Standing Waves and Resonance
Standing Waves
We have discussed the fact that musical instruments absorb sound at their resonant frequencies, and have also discussed the fact that the sound is stored in a pressure wave that oscillates above and below ambient pressure within the instrument, while little bits of energy leak out and lead to the sound of the instrument. Now we will go into more detail about those internal waves.
It turns out the waves that are inside the instrument's air cavity (or on the strings or sound board) manage to add constructively to one another in an interesting phenomenon called standing waves. We call them that, but they are really made up of two waves traveling in opposite directions at the same time. Take a look at the graphic below. When you push the play icon you'll see the red and orange traveling waves give the effect of a wave that stands still in purple.
Requirements of Standing Waves
Why do we get standing waves inside instruments? In other words, why are waves traveling both ways, and what makes them persist? Well we know that sound waves echo off walls. The cavity of a musical instrument is a great place for sound to echo or bounce around inside. So it's easy to see that we will have waves traveling both in all sorts of directions at once, but as they continue to bounce again and again, over and over they must somehow not cancel out. Unless there is a very special relationship between the wavelength of the wave and the size of the cavity, the relative phases of the waves would cause them to cancel one another out more often than not.
So what conditions are required? Let's discuss this as a linear problem that is analogous to a wave on a guitar string. Suppose we start with a single sine wave traveling to the right, and imagine it's reflected off a wall. Now you have two sine waves meeting - the incoming wave meeting the outgoing one. That meeting will always be constructive since the wave will not flip when bouncing off a solid in which sound travels faster than in air. Next, that reflected wave will reflect again and meet back up with the original wave that started it all. This is animated below. The red wave starts it all, leads to the blue reflected wave and then the blue reflected wave leads to the purple reflection of itself - a doubly reflected wave. Notice that if the purple reflection of a reflection is out of phase with the original red wave, that typically there would be an addition of waves that may cancel much or all of the intensity of the original wave. This wavelength would not persist in an instrument cavity for more than a moment. You can move the sliders so that the purple wave becomes in phase with the red one. That wave would bounce back and forth amplifying itself each time. That is how resonance leads to standing waves inside the cavity of a guitar, for instance.
Phasors and Standing Waves
In order for the wave to persist in a cavity or along a string, we need the doubly reflected wave (purple) to be in phase with the original (red) wave. The reason for this requirement may not be obvious at first. After all, even when not perfectly in phase, it looks like the waves could add up anyway to something larger, and that seems good. But that line of reasoning ignores the fact that the purple wave will also generate a reflection and that its reflected wave will reflect. This process would go on and on. Ultimately we'd end up with lots of waves all traveling together and all slightly phase shifted with respect to one another. While it may not seem obvious when looking at a graph like the one below, it actually causes the waves to cancel one another when that happens. So with nearly never-ending reflections meeting one another, we need them all to be in phase with one another in order for anything to survive.
This will only occur for certain sizes of cavity. The relationship is a simple one. The doubly reflected wave will travel an extra distance of twice the cavity length before meeting back up with the original wave. This means in order for the path length difference to lead to constructive interference. Plugging in and simplifying leads to . This equation relates the cavity size L with the wavelength of the sound that will naturally resonate in the cavity. Notice that by fixing L and solving for , we can find all the wavelengths of sound that would resonate in a given cavity, or on a string of length L.