The Physics Philes, lesson 106: The Sound of Physics
We’ve just spent basically forever getting our hands dirty with mechanical waves. Now it’s time to get our hands even dirtier with a specific type of mechanical wave: the sound wave.
Sound waves are something most of us are familiar with. The general definition of sound is actually quite simple. Sound is just a longitudinal wave in a medium. Sound waves can travel though anything – gas, liquid, or solid – but for our purposes we’re going to mainly deal with sound waves through the air.
Like the other mechanical waves we’ve discussed, the simplest sound waves are sinusoidal. Sound waves are in the audible range if they are between 20 to 20,000 Hz. This is the range the human ear can recognize. However, the term “sound” is also used for longitudinal waves through a medium with frequencies above and below the audible range. These waves are called ultrasonic and infrasonic, respectively.
Sound waves usually travel in all directions from the source, but for simplicity we’re going to assume that we have an idealized sound wave that propagates along the positive x-axis. We know that such a wave can be described by a wave function that gives the instantaneous y-displacement of a particle in a medium at position x and time t. If the wave is sinusoidal, we know the wave function:
Remember that, unlike transverse waves on a string, the displacement of particles in a longitudinal waves is parallel to the direction the wave is traveling. That means that distances x and y are parallel to each other. The amplitude A is the maximum displacement of the particle from its equilibrium position; it’s also called displacement amplitude.
We can think about sound waves in terms of displacement, but it might be more useful to think about them in terms of a difference of pressure. It has a lot to do with how our ears work. Human ears sense pressure fluctuations. A sound wave enters the ear and exerts a fluctuating pressure on one side of the eardrum. The air on the other side is at atmospheric pressure and the difference in pressure on the two sides of the eardrum sets the eardrum in motion. Luckily, there is a relationship between the description of a wave in terms of pressure and the description of a wave in terms of displacement.
Let p(x,t) be the instantaneous pressure fluctuation in a sound wave at any point x at time t. The see how this is related to the displacement y(x,t), think about a cylinder filled with a medium with a cross sectional area S and an axis along the direction of propagation. With no sound present, let’s say the cylinder has a length of Δx. In that case, the volume would be SΔx. The ends of the cylinder are at x and x + Δx.
Now, when a wave is present, at time t the end of the cylinder initially at x has been displaced by y1 = y(x,t) and the other end has been displaced by y2 = (x + Δx, t). If y2 is greater than y1, the cylinder’s volume has increased, which will cause a decrease in pressure. If y2 is less than y1, the cylinder’s volume has decreased and the pressure has increased. If the two displacements are equal, then the cylinder was just shifted and there is no volume change and no fluctuation in pressure. The pressure fluctuation is dependent on the difference between the displacement of neighboring points in the medium.
We can represent the change in volume using some math:
If we take the limit as Δx goes to zero, the change in volume divided by the original volume (the fractional change in volume) is equal to the partial derivative of y(x,t) with respect to x. This fractional change in volume is related to the fluctuation of the bulk modulus B. (The bulk modulus, remember, is a substance’s resistance to compression.) The bulk modulus is defined as
If we solve for p(x,t), we get
We get a negative sign when the partial derivative factor is positive and the displacement is greater at x + Δx than at x, which corresponds to a decrease in pressure.
When we evaluate the partial derivative of the displacement wave function with respect to x for the sinusoidal wave equation, we get
The functions p(x,t) and y(x,t) describe the same wave, but they are a quarter out of phase. Note that both points of greatest pressure and density and points of lowest pressure and density are also point of zero displacement.
Note also that the quantity BkA represent the maximum pressure fluctuation. This is called the pressure amplitude. This pressure amplitude is directly proportional to the displacement amplitude A and the wavelength. Waves that have shorter wavelengths have greater pressure variations for a given amplitudes because the maximum and minimums are squeezed closer together. Media with a large bulk modulus requires a relative large pressure amplitude for a given displacement. (This is because a large value for bulk modulus means that the medium is resistant to compression.)
OK, that’s it for now. Next week we’ll talk about how sound is perceived.
Featured image credit: Travis Isaacs via Flickr