# Physical Modeling: Waveguides

## Wave Equation for Virtual Strings

The wave-equation for the one dimensional ideal string:

$\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2}$

Solution without losses (d'Alembert):

$y(x,t) = y^+ (x-ct) + y^- (x+ct)$

• $y^+$ = left traveling wave
• $y^-$ = right traveling wave

## Tuning the String

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

$c^2 = \sqrt{\frac{K}{\epsilon}} = \sqrt{\frac{K}{\rho S}}$

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:

$f = \frac{c}{2 L}$

## Make it Discrete

$y(m,n) = y^+ (m,n) + y^- (m,n)$

$t = \ nT$

$x = \ mX$

Spatial sample distance $X$ depends on sampling-rate $f_s = \frac{1}{T}$ and velocity $c$:

$X = cT$

An ideal, lossless string is represented by two delay lines with direct coupling. ### Losses

Losses can be implemented by inserting filters between the delay lines. Contents © Henrik von Coler 2020 - Contact