constructive interference** happens when two waves overlap in such a way that they combine to create a larger wave. Destructive interference happens when two waves overlap in such a way that they cancel each other out.

**What are two examples of wave interference? ›**

One of the best examples of interference is demonstrated by the **light reflected from a film of oil floating on water**. Another example is the thin film of a soap bubble (illustrated in Figure 1), which reflects a spectrum of beautiful colors when illuminated by natural or artificial light sources.

**What is an example of wave interference? ›**

**A rainbow** is an ideal example that is formed by light. The echo is the result of the reflection of sound. The sound felt while striking a tuning fork.

**What are the 3 equations for wave speed? ›**

The relationship between the parts of a wave is given by the wave speed or wave velocity equation. A wave speed is the distance traveled by a given point on a wave in a given interval of time. It is expressed as **v = λ T or v = λ f** , where v is the wave speed, is the wavelength, T is the period, and f is the frequency.

**What is the formula to solve for wave speed your answer? ›**

Wave speed is related to both wavelength and wave frequency by the equation: **Speed = Wavelength x Frequency**.

### What is an example of a mechanical wave? ›

A **sound wave** is an example of a mechanical wave. Sound waves are incapable of traveling through a vacuum. Slinky waves, water waves, stadium waves, and jump rope waves are other examples of mechanical waves; each requires some medium in order to exist.

**What is simple interference of waves? ›**

What is Interference? , **The phenomenon in which two or more waves superpose to form a resultant wave of greater, lower or the same amplitude**. The interference of waves results in the medium taking on a shape resulting from the net effect of the two individual waves.

**What is wave interference of 3 waves? ›**

The waves are shown as the displacement as a function of position at a fixed instant in time. The resulting wave is created by the “interference” of the three waves, and mathematically is simply **a sum of the three individual waves at each position (and instant in time)**.

**What is the law of interference physics? ›**

interference, in physics, **the net effect of the combination of two or more wave trains moving on intersecting or coincident paths**. The effect is that of the addition of the amplitudes of the individual waves at each point affected by more than one wave.

**What is it called when two waves interfere? ›**

**Constructive interference** occurs when the crests of one wave overlap the crests of the other wave, causing an increase in wave amplitude. Destructive interference occurs when the crests of one wave overlap the troughs of the other wave, causing a decrease in wave amplitude.

**What if two waves interfere with each other? ›**

When two waves meet in such a way that their crests line up together, then it's called **constructive interference**. The resulting wave has a higher amplitude. In destructive interference, the crest of one wave meets the trough of another, and the result is a lower total amplitude.

**What wave pattern forms when waves of equal wavelength and amplitude but traveling in opposite directions continuously interfere with each? ›**

**standing wave**, also called stationary wave, combination of two waves moving in opposite directions, each having the same amplitude and frequency. The phenomenon is the result of interference; that is, when waves are superimposed, their energies are either added together or canceled out.

**When these two waves combine what kind of interference will occur? ›**

**Constructive interference** happens when two waves overlap in such a way that they combine to create a larger wave. Destructive interference happens when two waves overlap in such a way that they cancel each other out.