The Physics Philes, lesson 110: Ring My Decibel
OK, friends. Let’s talk about wave intensity.
Wave intensity (denoted with an I) is the average rate of time energy is transported per unit area across a surface that is perpendicular to the direction of propagation. So, if we’re talking about a transverse wave on a string, the wave might be propagating right to left. That means that the relevant surface for calculating wave propagation is the surface that goes up and down. We can express the wave intensity by using concepts we already understand: displacement amplitude and pressure amplitude.
Let’s start with displacement amplitude. We have a sound wave moving in the positive x-direction. We know that the power of a sound wave is the product of the force and velocity. That means that the power per unit area is the product of the force per unit area and the particle velocity. In math, it looks something like this:
Remember that particle velocity is the velocity at a particular time t of a portion of the wave medium at a particular coordinate x. It’s important to keep this concept distinct from wave velocity. A wave continues to move through a medium, but the individual particles just slosh up and down or side to side a bit. The particles don’t move along with the wave. Also, the max wave speed and the max particle speed can be very different. So don’t mix them up!
The wave intensity is the time average of the above equation. So, for any x value the average of the sin^2 function over one period T = 2π/ω is a half. If we weave that into the product of the force per unit area and the particle velocity, we get:
We can now use a couple of relationships – ω = vk and v^2 = B/ρ – to further transform the equation:
This is the equation for intensity of a sinusoidal sound wave in terms of displacement amplitude!
But there is another, more useful way to express wave intensity. This time, we’ll use pressure amplitude. Finding this equation is comparatively easy. All we need is the equation for pressure amplitude and ω = vk.
If you’ll remember, the pressure amplitude is
Using this equation and ω = vk, we can rewrite the wave intensity equation:
Now we can use the wave speed relationship v^2 = B/ρ to further rewrite the equation:
This is the intensity of a sinusoidal sound wave in terms of pressure amplitude!
If sound waves are emitted in all directions equally we need to deal with something called the inverse square law. The intensity of the sound wave is proportional to 1/r^2, where r is the distance from the source of the sound wave. However, if the sound goes mostly in one direction, the inverse square law doesn’t apply. In that case, the intensity of the sound wave will decrease more slowly. In addition, the inverse square law doesn’t apply indoors because the sound will bounce off the walls and ceiling.
Finally, let’s talk briefly about the decibel scale. The decibel scale is a logarithmic intensity scale. The sound intensity level β is defined as
Sound intensity levels are expressed in decibels, which is one-tenth of a bel. We can use sound level meters to measure the sound intensity level, because the human ear isn’t equally sensitive to all frequencies, some meters weight the frequencies unequally. This is called the dBA scale, and it does not emphasize very low and very high frequencies because we aren’t as sensitive to those wavelengths.
Welp, we made it through sound intensity! I’m proud of us. Next week we’ll take what we know about normal modes and apply it sound waves. It’ll be fun!
Featured image credit: flattop341 via Flickr