Two waves with the same wavelength and frequency move in the same direction, with one wave leading the other by one whole wavelength. Is the interference between the waves constructive, destructive, or neither constructive nor destructive?
Okay, so, first of all, in this question, we’ve been told that we’ve got two waves each having the same wavelength and frequency as the other. So, let’s start by drawing out two sinusoidal waves, in other words, waves shape like sine curves, where these two waves have the same wavelength and frequency as each other. Now we can see that for the orange wave, the wavelength is, for example, the distance between this peak and the next adjacent peak. So, that’s this distance here. And then for the pink wave, it’s the distance between this peak and the adjacent peak here. So, in other words, this distance.
Now what we’re saying is that if the orange wave has a wavelength 𝜆, then the pink wave also has a wavelength 𝜆. Now as well as this, we’ve been told that the two waves have the same frequency. So, if the orange wave has a frequency 𝑓, then the pink wave also has a frequency 𝑓. In other words, if we were to stand at a certain point along which the orange wave is travelling, and as the orange wave moved in this direction, we counted the number of peaks that passed that pink point per second, then we would have the frequency 𝑓. And also, wow, that was a lot of ps in one sentence.
But anyway, so, what we’re saying here is that not only do these two waves have the same wavelength 𝜆, but also the number of cycles of the orange wave that pass a particular point every second is the same as the number of cycles of the pink wave that pass a point per second.
Therefore, for our purposes, these two waves are travelling at exactly the same speed in the same direction. And, we know this because we’ve been told that they’re travelling in the same direction. But also, we can recall that wave speed is defined as the frequency of the wave multiplied by the wavelength. And we’ve been told that both the orange and the pink wave have the same frequency and the same wavelength.
Now the only difference is that we’ve been told that one wave is leading the other by one whole wavelength. In other words, the way that we’ve currently drawn the two waves, we see that they’re basically in sync. This peak lines up with this peak. This trough lines up with this trough. This peak lines up with this peak, and so on and so forth. However, we want one of the waves, let’s say the orange wave, to be leading the pink wave by one whole wavelength.
Now one whole wavelength is the distance between two adjacent peaks or two adjacent troughs, or so on and so forth. And therefore, what we could do is to move the orange wave along so that this peek here now moves along to where this peak is. And this peak has moved along the same distance forward. And this trough has moved the same distance forward, and so on. And this is what that looks like. We’ve now got this orange wave leading the pink wave by one whole wavelength.
But now what we see is that a trough is still corresponding to a trough on the pink wave. And this peak here on the orange wave corresponds to another peak on the pink wave. And if we were to draw more of the pink wave and more of the orange wave, then we can see that peaks always corresponds to peaks, albeit badly drawn in this case, and troughs always correspond to troughs. And midpoints always correspond to midpoints. And we can see that even though the orange wave is supposed to be leading the pink wave by one whole wavelength, the fact is that the two waves are in-phase with each other.
And so, when we talk about the interference between two waves, what we’re essentially talking about is the net result when we add these two waves together. Now, of course, in reality, if the two waves are to interfere, then they must be occupying the same region or space. In other words, if we zoom out and look at the orange wave doing this, then the pink wave must actually be on top of the orange wave. However, just for clarity, we’ve drawn one above the other.
So, to work out the net result of the interference between these two waves, what we need to do is to add the vertical displacement of each wave at every single point along these waves. And because these two waves are in-phase with each other, we can see that when we add the vertical displacement of this peak to the vertical displacement of this peak, the net result is going to be that we’re going to get an even bigger peak.
And similarly, at a midpoint we see that the vertical displacement from an imaginary axis that we can draw is zero. And we can draw that imaginary axis on the pink wave as well. And so, at any midpoint where the vertical displacement is zero, we see that the net result is that the vertical displacement will be zero as well.
Zooming out slightly, we can see that when we add the orange wave to the pink wave, the net result is going to be a wave which has an amplitude that’s equal to the sum of the amplitudes of the orange and pink waves. And remember, in this case, the amplitude of the wave is the maximum vertical displacement of the wave. That could be the distance from the centre line to the peak or the distance from the centre line to the trough. And this is true for the orange wave and the pink wave.
And so, what we’re saying is that the amplitude of the blue wave, that’s this distance here, is equal to the amplitude of the orange wave plus the amplitude of the pink wave. And because we’ve got two waves adding together to result in an even bigger wave, what we have here is constructive interference. And it’s important to remember that constructive interference occurs when two waves are in-phase with each other. And hence, we found our final answer. The interference between these two waves is constructive.