# The Physics Philes, lesson 102: Reflections

We’ve been talking about waves for a while now. But we’ve been assuming that waves will just travel in one direction forever. That’s not very realistic, is it? In real life, waves are bouncing all over the place. They are hitting the boundaries of their mediums, reflecting back, and interfering with each other.

Even if you’ve never thought about waves in these terms, I bet it makes perfect sense intuitively. Think about an echo. It’s just the sound waves of your voice bouncing back to you off a surface. That wave has hit a medium boundary and reflected. The wave is reflected back over the same part of the medium that the initial wave traveled, which is called wave interference. See? It’s not too difficult to understand. But let’s look into it a bit more deeply.

Let’s think about a transverse wave on a string. Pretend that one end is fixed securely to a wall. Now think about flicking the unsecured end of the string to create a transverse wave. What happens when the wave hits the end? The other end is fixed. But the wave doesn’t just stop. The wave exerts a force on the wall. In response, the wall exerts a force on the string and causes a wave to travel in the opposite direction. Not only that, but the pulse is also inverted. That is, if the pulse is in the positive y-direction, the reflected pulse will be in the negative y-direction.

You may have already surmised, but this is called reflection. It’s not the only way a wave can be reflected, however. What happens if the wave hits a boundary, but the end isn’t fixed? Let’s think about our string again, except this time think about one end of the string being attached to a ring that is around a pole so the ring can freely move up and down. The ring and rod still maintain tension, but doesn’t exert a transverse force like the the wall in the previous example would.

OK, not think about flicking the other end of the string, just like in the previous example. When the wave reaches the end, the ring will move and reach maximum displacement. The ring will be momentarily at rest, and eventually be pulled back down. However, instead of just stopping, a wave in reflected in the opposite direction, just like before. This time, however, the displacement will be in the same direction as the initial wave.

Both of these examples are ways that waves can be reflected. Whether the end of the string is fixed or free is called the boundary condition.

OK, fine. But what happens if two waves hit each other from opposite directions? Do they fight a fight to the death? No, that would be madness. But we can look at a couple of situations to figure out what goes on.

Think about a string with two waves flowing through it. Going from left to right, we have a wave whose displacement is in the negative y-direction. Going from right to left, we have a wave whose displacement goes in the positive y-direction. Something like this:

These pulses are supposed to be more or less the same shape (probably less, but just go with it). To figure out what the total displacement of the string is as the pulses overlap and pass each other, we just add up the displacements. Because the two pulses have the same shape, the displacement at point O is always zero. The motion on the left half is the same if we cut the string at point O, threw the right side away, and held point O as fixed. The pulses on the left side correspond to the incident and reflected pulses.

Of course, that won’t always happen. Sometimes, waves that are not inverted with respect to each other. In that case, the displacement at point O would not be zero, but it’s slope will be zero. In that case, the wave will look more like this:

This is supposed to be a representation of a couple of wave pulses as they travel through a string in opposite directions. Do you see those pink circles? Inside those circles, the slope of the string is zero. The string at that point acts like the string attached to the ring and pole.

Adding up the displacements, like we do here, is an example of the principle of superposition. Basically, that means that when two waves overlap, the actual displacement at any point on the string at any time is the sum of the displacement the point would have if just the first wave went through and the displacement the point would have if just the second wave went through. We can just add up the wave functions of the two waves.

It’s important to note that this works because wave functions are linear. This won’t work if the medium the wave is going through doesn’t obey Hooke’s law (which would mean it’s not linear). Still, the principle of superposition is incredibly important when we’re talking about all kinds of waves. For example, superposition is the only reason you can distinguish two sounds at once. If waves didn’t combine in this way, you wouldn’t be able to have a conversation with someone and listen to music at the same time. You would be caught in a cacophony.

Keep this concept of superposition in mind. You’ll need it next week when we start talking about standing waves.

Featured image credit: narcah via Flickr