5.1 General Properties of Waves

Now that we have learned quite a bit about waves and how they relate to music and musical instruments, it is useful to go back and discuss waves from a more general, and more technical, perspective. Many of the concepts will not come as a surprise, as you have begun to think about waves more carefully.

Waves in General

1. Waves require a medium.
Perhaps one of the most unusual features of waves, as compared to objects, is that waves cannot exist on their own. They require a medium to travel in. For example, sound waves need air. If you place an alarm clock in a large jar and pump out all of the air, you cannot hear the alarm clock ring. The sound waves simply do not exist in a vacuum. The waves in the ocean only exist because of the water. The waves are not objects in themselves; they don’t exist without the medium. In fact, waves represent a disturbance of the medium.
2. A medium that supports waves must have a stable equilibrium.
An equilibrium is a state of any system that does not change over time. For example, a perfectly flat pond is in a state of equilibrium and no waves are present on the pond. However, if we throw a rock into the pond, the water (or the medium) is disturbed, creating ripple around the point that the rock hit the water. The ripples are NOT static and move through the water. It is this disturbance of the medium that IS the wave. There are two kinds of equilibriums: stable and unstable. If we push a system way from equilibrium and it tends to return to its equilibrium, it is said to be stable. For example, let’s consider a marble at the bottom of a bowl:

While the marble sits at the bottom of the bowl, it is in equilibrium – it will not move. If we displace the marble slightly from the bottom, the marble will tend to roll back to its equilibrium point. This is the defining characteristic of a stable equilibrium. Of course, it will overshoot this point and roll up the other side of the bowl. But, now, it will still tend to come back towards the equilibrium and will end up oscillating back and forth around the equilibrium point. This ability to oscillate, as you know, is a critical property of a wave.

Now consider an inverted bowl with a marble on top:

In this case, the marble will again sit on the top of the bowl without moving. However, if we displace it slightly, it will tend to just roll of the bowl altogether. It will certainly not oscillate or even try to come back. This is an unstable equilibrium. A medium with an unstable equilibrium will not support a wave.
3. Restoring force.
A stable equilibrium exists because there is a restoring force returning the medium to its equilibrium. The restoring force always points back to the equilibrium point and is a property of the medium. For sound waves, the medium is air and the restoring force is air pressure. For water waves, the medium is the water and the restoring force is either surface tension or gravity.
4. A wave, or a disturbance of a medium does not stay localized. The wave spreads out in one, two or three dimensions. This aspect is also quite different from normal objects. An object like a baseball generally stays together. If you place it on a table it just sits there. In contrast, a wave is always in motion – sound waves or water waves do not just stand still. Also, waves tend to spread out. If you throw a baseball, the baseball arrives as a whole at some other location; it does not spread out over a large area. Waves do not stay together; they spread out. If sound waves stayed together like an object, I could only lecture to one person at a time. The sound waves that I create as I talk would only get delivered to one location. In fact, waves spread out, so that the sound waves I produce fill the entire room and everyone in the room can hear me.
5. A wave, or a disturbance of a medium, carries energy. This is a very important point. If waves do not exist on their own and are only a disturbance of a medium, one might wonder if they really have a separate existence. The reason we know that waves exist is that they carry energy. It takes energy to move something from its equilibrium. This is the initial disturbance. If the disturbance moves, so must the energy. Therefore, the energy moves along with the wave. If waves did not carry energy, you could not hear me speak. I create sound waves, the wave travel to your ear, and the waves make the eardrum vibrate. However, it takes energy to make the ear drum move, so the sound wave must have energy. A more dramatic example of waves carrying energy is a tsunami. An earthquake in Japan can produce a enormous wave in the ocean called a tsunami. The tsunami can travel across the entire Pacific Ocean and crash on the North American coast and destroy a town. This represents a huge amount of energy traveling a very long distance. Thus, it should be clear that waves have a physical existence, just like objects - waves just follow different physical laws that we are trying to understand.
6. Longitudinal and transverse waves.
There are actually two very distinct kinds of waves or disturbances, depending on the medium. The type of wave has to do with what direction the medium is moving as compared to the direction that the wave is moving. Consider a wave on a string. The wave moves along the string, but the string, itself, moves up and down or back and forth. In this situation, the medium (the string) is moving in a direction perpendicular to the direction that the wave moves. This is called a transverse wave.

Sound waves are rather different. Looking back at Section 3.6, we considered sound waves traveling up and down a tube. Now think about what the individual air molecules are doing. They are vibrating back and forth along the tube, as well. So, in this case, the medium is moving in the same direction as the wave. This is called a longitudinal wave.

Notice that for transverse waves, we mentioned two possible directions for the string: up and down and back and forth. This is true of all transverse waves: there are always two directions for the medium to move that are perpendicular to the direction of the wave. The particular direction that the medium moves is referred to as the polarization of the wave. Transverse waves have two possible polarizations. The idea of polarization is quite important in discussing the properties of light – a kind of wave that we will explore later in this chapter.

For longitudinal waves, the medium has only one direction that it can move – that is parallel to the direction of the wave. So, in this case, there is no need for the concept of polarization.
7. Wave velocity
We have already discussed a fair amount the velocity of a wave. As mentioned above, this is a curious property of a wave – it is always in motion and always travels with the same speed, the wave velocity. The velocity of a wave depends on properties of the medium is a rather simple way:

We discussed the restoring force above – that is the force tending to return the medium to is equilibrium. The density of the medium is just what it sounds like. Waves in a thick liquid, like molasses, will travel more slowly than waves in a thin liquid, like water (assuming, of course, that the restoring force is the same). The units on restoring forces and densities are a bit complicated, but the combination of the square root of the ratio of the two does work out to be length/time, as it should.

So, the speed of sound is given by:
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Similarly, the speed of a wave on a string is given by:
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The restoring force for a string is the tension on the string. Looking back at Section 3.5, we can now see in a more general way where the expression for the velocity of a wave on a string came from.
8. Waves are not solid.
Perhaps the most unusual aspect of waves is that they can move through each other without affecting the other wave. This is very different from solid objects. If two solid objects try to pass through each other, either they bounce off each other, or one or both objects break up. If waves bounced off of each other, conversations would be very difficult! It would be possible to block a sound wave with another sound wave. This does not happen.
9. Interference.
Based on the previous point, it can happen that two waves can be at the same place at the same time. How do we describe this situation? Instead of displacing each other or bouncing off each other, the waves simply add together. This is called interference. Although this statement is easy to make, that two waves in the same space at the same time simply add together, actually calculating the effects of this can be quite tricky. A number of unusual things can happen because of this property of waves, and are generally called interference effects. We will explore several consequences of interference in the next few sections.