Motion Captured – Making Waves

November 1, 2014

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This time we are going to look at waves. The animation below lets you create some waves by clicking to create a disturbance and watching it propagate outwards, like rain drops in a pond. There are some options for you to play with. “Impulse” gives an approximation of an infinitely sharp infintely short pulse while “Gaussian” is a smoothed out initial disturbance. The slider lets you change the damping, $\eta$. For explanations read on, but the initial set up gives nice looking waves.

The wave equation was first written down and solved by d’Alembert during the course of his research on the problem of a vibrating string, a hot topic in the 18th century. In one dimension it looks like
\[ \frac{d^2u}{dt^2} = c^2 \frac{d^2u}{dx^2} \]
the quantity $u$ refers to the displacement of the string as a disturbance moves along it, $c$ is a constant that turns out to have the units of velocity. The equation has an infinite number of possible solutions, but most interesting are the so called normal modes. This is a smaller set of solutions (but still infinitely many!) from which any other solution can be built. They are solutions that oscillate with a fixed frequency.
\[ A e^{-i(kx + \omega t)} + B e^{i( k x – \omega t)} \]
Studying these normal modes could lead us to talk about Fourier analysis, which has applications in almost every field of science that involves mathematics; from making a jpeg to comparing DNA samples. As such there are a lot of good places to read about Fourier series, instead I will talk about something slightly more obscure: Green’s functions.

George Green himself is an unusual story, see wikipedia for a full biography. He was largely self taught and from ‘the provinces’ so his first and most famous publication “An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism” was initially read by a subscription list of 51 people, most of whom almost certainly did not understand it. Later in life he did manage to get accepted as a Cambridge fellow and read by some people who could understand him before he finally “succumbed to alcohol”.

The idea of a Green’s function (or a Green function, the naming convention is controversial cf. Bessel functions) is based on calculating the response to an impulse. An impulse is an infinitely brief but infinitely strong pulse, such that the area under the force curve is finite. It’s an idealization of hitting something with a hammer or the sound of a starting pistol. For a certain type of system (linear, time invariant) the response to an impulse characterises the response of the system to any input. By weighting the solutions for different impulses and summing we can construct the solution for any type in input or source. The Green’s function gives the weights.

The Green’s function $G(y,z)$ is the response at $y$ due to an impulse at $z$. $z$ is the source and $y$ is the detector. For example if we have a system described by a wave equation (say the surface of a pond), we hit it with an impulse at $z$ (drop a rock in) the displacement of a lily pad located at $y$ is given by $G(y,z)$.

Mathematically an impulse is easy to express using the Dirac delta function. An impulse at position $z$ is represented by
\[ \delta(x – z) \]
this function is zero everywhere except for the point $z$, where it diverges. However it diverges in such a way that
\[ \int \delta(x – z) = 1.\]
Just the property we want an impulse to have. We then write the equation to solve in the form
\[ L(x,p) u(p) = 0 \]
which describes the situation without any sources of disturbance. $L$ is a linear differential operator. It could be $\frac{d^2}{dx^2} – c^2 \frac{d^2}{dt^2}$ as in the wave equation or in two dimensions $\frac{d^2}{dx^2} + \frac{d^2}{dy^2} – c^2 \frac{d^2}{dt^2}$ . The arguments of $L$ are a little sloppy, take the form below as a kind of schematic, with the bonus that if we discretise and map function arguments to indices the equations are all correct.

With a delta function as a source we have,
\[L(x,y) G(y,z) = \delta(x – z)\]
$G(y,z)$ is the Green’s function of the operator $L(x,y)$. In a discrete system, like the simulation above, $L$ is a matrix and $G$ is its inverse. Now a cute trick using the magical properties of a delta function lets us take any ‘source’ function $f(z)$ and see that
\[ \int L(x,y) G(y,z) f(z) dz = \int \delta(x – z) f(z) dz = f(x) \]
which implies
\[ L(x,y) \left( \int G(y,z) f(z) dz \right) = f(x). \]
The thing in brackets gives the solution with the source $f(x)$. Integrals are simple to do, at least numericaly, so given $G$ we can solve the system for any source. This is used in audio signal processing where there is a technique called convolution reverb. A room is ‘excited’ by an impulse (a balloon popping) and the reverberations are measured by a microphone. The measurement gives, roughly, $G(y,z)$ where $z$ is the balloon pop location and $y$ is the mic location. Then the response of the room to other, nicer sounds can be modelled. Read more about it here.

The simulation above lets you play with impulses (using discrete deltas rather than continuous ones) and also smoother sources, gaussian disturbances in the ‘membrane’ that are allowed to evolve. The solutions could be obtained by convoluting the initial gaussian source with the solution due to an impulse, but it is easier numerically to just solve the equation. Real water drops are much more complicated, the process of dropping water into water proceeds in several stages. Speaking of drops, watching your tap drip is a cheap way to simulate nuclear fission. The liquid drop model of the nucleus predicts that atoms split in much the same way as a drop falls. A long ‘neck’ develops in the nuclear fluid which distorts into a dumbbell shape. If the neck stretches far enough the nucleus can split. The liquid drop model even predicts that some small droplets (i.e. neutrons) fly away from the neck after the split, much like the small droplets you can see when your tap drip finally breaks.

Our simulation is not as complicated as a dripping tap but it is easier to stop.