In this video, we’re going to learn about the propagation of sound waves. We’ll see how sound waves move and what factors affect their motion through different materials.
To start out, imagine that you wanted to get a message to your friend as quickly as possible. One option of course is to say the message out loud and let the sound transmit through the air between you. Another option is to speak into the open end of an empty tin can which is connected to a cord which on the other end is tied to a can up to your friend’s ear. To maximize the speed at which your message travels to your friend, which way would you choose? To understand the answer, we’ll want to know something about the propagation of sound waves.
Imagine you are standing outside on a pleasant spring afternoon. All around you, present but invisible to the eye are air molecules. Based on their temperature, these molecules move around at high speeds and collide with one another, roughly a billion times every second. It’s these molecules as disordered and high-speed as their motion is that allow the transmission of sound from your vocal cords out your mouth and into the air to a listener’s ear. When we open our mouth in speech, based on the direction of our mouth, we create a pathway that we transmit energy into.
As a longitudinal wave, the sound energy that we release has an effect on the air molecules in its path as the wave is passing through those molecules. The air molecules in this path continue to move around as they were before, bouncing off one another at high speed. Yet while this wave is passing through the molecules, they’re temporarily influenced to alternately bunch together or spread out. That is, while the sound wave is passing through this swathe of air, the air molecules begin to take on the shape of the wave. The areas where air molecules begin to bunch together are called compressions. These are like peaks in the wave. And the places where molecules are few and far between are called rarefactions. In those places, the air molecules are rare while the wave is passing through. Compressions and rarefactions are the peaks and troughs of this wave, respectively.
We’ve seen that if we have a transverse wave, one whose peaks and troughs are perpendicular to the direction of the wave’s motion, then one wavelength for that wave is equal to the distance from one peak to another one or one trough to the one next to it. Looking at our longitudinal wave then, we can say that the distance in space between adjacent compression points or maxima is equal to the wavelength 𝜆 of this sound wave. Another property of this sound wave we might be interested to learn about is its speed. Just how fast does the sound travel through this air?
One of those factors is referred to as capital 𝐵. It stands for bulk modulus. This is a measure of how hard it is to compress a material. It could be a gas or a liquid or a solid. As an example, if we consider air at standard temperature and pressure, we can see that that air isn’t too difficult to compress. Every time we inflate our bike tires with a hand pump, we can press atmospheric air. To consider how this factor, bulk modulus, affects the speed of a sound wave. Let’s consider examples of a gas, a liquid, and a solid, side by side by side.
If we were to compare the molecular spacing of a gas and a liquid and a solid, we would see that as we go from gas to solid, that spacing gets smaller and smaller. As a material becomes less compressible, like our solid is less compressible than a liquid or the gas, its bulk modulus 𝐵 goes up. Now let’s consider what it would be like for a sound wave to travel through these three different materials. Since sound waves require a physical medium to pass through, that means these waves rely on the interactions of individual molecules within a material. The more interactions, the more easily the wave will travel through.
Now which type of material would have more intramolecular interactions, one with lots of space between the molecules as in our gas or one where the molecules are very tightly packed as in our solid? When molecules in a material are very close together, the signal transmits that much more efficiently and therefore quickly in its direction of motion. Considered from this perspective then, we would expect that our wave speed of our sound wave would go up the higher the bulk modulus we have and the more dense our material. In naming material density 𝜌, we’ve actually named that the second parameter that affects sound wave speed. But density affects the speed of a sound wave in a way we might not expect.
We know that in general solids are more dense than liquids which are more dense than gases. But consider this: when a sound wave moves through a material, it moves the elements or the molecules of that material forward and backward while the wave is passing through. Remember how when we speak into the air, we push air molecules forward to bunch them together and then as the wave passes they slide on back. So for the wave to pass through, the molecules need to move temporarily. But this is hard to do in a material that’s very dense. This material has lots of inertia, which resists motion. So higher density actually works against wave speed. The higher the density of a material, the more it resists moving in the motion of the wave that passes through it. This means that our two parameters affecting wave speed, bulk modulus and density, work against one another. So we wonder which effect wins out.
If a material such as a solid has both a high bulk modulus and a high density, which one has a greater effect on the speed of sound through it? It turns out it’s the bulk modulus, a material’s compressibility, that is the driving or dominant factor affecting wave speed. In other words, as we go from a gas to a liquid to a solid, the gains in density which result in greater inertia in the material and therefore slower transmission of sound are outweighed by the ease with which a sound wave moves through a more interconnected lattice.
We can summarize all this discussion with a relationship that says the speed of sound through a material equals the square root of its bulk modulus divided by its density. Let’s consider examples of what this mathematical relationship means for the speed of sound in solids, liquids, and gases.
If we were to enter the values of bulk modulus and air density at standard temperature and pressure, we would find that the speed of sound in air is about 340 meters per second. Air has a relatively low bulk modulus and a low density. Now, imagine we enter the values for bulk modulus and density for water. We find that in water sound travels roughly four times as fast as it does in air. Then going further, what if we enter the bulk modulus and density of a solid material, such as brick? The speed of sound through a solid like this is about 3650 meters per second.
When we see these three different speeds of sound arrange like this, that may raise a question in our minds. What if once more we were standing outside and we create a sound wave with our vocal chords? And say we’re able to create a wave at a very specific frequency. Let’s say it’s a 200-hertz-frequency wave. According to the results that we’ve just calculated, that means that if this wave traveled from air into water, its speed would change. And then, if it travelled from water into brick, its speed would change again. As strange as that may seem for one sound wave to change speed as it enters different media, that’s in fact what happens.
We can recall that the speed of a wave 𝑣, in this case a sound wave, is equal to the wave’s frequency times its wavelength. This means that as our wave moves from traveling through air to travelling through water—that is, as its speed goes up—something on the right-hand side of this equation needs to change so the equation is still correct. What we find when we look at situations like this experimentally is that wave frequency 𝑓 stays the same regardless of what medium it passes into and it’s the wavelength 𝜆 that changes. As the speed of our sound wave increases as we transition from air to water and then from water into brick, the wavelength of our wave gets longer and longer to keep pace with that increase in speed. All the while, our wave has a steady frequency of its original 200 hertz.
We can show how this works mathematically. Rearranging our equation for wave speed, we see that the wavelength is equal to wave speed over frequency. In the case of air, the wavelength in air of our sound wave is equal to the speed of the wave in air divided by 𝑓, which is about 1.7 meters. When we go to calculate the wavelength in water of our sound wave, we use the speed of our wave in water divided by that original frequency. And we find a result of about seven and a half meters. And then calculating the wavelength of our wave when it moves through brick, we find a result of about 18 meters. As our wave propagates along then, starting from our mouth and moving through the air into the water and then into the brick, we can represent its wavelength by crests spaced further and further apart.
Interestingly then, when sound waves move from one material to another, their frequency is constant, but their wavelength and wave speed change. And this change in wave speed we noted has to do with the two material properties of bulk modulus and density. Let’s summarize what we’ve learnt so far about the propagation of sound waves.
We’ve seen that sound waves are longitudinal waves that require a material to travel through in order to move. We’ve also seen that the speed of sound in a material depends on the material’s compressibility, called its bulk modulus, as well as its density. Written as an equation, the speed of sound in a material is equal to the square root of the material’s bulk modulus divided by its density. And finally, we saw that when sound waves cross into a new material, wave frequency stays the same while wavelength and speed change.