Video Transcript
In this video, we’re going to learn about wave superposition. We’ll come to understand wave motion better. And we’ll see what happens when multiple waves overlap or interfere with one another.
To get started, imagine that one warm summer day, you and three of your friends gather in your backyard around your above-ground swimming pool. You notice that, at the very center of the pool, your younger brother is taking a nap. Working together with your friends, your goal is to create a wave at the center of the pool big enough to wake your little brother up.
To make the biggest wave possible at the center of the pool, you want to know something about wave superposition. The basic question of wave superposition is this. If I send a wave from the left side moving to the right and then I send a different wave from the right side moving to the left, what will happen when these waves meet?
We can imagine a few different possibilities for what happens when these waves overlap. One possibility is that when these two waves overlap, they don’t combine in any special way. They could simply move straight through one another with no combined effect while they overlap. We can use our experience though to guess that this might not be so. Think of water waves that you’ve seen mixing together, perhaps in a pool or in a small body of water. In fact, these two waves we’ve drawn could be examples of waves of water that mix together.
When mixing of waves happens, for the time that the waves are overlapping one another, accumulative or net wave is created. This cumulative or resultant or net wave has an interesting property. At each point along this wave, the amplitude of the wave is equal to the sum of the amplitudes of the two waves combining to create it. If we call the wave that’s moving to the right, the green wave, wave one and the blue wave wave two and our resultant wave 𝑤 sub 𝑟, then we can write that 𝑤 sub 𝑟 is simply equal to 𝑤 sub one plus 𝑤 sub two.
This means that if we wanna solve for the value of the resultant wave at any point in space, we draw a vertical line. And we add together the value of 𝑤 one on that line with 𝑤 two. The sum of those two numbers is equal to 𝑤 sub 𝑟, the resultant wave amplitude. And we follow that same process at all points along our overlapping waves.
Combining the amplitudes of two waves that are mixing works just like this. And interestingly, if we add more waves, we find more of the same. Say that we had a third wave, call it 𝑤 sub three, added in to this mix of waves. Once again, to find the amplitude of the resulting wave, we would just add 𝑤 one, 𝑤 two, and 𝑤 three’s amplitudes at each point. This relationship for finding the resultant wave amplitude, when we add waves that overlap, is true no matter how many overlapping waves we have. This is called the principle of superposition. Superposition just means waves that overlap or interfere with one another.
To use this principal practically, we’ll need to be able to solve for the amplitudes of each of the waves involved. And to do that, we’ll want to understand their wave equations. Say that we draw a few wavelengths of a sinusoidal wave, where we plot the wave height 𝑦 as a function of its horizontal position 𝑥. We’ve seen that the height 𝑦 of a harmonic wave that varies with position and time can be written as the amplitude of the wave multiplied by the sine of the wavenumber, 𝑘, times its position, 𝑥, minus the angular frequency of the wave, 𝜔, times time 𝑡.
This wave changes its height in both position and time. And in order to develop some intuition about traveling waves overlapping, let’s consider this minus sign here in the sine argument as a way of better understanding wave motion. When we look at this wave drawn as is, we see it’s a snapshot in time of the wave’s position. But we don’t know, just looking at this, which way the wave is moving.
To get clear on that, we can draw in the third axis, the time in seconds, showing the wave’s evolution in that dimension. We’ve already drawn in two wavelengths of the wave when time is equal to zero. In other words, we froze in time and let position vary. So now, let’s alternate. Let’s let the position 𝑥 be frozen at zero and let the time 𝑡 vary. And we’ll see what our wave looks like.
If we were to fix 𝑥 at zero and let time 𝑡 vary, then the wave that will be created along the 𝑡-axis will look something like this. The important thing to notice here is that when we fixed 𝑡 and let 𝑥 vary, we started at zero and then went up to positive values for our wave amplitude. While now, when we fix 𝑥 and let 𝑡 vary, we start at zero but have negative values following that. The reason that this gold curve is inverted compared to the blue curve has to do with this minus sign in the argument for our sine function.
The wavenumber 𝑘 and angular frequency 𝜔 are both positive. So the minus sign in front of our angular frequency means that, right after our curve starts out at zero, at 𝑡 is equal to zero, it will have a negative height. Given this curve, showing the height of our wave when 𝑥 is equal to zero, we want to know which way in space our wave is moving. Is it moving to the right or is it moving to the left?
We can figure this out by sketching in a few different wave iterations at different values of 𝑡. Our drawing quickly starts to get cluttered. But with just a few curves drawn in, we can see that, overall, this wave, as time evolves in the positive direction down the 𝑡-axis, is moving in the positive 𝑥-direction, to the right. So that means that when we see a minus sign in front of our 𝜔𝑡 in the argument of our sine function, our wave is moving in the positive, not negative, 𝑥-direction.
We might wonder though, how would our wave move if, instead of a minus sign, we had a plus sign there? When we have a positive sign, then our curve, when we fix 𝑥 equal to zero, flips around the 𝑡-axis. So that compared to the curve when we had a minus sign, it’s inverted. In this case, when we have this positive sign, we see that the peak of our wave moves to the left in the negative 𝑥-direction.
If we look at a general equation for our wave height then, if we have a positive sign separating the 𝑘𝑥 and 𝜔𝑡, that means our wave moves in the negative 𝑥-direction. But a negative sign means it moves in the positive 𝑥-direction. Speaking of a general equation for wave height, there’s one more aspect we can add to the argument of the sine function to fully generalize this expression.
So far, we’ve considered waves which start at the origin at 𝑥 and 𝑡 equal zero. In other words, a wave height of zero corresponds to those 𝑥- and 𝑡-values. But in general, the wave could start at any point within its amplitude. We could have it begin at its maximum value or somewhere in between. It all has to do with a slight shift, left or right, of the wave. This is called a phase shift. And the way we represent a phase shift in our equation is to have this Greek letter 𝜙 in the argument of our sine function.
Depending on the value of 𝜙, which will be given in radians, our curve will move to the left or to the right. Now that we’ve included the phase angle 𝜙, we have a truly general equation for the height of our sinusoidal wave at any position 𝑥 and any time 𝑡. It’s waves that are described by this relationship that mix and combine and let us use the principle of superposition.
Say that we have two waves. We’ll call them 𝑦 one and 𝑦 two. 𝑦 one, the first wave, has an amplitude of two meters, a wavenumber of six inverse meters, and an angular frequency of four inverse seconds. And since there’s a minus sign in front of the angular frequency, that means that this wave is moving in the positive 𝑥-direction.
Let’s say that 𝑦 two, the second wave, has the same amplitude, the same wavenumber, and the same angular frequency. But this wave is traveling in the negative 𝑥-direction. And it has a phase shift of 𝜋 over two radians. Say that these two waves combine, 𝑦 one coming from the left and 𝑦 two coming from the right. At any given position and time value, we can use the principle of superposition to solve for the resulting wave height.
Imagine we pick 𝑥 is equal to two meters and 𝑡 is equal to one second, as the 𝑥- and 𝑡-values where we wanna solve for the resulting wave height. That wave height is equal to 𝑦 one, with those 𝑥- and 𝑡-values, plus 𝑦 two, at those same 𝑥- and 𝑡-values. Plugging in for these 𝑥- and 𝑡-values into our 𝑦 one and 𝑦 two equations, in the sine arguments of both equation, the units cancel one another out. And we’re left with a unitless result. And our result simplifies to two meters, the wave amplitude, times the sin of eight radians plus two meters times the sin of 16 plus 𝜋 over two radians.
When we add these two numbers together, we get a result of about 0.06 meters. That’s the wave height of the resulting wave from adding 𝑦 one and 𝑦 two together and an 𝑥-value of two meters and a 𝑡-value of one second. In general, when two waves overlap, they can have any phase relationship between them, from zero to two 𝜋 and anywhere in between.
Using the principle of superposition, we can solve for the resulting wave height at any 𝑥- and 𝑡-value for these overlapping waves. But there are two special phase relationships that waves can have when they overlap with one another. Imagine that we take our green wave and we shift it to the left so that it overlaps perfectly on our blue wave. Since these two waves have the same wavenumber and period, when they’re also in phase, they perfectly overlap one another, the green right on top of the blue.
When these waves combine like this, the principle of superposition tells us that the resulting wave height will equal the sum of the wave heights of each individual wave. Sketching in the wave that results from adding together the blue and the green waves, we see its amplitude is significantly increased. When two overlapping waves have peaks that line up as well as troughs that line up, they’re said to be in phase with one another.
When we add together waves that are in phase with one another, those waves are said to constructively interfere. That means that the resulting wave has a greater amplitude than either of the ones that combine to make it. In order to create this constructive interference, we shifted our green wave so it perfectly lined up with the blue. You can probably imagine another scenario that’s a special case of wave interference.
What if we kept moving our green wave to the left so that now, instead of perfectly overlapping with the blue, the peak of the green wave corresponded with the trough of the blue, and vice versa? If we added these two waves together, what would we get? You can probably see we would get a flat line of amplitude zero. The waves perfectly cancel one another out. When interfering waves have a phase difference like this, that is, a phase difference of 180 degrees or 𝜋 radians, they’re said to be out of phase. The peaks of one wave overlap with the troughs of the other.
Since the resulting wave has an amplitude of zero, this is said to be destructive interference, when two waves overlap so that their combined effect is to null one another out. These special overlapping conditions, in particular the condition of constructive interference, is the one that will help us create the biggest wave in our backyard swimming pool to upset our napping brother.
Let’s summarize what we’ve learned so far about wave superposition. We’ve seen that when waves overlap, when they interfere with one another, they follow the principle of superposition. This principle says that the wave height that results from a bunch of waves mixing is equal to the individual heights of each one of those waves. We’ve also seen that, in general, our harmonic wave is described by the equation. The wave height 𝑦, as a function of wave position and time 𝑡, is equal to the wave amplitude multiplied by the sine of the wavenumber times its position minus the angular frequency times time, where this minus sign indicates wave direction in the positive 𝑥-direction, plus a phase angle or a phase shift 𝜙.
It’s waves of this description that make up the individual waves we combine according to the principle of superposition. We’ve also seen that when waves interfere, they can do it constructively, that is, in phase with one another; destructively, out of phase; or at any phase in between. When waves interfere constructively, the resulting wave amplitude is increased. And when they interfere destructively, the resulting wave amplitude is zero.
