The Physics Philes, lesson 105: Physics Makes Beautiful Music
Over these past several weeks, we’ve learned a lot about mechanical waves. However, what we’ve learned is a simple version of the waves we come in contact with every day. For example, the sound string instruments make are complex standing waves.
To illustrate this, let’s think about what would happen if a string vibrated at one frequency. If we displace a string so the shape it makes is the same as one of the normal modes, when we release it it would vibrate at the frequency of that mode. This vibration would cause the air to vibrate with the same frequency, which would produce a sinusoidal wave, which we will hear as a pure tone. However, in real life, that’s not generally what happens. When you pluck a guitar string, the shape of the displacement isn’t as simple. There will be the fundamental frequency as well as overtones, and the result will be a combination or superposition of the many normal modes.
The sound produced by this combination of vibrations is also the superposition of traveling sound waves. We hear this, not as a pure tone as before, but as a rich, complex tone with a certain fundamental frequency. The standing wave on a string and the traveling sound wave have a similar harmonic content. That is, the waves are similar in the extent to which the frequencies are higher than fundamental frequency are present. This harmonic content is determined by how the string is set in motion. For example, if you pluck the string of a guitar over the sound hole, it will produce a sound with different harmonic content than if you plucked it somewhere along the fretboard.
We’re not left without tools to analyze these waves, but the method is a little bit above my pay grade at this point. Suffice it to say that it is possible to represent every possible motion of the string as some superposition of normal mode motions. This is called harmonic analysis, and the sum of all the sinusoidal functions that represents a complex wave is called a Fourier series. In another universe, I might to explain this. But series are by far my calculus weak point so I’m just going to leave it there.
There is more physics of string instruments than that, though, as we can see from equations we’ve already talked about. We know that the fundamental frequency can be represented as f1= v/2L, or the wave speed divided by twice the length of the string. we also know that the wave speed is equal to the square root of the tension F divided by the mass per unit length μ. If we combine those equations, we get
This is also the fundamental frequency of the sound wave created by the vibrating string. What can this equation tell us about the sound we’ll hear from string instruments? Well, you know how a bass or cello has a deeper sound than a viola or violin? That’s the inverse relationship to the length L talking. The longer the string, the lower the frequency, the deeper the sound. Also, you can vary the pitch of an instrument like a violin or guitar by pressing your fingers down on the string along the fretboard at various places. This changes the length of the string that vibrates. We can also change the pitch by increasing or decreasing the tension of the string. If we increase the tension, we increase the pitch. I mean, adjusting the tension is how you tune string instruments, after all. And what about μ? Well, if you increase the mass per unit length, it will decrease the wave speed, thus decrease the frequency. That’s why strings for lower notes often have thicker strings.
Physics! It’s everywhere!
Welp, here we are. Done with mechanical waves. But don’t forget everything you’ve learned just yet. We’re now moving from the standing waves on the string of the violin to the traveling sound waves that hit our ears. Over the next several weeks we’re going to delve more deeply into sound waves and their properties. Don’t miss it!
Featured image credit: woodleywonderworks via Flickr