Florilegium Mathematica-Sonus (FloMaS)

The Wave Equation

The most elegant thought to come out of France—a country bursting with elegance—is the wave equation. This little pastry from the mind of d'Alembert describes a wide range of phenomena—including the propogation of sound and light. Its application could be considered encyclopdic (which is expected for the great encyclopedist d'Alembert).

It is interesting that many sound artists and musicians have never heard of it. Sound is fundamental to these arts so it peculiar that artists are not as aware of the math bedind sound's propagation as a wave.

The Equation

Here is the wave equation:

$u_{t t} = c^{2} u_{x x}$

It describes how a continum, $u (x, t)$ , which varies in space, changes according to these variations. It includes the paramter $c$ , which is the speed that a wave propogates in the medium.

To be hand-wavey, $u_{x x}$ decribes which way a medium bends (or fluctuates for air columns), and $u_{t t}$ describes which direction the medium will begin to move at each point in response to this bending (or the accelration at each point in reponse to the bending). $c$ therefore predicts a faster movement in response to bending or fluctuation.

To hear this equation, we further need to specify initial conditions and boundary conditions. We will decide that $u$ at $x = 0$ and $x = l$ (where $l$ is the length of the medium) is zero. This means that the medium is fixed at both ends. We will decide that the shape of the medium at $t = 0$ is a triangle.

The Sound of the Wave Equation

Here is what this sounds like:

We clearly have an oscillation, but the sound does not evolve. This is because the wave equation is conservative. In a real system, there are usually loses that cause the sound to decay; ff we pluck a string, for example, the sound will die away. Here we have nothing to model the losses seen in a real system in this equation, so the sound continues forever.

We can add losses by adding a term to the wave equation:

$u_{t t} = c^{2} u_{x x} - α u_{t}$

Here is what is sounds like:

Interestingly, however, this does not change anything about the sound other than its amplitude. What is missing from this equation is the nonlinearity of the system, the frequency dependence of the attenuation, the change in frequency response of the amplifying medium (i.e. the soundboard), et cetera.

The lesson for the sound artist, perhaps, is that elegance only gets you so far.

Postscript

You can find out more about how the equations were made into sound by learning about Finite Difference Methods. An excellent book that talks about this in the context of sound is Numerical Sound Synthesis by Stefan Bilbao. There are many difficulties in making these equations sound so that there is not room to discuss this here. The reader is advised to look into the Courant-Friedrichs-Lewy condition if they wish to attempt this experiment.

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