Projects@anabrid · work in progress
The wave equation on strings
A plucked string is the simplest system that carries a wave, and the only one of these problems with an exact solution to be wrong against. That makes it the machine's report card rather than its demonstration.
Every other problem in this section is hard because nobody knows the answer. This one is useful for the opposite reason. A string clamped at both ends and released from rest has an exact solution, written down in the nineteenth century. A machine claiming to solve it can therefore be checked against the truth, rather than against another solver.
The equation
A displacement $u(x,t)$ on a string under tension obeys
with $c$ the wave speed set by tension and mass per unit length. Clamp both ends,
pull the string into a corner and let go, and the motion is a sum of standing modes whose frequencies are exact integer multiples of $c/2L$. That integer relationship is why a string sounds like a note rather than a noise, and it is the thing a discretised solver gets wrong first.
What discretisation does to it
Cut the string into $N$ nodes a distance $h$ apart and replace the spatial derivative with its three-point difference. Each node becomes a second-order oscillator driven by its two neighbours,
On the machine that is a pair of integrators per node, one carrying velocity and one carrying position. They are wired to their neighbours through a tridiagonal stencil. No multipliers: the wave equation is linear, so the whole string is integrators and coefficients.
The discretisation exacts a price, and it is a specific and predictable one. The modes of the chain are no longer exact integer multiples of the fundamental. The higher the mode, the further it falls below where it should be. This is numerical dispersion, and it is not a bug in the solver. It is what replacing a derivative by a difference does, on any machine.
That makes dispersion the right thing to measure. It is a known, calculable disagreement between the chain and the continuum, so anything on top of it belongs to the hardware. A string is a spectrometer for a solver's errors.
Why put it on an analog computer
The chain is exactly the shape the machine likes: linear, local, and built from integrators. It also has the property this section keeps returning to, which is a quantity that ought to be conserved. An undamped string cannot lose energy. If the circuit's string decays, the circuit is wrong, and by how much is a number.
And a string has one output no other problem here has. It can be played. The node traces are a pressure signal, so the machine's answer can be listened to next to the analytic one.
We have run this. A plucked string on the hardware, measured against both the discrete and the continuous analytic solutions, with the dispersion relation, the spectrum and the error maps taken, and the result rendered as sound. What came out is behind the password.