Elementary Digital Waveguide Models for Vibrating Strings

Julius Smith and Nelson Lee

RealSimple Project¹
Center for Computer Research in Music and Acoustics (CCRMA)
Department of Music, Stanford University
Stanford, California 94305

Detailed Contents (and Navigation)

• Ideal Vibrating String
  • String Wave Equation Derivation
  
  
• Traveling-Wave Solution
  • Infinitely long string plucked simultaneously at three points marked `p'
  
  
• Sampled Traveling Waves in a String
• Sampled Traveling Waves in any Digital Waveguide
  • Lossless digital waveguide with observation points at $x=0$ and $x=3X=3cT$
  
  
• Other Wave Variables
• Rigidly Terminated Ideal String
• Moving Termination: Ideal String
  • Waveguide ``Equivalent Circuits'' for the Uniformly Moving Rigid String Termination
  
  
• Doubly Terminated Ideal Plucked String
  • Digital Waveguide Plucked-String Model Using Initial Conditions
  • Acceleration-Wave Simulation
  • Ideal Struck-String Velocity-Wave Simulation
  • External String Excitation at a Point
  • Delay Consolidated System (Repeated):
  • Algebraic Derivation
  
  
• Damped Plucked String
  • Equivalent System: Gain Elements Commuted
  • Computational Savings
  
  
• Frequency-Dependent Damping
  • Simplest Frequency-Dependent Loop Filter
  • Next Simplest Case: Length 3 FIR Loop Filter
  • Length 3 FIR Loop Filter with Variable DC Gain
  • Karplus-Strong Algorithm
  • Interpretations of the Karplus-Strong Algorithm
  • KS Physical Interpretation
  • KS Sound Examples
  
  
• Extended Karplus-Strong (EKS) Algorithm
  • EKS Sound Example
  
  
• Loop Filter Identification
  • Plucked/Struck String Pitch Estimation
  
  
• Nonlinear ``Overdrive''
  • Hard Clipper
  • Soft Clipper
  • Amplifier Distortion + Amplifier Feedback
  
  
• About this document ...