# 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 \(c\).

\(t = \ nT\)

\(x = \ mX\)

\(X = cT\)

### References

#### 2019

- Stefan Bilbao, Charlotte Desvages, Michele Ducceschi, Brian Hamilton, Reginald Harrison-Harsley, Alberto Torin, and Craig Webb.
**Physical modeling, algorithms, and sound synthesis: the ness project.***Computer Music Journal*, 43(2-3):15–30, 2019.

[details] [BibTeX▼]

#### 2004

- Chris Chafe.
**Case studies of physical models in music composition.**In*Proceedings of the 18th International Congress on Acoustics*. 2004.

[details] [BibTeX▼]

#### 1995

- Vesa Välimäki.
*Discrete-time modeling of acoustic tubes using fractional delay filters*. Helsinki University of Technology, 1995.

[details] [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.

[details] [BibTeX▼]

#### 1993

- 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.

[details] [BibTeX▼]

#### 1992

- Julius O Smith.
**Physical modeling using digital waveguides.***Computer music journal*, 16(4):74–91, 1992.

[details] [BibTeX▼]

#### 1971

- 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.

[details] [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.

[details] [BibTeX▼]