Physical Models

Lecture 9 · From the wave equation to modes, waveguides, and instruments

Prof. Dr.-Ing. Sebastian J. Schlecht · Dr.-Ing. Maximilian Schäfer

Where we are

One question, asked of every method

Block	Representation	Generator
Filters and effects	LTI and time-varying systems	hand-designed
Classical synthesis	samples, partials, modes, physical state	hand-designed
DDSP	the same structured models	parameters learned
Reverberation	a recursive dynamical system	hand-designed, fittable
Diffusion	a learned representation	learned

The four questions

How is the sound represented?
What are the controllable parameters?
What structure is built in?
What is fixed by design, and what is learned?

Lecture 7 wrote down the spectrum, lecture 8 kept the recording. Neither could answer “what if I pluck it somewhere else?” Today: model the mechanism, and the answer is a parameter.

Listen

Ask the string a question

The riff again
target: the real guitar
lecture 7: additive, ideal string
today: modal string, struck by a pick near the bridge, damped, stopped by the next note

One string, three questions a recording cannot answer:

pluck in the middle
pluck at the bridge
strike with a hard hammer

Listen first

What changed between the lecture-7 riff and today’s? Where do the attack and the short, muted notes come from?

Represent: the physical state

Simulate the mechanism

Every method so far models the sound. A physical model simulates the object: a string, a tube, a membrane, including where it is struck and where it is heard.

The representation

Not samples of the output, but the state of the object: the displacement and velocity of every point of the string. The sound is what a pickup reads from that state.

What a real string does

The first milliseconds after a pluck: the deflection \(y(x,t)\) is a function of space and time.

initial conditions: pluck \(y(x,0)=y_0(x)\), strike \(\dot y(x,0)=v_0(x)\)
boundary conditions: what happens at the two ends, \(x=0\) and \(x=\ell\)
a PDE for everything in between

One equation, three effects

\[\underbrace{\rho A\,\ddot y - T_s\,y''}_{\text{wave equation}} \;+\; \underbrace{EI\,y''''}_{\text{stiffness}} \;+\; \underbrace{d_1\,\dot y - d_3\,\dot y''}_{\text{damping}} \;=\; f_e(x,t)\]

Wave equation only: pulses travel unchanged at \(c_0 = \sqrt{T_s/\rho A}\)

+ stiffness: high frequencies travel faster, so the pulse spreads (dispersion)

+ damping: frequency-independent (\(d_1\)) and frequency-dependent (\(d_3\)) losses

Hearing stiffness

Same string, stiffness increasing from left to right. Listen to the attack and the “twang”:

reference

none

more still

extreme

Stiffness stretches the partials: \(f_\mu \approx \mu f_1\sqrt{1 + B\mu^2}\). In a piano this inharmonicity is why octaves are tuned slightly wide, and why bass strings sound “metallic”.

Structure: from grid to modes

Finite differences: sample space and time

Grid \(x_m = mh\), \(t_k = kT\); replace derivatives by central differences. For the wave equation \(\ddot y = c^2 y''\):

\[y_m^{k+1} = 2(1-\lambda^2)\,y_m^k + \lambda^2\big(y_{m+1}^k + y_{m-1}^k\big) - y_m^{k-1}\]

with the Courant number \(\lambda = cT/h\). Ends fixed: \(y_0^k = y_{N}^k = 0\).

Add the two loss terms and pluck at 0.3 (200 grid intervals, ≈ 108 Hz):

The triangle splits into two pulses that travel, reflect inverted at the fixed ends, and recombine once per period.

Failure case: the Courant limit

The explicit scheme is stable only for

\[\lambda = \frac{cT}{h} \le 1\]

A wave may not travel more than one grid step per time step. One percent over, and the string explodes within 2 ms.

Fixing \(f_s\) and the pitch fixes \(c/h\): a stiffer or higher string needs a coarser grid, and the grid sets how many modes you get.

FDTD is a state-space system

Stack the grid into a vector \(\mathbf y^k\). The update is a matrix recursion with the discrete Laplacian \(\mathbf L = \operatorname{tridiag}(1, -2, 1)\):

\[\mathbf y^{k+1} = \big(2\mathbf I + \lambda^2 \mathbf L\big)\,\mathbf y^k - \mathbf y^{k-1} \qquad\Longleftrightarrow\qquad \underbrace{\begin{bmatrix}\mathbf y^{k+1}\\ \mathbf y^{k}\end{bmatrix}}_{\mathbf x[k+1]} = \underbrace{\begin{bmatrix} 2\mathbf I + \lambda^2 \mathbf L & -\mathbf I\\ \mathbf I & \mathbf 0\end{bmatrix}}_{\mathbf A} \underbrace{\begin{bmatrix}\mathbf y^{k}\\ \mathbf y^{k-1}\end{bmatrix}}_{\mathbf x[k]} + \mathbf b\, u[k]\]

the state \(\mathbf x\): the whole string, now and one step ago
the input \(\mathbf b\,u[k]\): where and how it is excited
the output \(\mathbf c^\top \mathbf x\): where the pickup sits

Hold on to this form

Lecture 11’s reverb (the FDN) is \(\mathbf x[n+1] = \hat{\mathbf A}\mathbf x[n] + \hat{\mathbf b}\,x[n]\) too. A string and a reverb are the same kind of object: a large, sparse, recursive linear system.

Diagonalize: the modes fall out

\(\mathbf L\) is symmetric: \(\mathbf L = \boldsymbol\Phi\boldsymbol\Lambda\boldsymbol\Phi^\top\), with discrete sines as eigenvectors. In modal coordinates \(\boldsymbol\eta = \boldsymbol\Phi^\top\mathbf y\) every mode runs on its own:

\[\eta_\mu^{k+1} = \big(2 + \lambda^2\beta_\mu\big)\,\eta_\mu^k - \eta_\mu^{k-1}\]

one second-order filter per mode, with poles on the unit circle at

\[\cos\Omega_\mu = 1 + \tfrac{\lambda^2}{2}\beta_\mu, \qquad \beta_\mu = -4\sin^2\!\frac{\pi\mu}{2N}\]

left: grid and filter bank give the same signal, to machine precision
right: the grid’s own error, numerical dispersion: at \(\lambda < 1\) high modes are flat

FTM: the same steps, done analytically

Discrete (FDTD notebook)	Continuous (FTM)
central differences	the PDE itself
\(z\)-transform in time	Laplace transform in time
eigenvectors of \(\mathbf L\)	Sturm–Liouville transform in space
\(\beta_\mu\) of a finite grid	exact eigenvalues, all terms of the PDE
resonator per mode	resonator per mode

Functional Transformation Method (FTM)

Transform the PDE with its boundary conditions into a set of independent second-order systems, then into discrete-time resonators. Exact modes, no spatial grid, no numerical dispersion. Stiffness and damping just move \(f_\mu\) and \(\sigma_\mu\).

L. Trautmann and R. Rabenstein, Digital Sound Synthesis by Physical Modeling Using the Functional Transformation Method, Kluwer, 2003.

A vibration is a sum of modes

full string motion

\(\mu = 1\)

\(\mu = 2\)

\(\mu = 3\)

\[y(x,t) = \sum_{\mu} \underbrace{\bar y_\mu(t)}_{\text{temporal: a damped sinusoid}} \;\cdot\; \underbrace{K_\mu(x)}_{\text{spatial: the mode shape}} \qquad \text{e.g. } K_\mu(x) = \sin\!\Big(\frac{\mu\pi x}{\ell}\Big)\]

One mode = one resonator

Each temporal part \(\bar y_\mu(t) = e^{-\sigma_\mu t}\sin(2\pi f_\mu t)\) is the impulse response of a second-order IIR filter:

\[H_\mu(z) = \frac{b_\mu z^{-1}}{1 - 2 e^{-\sigma_\mu T}\cos(2\pi f_\mu T)\, z^{-1} + e^{-2\sigma_\mu T} z^{-2}}\]

The same structure as formant synthesis and as lecture 7’s additive pluck, but now every \(f_\mu\), every \(\sigma_\mu\) and every weight comes from the physics, and where you pluck and listen is a parameter.

Structure: traveling waves

Traveling waves

The lossless wave equation \(\ddot y = c_0^2 y''\) has d’Alembert’s solution \[y(x,t) = y_l(x + c_0 t) + y_r(x - c_0 t)\] two shapes traveling left and right. Sample with \(x = mh\), \(t = kT\) and \(h/T = c_0\) (that is, \(\lambda = 1\)): \[y[m,k] = y_l[m+k] + y_r[m-k]\]

A traveling wave is a delay line. Two delay lines in opposite directions simulate the lossless string exactly, at the sample points, for one addition per output sample.

J. O. Smith III, Physical modeling using digital waveguides, Computer Music Journal, 1992.

The FDTD figure showed exactly this: two pulses, traveling and reflecting.

Losses, dispersion, and the bridge as filters

Insert filters \(H(z)\) into the lines:

\(H(z)\)	models
\(1\)	lossless
\(0 < g < 1\)	frequency-independent loss (\(d_1\))
low-pass	frequency-dependent loss (\(d_3\))
all-pass	dispersion (stiffness)

Boundaries reflect the waves: \(R(z) = -1\) for a rigid end, \(|R|<1\) for a lossy one, a filter for a bridge coupled to a soundboard.

Commute and combine

With one input and one output, all the delays merge into one delay line \(z^{-M}\) and all the filters into one loop filter. Add a fractional-delay filter to tune the pitch exactly.

Why Karplus–Strong sounds like a string

Max built it from a feedback comb filter: a delay line, and a two-point average in the loop.

The waveguide explains why it works:

Karplus–Strong	the string
delay line \(M\)	one round trip, \(2\ell / c_0\)
noise in the delay line	the initial displacement
averaging filter	\(d_3\): high partials lose more per trip
sign of the loop	the two inverting reflections

From accident to model

Karplus and Strong (1983) found the sound by experiment. The waveguide (Smith 1992) shows it is the commuted, lossy string with the crudest possible loop filter. Replace the average by a designed low-pass and all-pass, and every parameter has a physical meaning.

The other half of the 1983 paper: a drum

The Karplus–Strong paper is titled “plucked-string and drum timbres”. One change in the loop: keep the averaged sample with probability \(b\), negate it otherwise, \[y[n] = \pm\tfrac12\big(y[n-M] + y[n-M-1]\big), \quad P(+) = b.\]

\(b = 1\): the string
\(b = \tfrac12\): a drum
\(b = \tfrac12\), \(M\) three times longer

Random signs destroy the periodicity, so there is no pitch, only a noise burst whose brightness and decay are set by the loop: \(M\) now acts as a decay-time knob, not a pitch. A cheap snare, and a hint that a drum is not a string with a different number.

Waveguides in products

Yamaha VL1 (1994) · Daniel Hartwig, CC BY 2.0, cropped

Waveguides were the physical model of the 1990s: Yamaha VL1/VL7 (1994), Korg Prophecy (1995), sound cards from Creative.

oboe → bassoon

trumpet

tenor sax

Strong for 1D systems: strings, bores of wind instruments. 2D plates and membranes need waveguide meshes, which lose most of the efficiency.

Sound examples: J. O. Smith, CCRMA

Which method, when?

	Waveguide	Modal / FTM	FDTD
Represents	traveling waves	standing waves	the field on a grid
Best for	1D, low loss: strings, bores	linear systems, any shape with known modes	nonlinear, coupled, complex geometry
Cost	lowest: a few delays and filters	one resonator per mode	every grid point, every sample
Weak at	2D/3D, strong dispersion	nonlinearity (e.g. tension modulation)	numerical dispersion, stability limits

Rule of thumb

Waveguides dominated when computers were slow. Now that they are fast, modal and FDTD methods win whenever the physics matters more than the budget. In every case: understand the physics of the instrument first.

Control

The excitation is the input

In state space the excitation is \(\mathbf b\) (or the initial state). Project it onto the modes: each excitation writes a different spectrum into the string.

pluck: a displacement with a corner, modes fall as \(1/\mu^2\)
hammer: a velocity; the narrower, the more high modes

Same string, same position
pluck
hard hammer (1 % wide)
soft hammer (10 % wide)

Pluck position is a parameter

A modal string, identical except for where it is plucked (pickup near the bridge):

middle \(x_e = 0.5\ell\) even modes have a node there: hollow, round

one fifth \(x_e = 0.2\ell\) every 5th mode missing

at the bridge \(x_e = 0.05\ell\) all modes excited: bright, thin

The weight of mode \(\mu\) is \(K_\mu(x_e)\,K_\mu(x_o) = \sin(\mu\pi x_e/\ell)\sin(\mu\pi x_o/\ell)\). A sample would need a new recording for every position; the model needs a new multiplication.

Where the linear model ends: the bow

Pluck and strike are inputs: they happen, then the string rings on its own. A bow is an interaction:

the bow sticks to the string, drags it, then slips: stick–slip friction
the friction force depends on the relative velocity of bow and string, which depends on the state of the string
the result is a sawtooth-like Helmholtz motion, sustained as long as the bow moves

No \(\mathbf b\,u[k]\) can describe this

The input depends on the state: a nonlinear feedback loop between bow and string. Hammers that compress (piano felt) and strings that stretch at large amplitude (tension modulation) are the same kind of problem.

The waveguide or modal string stays; a small nonlinear junction is solved every sample at the contact point (McIntyre, Schumacher & Woodhouse 1983).

Make it playable

Which knobs would you give a performer?

The string has tension, density, length, stiffness, two damping constants, excitation shape, position and pickup position. A player gets four.

Macro	Physical parameters behind it
position	\(x_e\) (and a fixed pickup \(x_o\))
hardness	excitation width, or hammer felt stiffness
decay / brightness	\(d_1\) and \(d_3\), i.e. \(\sigma_0\) and how fast \(\sigma_\mu\) grows with frequency
material	stiffness \(B\) and the ratio of mode frequencies: string, bar, tube, plate

Here the mapping is not designed: physics says which parameters belong together. That is the real advantage over lecture 7’s knobs.

Beyond the string

Figure: Maximilian Schäfer Membranes, plates, tubes: same filter bank, only the mode shapes \(K_\mu\) change.

Excitations plug in as nonlinear inputs: a hammer (piano), a bow (violin), a pluck.

blues

jazz

A drum is a membrane

Circular membrane, radius \(a\), fixed rim: modes \(J_m(j_{mn} r/a)\cos(m\theta)\), \[\frac{f_{mn}}{f_{01}} = \frac{j_{mn}}{j_{01}} = 1,\ 1.59,\ 2.14,\ 2.30,\ 2.65,\ \dots\] with \(j_{mn}\) the zeros of the Bessel function \(J_m\): inharmonic, so no clear pitch.

Same drum, struck
at the centre
off centre, \(0.4a\)
near the rim, \(0.85a\)

At the centre every mode with \(m > 0\) has a nodal line: only the circular modes ring, a dull thud. The pluck-position slide in 2D: drummers strike off centre.

FTM tubular bells

FTM-Tube VST plugin

R. Rabenstein, T. Koch, and C. Popp, Tubular bells: a physical and algorithmic model, IEEE Trans. Audio, Speech, Lang. Process., 2010.

Physical parameters become sound-design knobs, far outside what real tubes can do:

glockenspiel geometry

extreme geometry

damping and excitation sweep

Products

AAS Chromaphone 3: modal resonators (plates, membranes, tubes) excited by mallets and noise

IK Multimedia MODO BASS: modal bass guitar; pluck position, pickups, and strings are parameters

Expressive E Imagine: physical modeling synthesizer plugin

Also: Pianoteq (Modartt), a physically modeled piano whose install is a small fraction of a sampled piano’s, with every string, damper, and resonance adjustable.

Match

Closer, but tuned by hand

target
lecture 7: additive, ideal string
lecture 7: subtractive
today: modal string

What is still missing?

And who chose \(B = 5\cdot 10^{-5}\), \(\sigma_0 = 6\), \(x_e = 0.15\)?

the body and the pickup: a filter we did not model
the pick and the fingers: contact noise, fret buzz
string interaction: here a new note simply stops the old one
every number was set by hand, from measured decays and listening

Next: fit them

The model is a differentiable function of \((B, \sigma_0, \sigma_2, x_e, x_o, \dots)\). Lecture 10 estimates those parameters from the recording by gradient descent; lecture 11 does the same for a room.

Wrap-up

The four questions, for physical models

	FDTD	Modal / FTM	Waveguide
Representation	displacement on a grid	modal amplitudes	two traveling waves
Controls	anything in the PDE, input anywhere	\(f_\mu, \sigma_\mu\), excitation and pickup position	loop filter, delay, excitation
Built-in structure	local physics, stability limit	linear, decoupled resonators	1D, lossless core plus filters
Fixed vs. learned	derived from physics	derived, or fitted to recordings	derived, or fitted

Take away

One system, three bases

FDTD samples the field, modes diagonalize it, waveguides split it into traveling waves. All three describe the same string; FTM gets the modes without a grid.

The input is a parameter

Excitation shape, position and pickup are vectors \(\mathbf b\) and \(\mathbf c\). Changing how the string is played costs a multiplication, not a new recording.

The same kind of object

A string, a reverb, and later a network layer: large, sparse, recursive state-space systems. Their parameters are physical, and fittable.