A visual, interactive companion to Julius O. Smith III's online book Physical Audio Signal Processing — the wave equation, digital waveguides, Karplus–Strong, reed and bow nonlinearities, FDTD, modal synthesis, commuted piano synthesis, and waveguide reverb.
Every musical instrument is a machine for converting small, slow gestures — a plucked finger, a bowed arm, a breath — into radiated air pressure variations. The machine exploits waves. A string shakes transverse waves along its length; a clarinet column pumps longitudinal waves through compressible air; a piano soundboard radiates plate modes into the room. Julius O. Smith's Physical Audio Signal Processing (PASP) is the canonical modern account of how those waves are modelled, discretised, and made to sing inside a real-time DSP kernel.
This companion is a browser-first visual walk through PASP's core ideas. The text is the authority — we link to it throughout. What the browser adds is interactivity: you pluck the string, you change the impedance, you blow harder on the reed and hear what happens. Every canvas here is computing real physics in real time; every audio button synthesises through the Web Audio API, not a pre-recorded sample.
Physical modelling rests on three convictions that are not shared by either sample playback or additive synthesis:
We begin with a short history, then derive the 1-D wave equation and D'Alembert's solution that makes digital waveguides possible. Chapters 4–6 build waveguides, Karplus–Strong, and wave-variable scattering. Chapters 7–8 add the two classical nonlinear excitations — the bow and the reed. Chapter 9 contrasts waveguides with FDTD on a 2-D membrane. Chapter 10 covers modal synthesis. Chapter 11 assembles the pieces into a small piano via commuted synthesis. Chapter 12 closes the acoustic loop with waveguide reverb, and Chapter 13 surveys the legacy.
Physical audio modelling has a 160-year lineage. It starts with Helmholtz peering down a vibration microscope at a violin string, passes through mid-century analogue analog-circuit emulation, crystallises in the early 1980s at Stanford, becomes a commercial product inside the Yamaha VL1, and arrives, in 2020, as a differentiable neural layer inside DDSP.
Almost every physical-audio model in this guide begins with the same partial differential equation. It describes transverse motion on an ideal flexible string and, with minor relabelling, longitudinal pressure in a thin tube. Writing it out and solving it by D'Alembert's method is the single most important step in the book — everything that follows is a finite, damped, nonlinear version of this starting point.
Take an ideal uniform string of linear mass density $\rho$ under tension $K$. Isolate an infinitesimal segment at position $x$ with length $\Delta x$ and transverse displacement $y(x,t)$. For small slopes the vertical component of tension at one end is $K\,\partial y/\partial x$; at the other end it is $K\,\partial y/\partial x + K\,\partial^2 y/\partial x^2\,\Delta x$. Newton's second law for the segment gives
Dividing by $\rho\,\Delta x$ and writing $c^2 = K/\rho$ yields the canonical form:
The constant $c$ is the transverse wave speed. For a bronze-wound steel violin G-string ($\rho \approx 10^{-3}$ kg/m, $K \approx 45$ N) one finds $c \approx 210$ m/s; at the string's fundamental the wavelength is twice its length.
The second-order equation factors:
Each factor is a first-order advection equation. The general solution, due to D'Alembert (1747), is the sum of an arbitrary right-going wave and an arbitrary left-going wave:
This is the idea on which every digital waveguide in this companion rests. The string's state at any point is the sum of two independent travelling waves; to simulate it, we only need to translate two 1-D arrays along a line and handle what happens at the two ends.
Physical strings are clamped at one or both ends, and the boundary condition decides the sign of the reflection:
A fixed end inverts the incoming wave; a free end reflects it unchanged. This single fact determines the harmonic series of every stringed instrument: fixed–fixed yields a full harmonic series $nf_1$; fixed–free yields only odd harmonics $(2n-1)f_1$.
Drop a Gaussian lump on the string and watch it split, bounce, and recombine. Toggle the right-hand boundary between fixed and free to hear the series change.
A digital waveguide is the discretisation of D'Alembert's solution. Sample the travelling waves $y^{+}$ and $y^{-}$ at rate $f_s$; use a spatial step $\Delta x = c/f_s$ so that one sample of time exactly equals one step of space; and the wave equation becomes two shift registers. That is the whole idea.
Discretise $y^{+}(x-ct)$ as $y^{+}[n,m] = y^{+}(m\Delta x, n\Delta t)$. Because right-going waves only translate to the right, we have
That is precisely a unit delay. Similarly $y^{-}[n+1, m-1] = y^{-}[n, m]$ is a backwards unit delay. A string of length $L = M\Delta x$ is therefore a pair of length-$M$ delay lines, wrapped back to each other at both ends by reflection filters.
In a real string, each sample of travel accrues a little frequency-dependent loss (air damping, internal friction) and a little dispersion (from bending stiffness). A waveguide hides this cost by commuting the loss from every sample into a single filter at the reflection point. As long as the filter's total loss per round trip matches the physical loss per round trip, the ear cannot tell the difference.
The simplest choice is the one-zero lowpass Karplus and Strong stumbled onto: $H(z) = (1+z^{-1})/2$. Better models use a one-pole or biquad lowpass so the decay time can be tuned per-frequency.
Displacement at spatial sample $m$ is read as $y[n,m] = y^{+}[n,m] + y^{-}[n,m]$. For perceptual purposes one typically reads from one fixed point on the string (the bridge); the radiated signal at the ear is this bridge velocity convolved with the soundboard response.
Two delay lines, one reflection lowpass, and a pick-position comb. Pluck the string at a chosen point and listen. The two slider knobs map to the physical attributes you hear: pitch (delay length), body (decay time), and tone (reflection-filter brightness).
Karplus–Strong is the simplest waveguide in the world. A single delay line of length $N$, a one-zero averaging filter, and a noise burst as excitation. Two lines of C code. The sound is a startlingly convincing plucked string — vaguely steel, vaguely nylon, and, if you squint, vaguely koto. It was discovered empirically at Stanford in 1978 by Alex Strong on a Samson Box, then mathematised by Kevin Karplus, and published in 1983 in Computer Music Journal. In the same issue, David Jaffe and Julius Smith extended it with pitch-accurate fractional-delay tuning, a pick-position comb, and a dynamic-level filter — effectively the first extended plucked-string waveguide.
The KS loop is a single difference equation. Let $y[n]$ be the delay-line output:
The mean-of-two filter inside the loop has transfer function $H(z) = \tfrac{1}{2}(1+z^{-1})$, a mild lowpass with a zero at Nyquist. It both damps the sound at higher frequencies (realistic) and shifts the loop's phase (introducing a small tuning error that Jaffe & Smith fix with an all-pass).
The classical KS excitation is a burst of white noise: the pluck is modelled as "all spectral content present simultaneously, then the physics removes what isn't allowed." A short impulse is also useful — it gives a cleaner attack at the cost of some perceptual "pluckiness." The pick-position comb filter $1-z^{-p}$ emphasises the idea that a pluck at fractional position $p/N$ along the string is silent at modes whose node sits at that position.
What happens when a wave arrives at the join between two strings of different mass densities, or between a string and a bridge? Part of the wave continues (transmission) and part of it comes back (reflection). The split is controlled by the wave impedance of each segment.
For a flexible string the wave impedance is
with units of force-per-velocity. Denser strings under the same tension have higher impedance. Strings of higher tension at the same density also have higher impedance.
At a lossless junction between two media of impedance $Z_1$ and $Z_2$, force continuity and velocity continuity give the Fresnel-style scattering formulas:
A clamped end is the limit $Z_2 \to \infty$, giving $r = 1$ for force but $r = -1$ for velocity — hence the velocity-wave inversion we used in Chapter 3. A free end is the opposite limit $Z_2 \to 0$.
When three or more waveguides meet at a point — a string splitting into a bridge and a nut, or a tube branching into two tone-hole sections — Kelly & Lochbaum's scattering matrix (originally developed in 1962 for speech-tract models) generalises the two-port formula:
where $Y_i = 1/Z_i$ is the port admittance and $p^{+}_i$ is the wave arriving at the junction from port $i$.
A lump travels from left to right through a junction where the impedance ratio $Z_2/Z_1$ is adjustable. Watch the transmitted and reflected components in real time and read the coefficients.
Add a nonlinear friction table to the string at the bow contact point and you have a bowed string. The table is the full physics: the horsehair sticks to the string when their relative velocity is small, and slips over it when the relative velocity is large. The transition is sharp, so the Helmholtz corner that McIntyre & Woodhouse made famous travels back and forth along the string between the nut and the bridge.
Let $v_{\mathrm{rel}}$ be the relative velocity between bow and string and $F_b$ the bow force normal to the string. A commonly used model (Smith, after McIntyre & Woodhouse) writes
Here $\mu_s$ is the static coefficient (stick), $\mu_d$ is the dynamic coefficient (slip), and $v_0$ sets the transition width. The table is tabulated once and looked up every sample.
Helmholtz's 1862 observation: a single "corner" in the string velocity waveform travels back and forth along the string at speed $c$, reflecting off both ends. The string alternates between a stick phase (bow drags the string with it) and a slip phase (corner passes; bow cannot catch up). To first order, the duty cycle of stick-vs-slip only depends on bow position; the period depends only on string length. All other Raman modes — multi-corner, double-slip, etc. — are unstable unless bow force is raised or lowered out of the Helmholtz regime.
The reed of a clarinet is a pressure-controlled nonlinear valve sitting at one end of a cylindrical waveguide. The air column is a pair of pressure delay lines; the reed is a table whose input is the instantaneous pressure difference across the reed and whose output is the resulting volume flow. Two delay lines plus a reed table produces a clarinet that starts, sustains, and overblows exactly as a real one does.
Pressure satisfies the same 1-D wave equation as the string (with $c$ now the speed of sound in air, $\approx 343$ m/s). A cylindrical tube, closed at the mouthpiece and open at the bell, gives odd-harmonic resonances at
The closed end reflects pressure without inversion; the open end inverts it and adds a small lowpass loss due to radiation into the room.
Let $\Delta p = p_m - p$ be the pressure difference between the player's mouth and the mouthpiece. Fletcher & Rossing's clarinet reed is a compliant valve: for small $\Delta p$ it is wide open; for moderate $\Delta p$ it narrows; for $\Delta p$ larger than a critical value it slams shut. A tidy analytical form (Smith, after McIntyre & Woodhouse) is
Wire this table into the reflection at one end of a cylindrical waveguide and the system self-oscillates at the tube's first odd harmonic. Blow harder (raise $p_m$) and it overblows to the third harmonic — the clarinet's famous "register jump" of a twelfth.
Digital waveguides are exact (to numerical precision) for strictly one-dimensional lossless propagation. Once the geometry branches, bends, or becomes two- or three-dimensional — a plate, a membrane, a room — the elegant D'Alembert decomposition no longer suffices, and we go back to finite-difference discretisation. FDTD is the generic tool for those problems. It replaces the derivatives in the wave equation with centred differences and steps the solution forward in time.
Discretise $y_{tt} = c^2 y_{xx}$ with sample period $T = 1/f_s$ and spatial step $X = c\,T / \lambda$, where $\lambda \le 1$ is the Courant number. The leap-frog update is
Setting $\lambda = 1$ (the so-called "magic time step") makes the stencil exact for band-limited waves on an ideal 1-D string; the FDTD update and the waveguide's two delay lines become algebraically equivalent.
In two dimensions the Laplacian is a five-point stencil:
For stability the Courant condition becomes $\lambda \le 1/\sqrt{2}$. With Dirichlet (clamped) boundary conditions the modes of a square $L\times L$ membrane are
Inharmonic, naturally — hence the characteristically drum-like, inharmonic timbre of any struck two-dimensional object.
Click anywhere on the canvas to inject a small Gaussian displacement. Watch the waves propagate, reflect, and interfere into mode patterns. A virtual microphone records the displacement at the red dot, and Listen plays it back.
A linear physical object — a bar, a plate, a bell — radiates a sum of damped sinusoids whose frequencies are its modes. If we know the modes, we can skip the wave equation entirely and just drive an oscillator bank. This is modal synthesis, the dual of waveguides: instead of propagating waves in space, we superpose eigenmodes in time.
For a linear, shift-invariant object, the response at observation point $x_o$ to a force at excitation point $x_e$ is
where $\phi_k$ is the $k$-th mode shape, $\omega_k$ its angular frequency, and $\alpha_k$ its damping. Each mode is one biquad resonator. A mode set of sixteen oscillators is enough to model a marimba bar credibly; a few hundred captures a church bell.
| Situation | Prefer | Why |
|---|---|---|
| 1-D propagation, low loss | Waveguide | Two delay lines, exact |
| Nonlinear interaction at one point | Waveguide | Table lookup at junction |
| Irregular 2-D/3-D geometry | Modal or FDTD | No travelling-wave decomposition |
| Few modes, cheap run-time | Modal | O(K) oscillators |
| Shape morphs continuously | FDTD | Modal bank would re-diagonalise |
Choose a preset (bar, plate, bell) or drag the sliders to sculpt your own mode set. Each marker in the top panel is one mode; vertical position = amplitude, horizontal position = frequency, marker radius = damping time.
The hammer–string–soundboard system of a piano is almost linear. The hammer contact is briefly nonlinear but after that the string is (approximately) linear, and the soundboard is always linear. Linear systems commute. Smith and Van Duyne (1995) noticed that the expensive soundboard convolution can be computed once, during excitation, and stashed inside a recorded impulse response. At runtime all that remains is a single lossy waveguide per note.
Record the combined hammer–soundboard impulse response once, offline, and you have a single pre-computed excitation that you feed into a waveguide at runtime. The soundboard — a messy, inharmonic, radiating plate — becomes part of the excitation, not part of the loop.
Twelve notes, each one a commuted waveguide: short noise-plus-tone excitation $\rightarrow$ hammer shape $\rightarrow$ soundboard IR $\rightarrow$ string waveguide. Click a key to strike.
The same delay-line machinery that models a string models a room. A room is a three-dimensional waveguide. Schroeder (1962) realised that a set of inter- connected delays with carefully chosen lengths and feedback coefficients can approximate the dense echo density and exponential decay of a real room — the parametric reverberator was born. Fifty years of refinement culminated in Jot & Chaigne's (1991) Feedback Delay Network (FDN), now the standard building block of every digital reverberator.
A Feedback Delay Network with $N$ delays is described by
where $\mathbf{m}$ is a vector of delay lengths in samples, $\mathbf{A}$ is a unitary (norm-preserving) feedback matrix, and $\mathbf{b}, \mathbf{c}, d$ are input/output/direct coefficients. Unitary $\mathbf{A}$ ensures the network is energy-preserving in the lossless limit; the actual decay is introduced by a per-line lowpass on each $\mathbf{s}_i$.
Four delay lines, a 4×4 Hadamard feedback matrix, and a one-pole lowpass on each line. Adjust the decay and colour and send a short click through it.
Smith's digital-waveguide framework shipped first inside the Yamaha VL1 (1994) and is now the mathematical engine under every physically-modelled synthesiser, including Pianoteq, Modalys, Arturia's CMI V physical-string mode, and the physical-modelling patches inside every contemporary modular platform. Since 2020 the waveguide has also become a differentiable layer: a neural network learns the excitation and the loop filter directly from audio, preserving the physical prior while absorbing what the physics misses.