Guitar Waves and the Sound of Shape

The string is modeled by the one-dimensional wave equation \(u_{tt}=c^2u_{xx}\) with fixed endpoints. The natural eigenfunctions are sine waves, and their frequencies are integer multiples of the fundamental frequency.

Use Play Natural Frequency Demo to step through the natural frequencies. Pluck mode is the default: choose a pluck position, then press Play pluck. The spectrum below shows how the initial pluck is built from Fourier modes.

This graph uses the same pluck position and amplitude from Concept 1. The blue curve is a simplified incoming string signal and the green curve is the body-filtered response.

The paired spectrum bars show the main idea: for each harmonic, the body changes the coefficient from \(A_k(a)\) to \(H(k)A_k(a)\). Press Play pluck in Concept 1 to animate both panels together.

One first model treats the air in the opening like a moving mass and the air inside the body like a spring. This is the Helmholtz oscillator model.

Its basic prediction is \[ f_H=\frac{c}{2\pi}\sqrt{\frac{A}{V L_e}}, \] where \(V\) is the cavity volume, \(A\) is the opening area, and \(L_e\) is the effective opening length.

This says area and volume are key factors in the body response, but they are not the whole story. This visual is a simplified, under-construction model; we need more research to improve it and hope to add better sound later.

Just like a string has natural frequencies and modes, a three-dimensional shape has preferred standing-wave patterns. This visual uses a simplified rectangular guitar body so we can see the idea clearly.

Choose mode numbers \(l,m,n\) from 1 to 5. Then move the slice sliders to see cross-sections of the wave passing through the body. In this simplified model, each mode has frequency \(F=\frac{c}{2}\sqrt{(l/L_x)^2+(m/L_y)^2+(n/L_z)^2}\).