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.

/images/Sound_Synthesis/physical_modeling/schematic_3.png

Losses

Losses can be implemented by inserting filters between the delay lines.

/images/Sound_Synthesis/physical_modeling/schematic_1.png

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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