# Digital Waveguides: Ideal String without Losses

## Ideal String with two Delay Lines¶

Based on the previously introduced discrete solution for the wave equation, an ideal, lossless string can be implemented using two delay lines with direct coupling. The left-traveling and right-traveling wave are connected end-to-end. Output samples of each delay line are direcly inserted to the input of the counterpart:

## The Excitation Function¶

The waveguides can be initiated - or excited - with any arbitrary function. This may vary, depending on the excitation principle of the instrument to be modeled. For a plucked string, the excitation can be a triangular function with a maximum at the plucking point $p$. Both waveguides (left/right travelling) are initiated with the same function:

$$y[i] = \begin{cases} \frac{i}{p} & \mbox{for } i \leq p \\ 1-\frac{i-p}{N-p} & \mbox{for } i > p \\ \end{cases}$$` `

## Oscillation¶

When both waveguides are shifted by one sample each $\frac{1}{f_s}$ seconds, the ideal string is oscillating with a frequency of $f_0 = \frac{f_s}{N}$. It will oscillate continuously. In fact it is a superimposition of two oscillators with the waveform defined by the excitation function.