# Digital Waveguides: Discrete Wave Equation

## Wave Equation for Ideal Strings

The ideal string results in an oscillation without losses. The differential wave-equation for this process is defined as follows. The velocity \(c\) determines the propagation speed of the wave and this the frequency of the oscillation.

A solution for the different equation without losses is given by d'Alembert (1746). The oscillation is composed of two waves - one left-traveling and one right traveling component.

- \(y^+\) = left traveling wave
- \(y^-\) = right traveling wave

## Tuning the String

The velocity \(c\) depends on tension \(K\) and mass-density \(\epsilon\) of the string:

With tension \(K\), cross sectional area \(S\) and density \(\rho\) in \({\frac{g}{cm^3}}\).

Frequency \(f\) of the vibrating string depends on the velocity and the string length:

## Make it Discrete

For an implementation in digital systems, both time and space have to be discretized. This is the discrete version of the above introduced solution:

For the time, this discretization is bound to the sampling frequency \(f_s\). Spatial sample distance \(X\) depends on sampling-rate \(f_s = \frac{1}{T}\) and velocity \(c\).

- \(t = \ nT\)
- \(x = \ mX\)
- \(X = cT\)

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