# 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.

### References

• Vesa Välimäki. Discrete-time modeling of acoustic tubes using fractional delay filters. Helsinki University of Technology, 1995.
[BibTeX▼]
• Gijs de Bruin and Maarten van Walstijn. Physical models of wind instruments: A generalized excitation coupled with a modular tube simulation platform*. Journal of New Music Research, 24(2):148–163, 1995.
[BibTeX▼]
• Matti Karjalainen, Vesa Välimäki, and Zoltán Jánosy. Towards High-Quality Sound Synthesis of the Guitar and String Instruments. In Computer Music Association, 56–63. 1993.
[BibTeX▼]
• Julius O Smith. Physical modeling using digital waveguides. Computer music journal, 16(4):74–91, 1992.
[BibTeX▼]
• Lejaren Hiller and Pierre Ruiz. Synthesizing musical sounds by solving the wave equation for vibrating objects: part 1. Journal of the Audio Engineering Society, 19(6):462–470, 1971.
[BibTeX▼]
• Lejaren Hiller and Pierre Ruiz. Synthesizing musical sounds by solving the wave equation for vibrating objects: part 2. Journal of the Audio Engineering Society, 19(7):542–551, 1971.
[BibTeX▼]

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