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 results in the frequency of the oscillation.

$\begin{equation*} \frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2} \end{equation*}$

With:

$\frac{\partial ^2y}{\partial t^2}$ : the second partial derivative with respect to time (the acceleration of the wave at a point).
$\frac{\partial ^2y}{\partial x^2}$ : the second partial derivative with respect to space (the curvature of the string at that point).
$c$ : the propagation speed of the wave, depending on the physical properties of the medium.

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. The displacement - $y(x,t)$ of the wave at position $x$ and time $t$ . can be expressed as:

$\begin{equation*} y(x,t) = y^+ (x-ct) + y^- (x+ct)$ \end{equation*}$

With:

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

Tuning the String

In an ideal string, the velocity $c$ depends on the tension $K$ and mass-density $\epsilon$ of the string:

$\begin{equation*} c^2 = \sqrt{\frac{K}{\epsilon}} = \sqrt{\frac{K}{\rho S}} \end{equation*}$

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

The frequency $f$ of the vibrating string depends on the velocity and the string length:

$\begin{equation*} f = \frac{c}{2 L} \end{equation*}$

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:

$\begin{equation*} y(m,n) = y^+ (m,n) + y^- (m,n) \end{equation*}$

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▼]

@article{bilbao2019physical,
    author = "Bilbao, Stefan and Desvages, Charlotte and Ducceschi, Michele and Hamilton, Brian and Harrison-Harsley, Reginald and Torin, Alberto and Webb, Craig",
    title = "Physical modeling, algorithms, and sound synthesis: The NESS project",
    journal = "Computer Music Journal",
    year = "2019",
    volume = "43",
    number = "2-3",
    pages = "15--30",
    publisher = "MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info\textasciitilde …"
}

2004

Chris Chafe. Case studies of physical models in music composition. In Proceedings of the 18th International Congress on Acoustics. 2004.
[details] [BibTeX▼]

@inproceedings{chafe2004case,
    author = "Chafe, Chris",
    title = "{Case studies of physical models in music composition}",
    booktitle = "{Proceedings of the 18th International Congress on Acoustics}",
    year = "2004"
}

1995

Vesa Välimäki. Discrete-time modeling of acoustic tubes using fractional delay filters. Helsinki University of Technology, 1995.
[details] [BibTeX▼]

@book{valimaki1995discrete,
    author = "Välimäki, Vesa",
    publisher = "Helsinki University of Technology",
    title = "{Discrete-time modeling of acoustic tubes using fractional delay filters}",
    year = "1995"
}

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▼]

@article{de1995physical,
    author = "de Bruin, Gijs and van Walstijn, Maarten",
    journal = "Journal of New Music Research",
    number = "2",
    pages = "148–163",
    publisher = "Taylor \\& Francis",
    title = "{Physical models of wind instruments: A generalized excitation coupled with a modular tube simulation platform*}",
    volume = "24",
    year = "1995"
}

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▼]

@inproceedings{Karjalainen1993towards,
    author = "Karjalainen, Matti and Välimäki, Vesa and Jánosy, Zoltán",
    booktitle = "{Computer Music Association}",
    pages = "56–63",
    title = "{Towards High-Quality Sound Synthesis of the Guitar and String Instruments}",
    year = "1993"
}

1992

Julius O Smith. Physical modeling using digital waveguides. Computer music journal, 16(4):74–91, 1992.
[details] [BibTeX▼]

@article{smith1992physical,
    author = "Smith, Julius O",
    journal = "Computer music journal",
    number = "4",
    pages = "74–91",
    publisher = "JSTOR",
    title = "{Physical modeling using digital waveguides}",
    volume = "16",
    year = "1992"
}

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▼]

@article{hiller1971synthesizing,
    author = "Hiller, Lejaren and Ruiz, Pierre",
    title = "Synthesizing musical sounds by solving the wave equation for vibrating objects: Part 1",
    journal = "Journal of the Audio Engineering Society",
    year = "1971",
    volume = "19",
    number = "6",
    pages = "462--470",
    publisher = "Audio Engineering Society"
}

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▼]

@article{hiller1971synthesizing2,
    author = "Hiller, Lejaren and Ruiz, Pierre",
    title = "Synthesizing musical sounds by solving the wave equation for vibrating objects: Part 2",
    journal = "Journal of the Audio Engineering Society",
    year = "1971",
    volume = "19",
    number = "7",
    pages = "542--551",
    publisher = "Audio Engineering Society"
}