Waves as physical objects

Plane-wave ansatz, dispersion, impedance, energy density, reflection, WKB.

The Math Foundations book contains the wave equation and its mathematics; the Sound book uses it to derive acoustic propagation and impedance. This chapter is the physical companion — the properties of waves that recur across acoustics, elasticity, electromagnetism, and quantum mechanics.

The plane-wave ansatz and dispersion relations

The plane wave

\psi(\mathbf{r}, t) \;=\; A\, e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}

is the basic eigensolution of every linear wave equation. The wavenumber $|\mathbf{k}|$ counts radians of phase per metre; the angular frequency $\omega$ counts radians per second. The relationship $\omega(k)$ — the dispersion relation — is the wave’s signature.

How does the dispersion relation come from a PDE? Substituting the plane-wave ansatz turns derivatives into multiplications: $\partial/\partial t \to -i\omega$ , $\partial/\partial x \to ik$ . The PDE becomes an algebraic relation between $\omega$ and $k$ .

PDE:

1. PDE

∂²ψ/∂t² = c² ∇²ψ

2. ansatz

ψ = A e^{i(kx − ωt)}

3. substitute

(−iω)² A e^{i(kx−ωt)} = c² (ik)² A e^{i(kx−ωt)}

4. dispersion relation
ω² = c² k²   →   ω = ±c k  (linear)

The plane-wave ansatz turns derivatives into multiplications: ∂/∂t → −iω, ∂/∂x → ik. The PDE becomes an *algebraic* relation between ω and k — the dispersion relation. Real ω means propagating waves; imaginary ω means decay (diffusion). The shape ω(k) determines whether the medium is dispersive (waves spread) or non-dispersive (waves propagate rigidly).

Three example equations: the wave equation gives $\omega = ck$ (linear, non-dispersive); the heat equation gives $\omega = -i\alpha k^2$ (purely imaginary, diffusive not wavelike); the Schrödinger equation gives $\omega = (\hbar/2m)k^2$ (quadratic, dispersive). The shape of $\omega(k)$ — linear, quadratic, sqrt, complex — determines whether the medium is non-dispersive, dispersive, or dissipative.

Phase and group velocity

Two velocities matter:

Phase velocity $v_p = \omega/k$ — speed of a single Fourier mode’s wave crest.
Group velocity $v_g = d\omega/dk$ — speed of an envelope built from a narrow band of modes near some carrier $k_0$ .

These need not be equal. They are equal only if $\omega$ is linear in $k$ . In any other case the wave is dispersive — different modes travel at different phase velocities, and a wave packet broadens as it propagates.

▶ Group velocity from a stationary-phase argument

A wave packet built from a narrow band of plane waves is

\psi(x, t) \;=\; \int A(k)\, e^{i(k x - \omega(k) t)}\, dk.

Expand $\omega(k)$ around the centre $k_0$ to first order:

\omega(k) \;\approx\; \omega(k_0) + (k - k_0)\, \omega'(k_0).

Substituting and factoring out the carrier,

\psi(x, t) \;\approx\; e^{i(k_0 x - \omega(k_0) t)} \int A(k)\, e^{i(k - k_0)(x - \omega'(k_0)\, t)}\, dk.

The integral is the Fourier transform of $A(k)$ evaluated at $x - \omega'(k_0)\, t$ — the envelope, displaced rigidly with velocity $v_g = \omega'(k_0)$ . The carrier propagates separately with velocity $v_p = \omega(k_0)/k_0$ .

Dispersion ω(k):

Carrier wavenumber k₀ = 1.00

The ω(k) curve is the dispersion relation. The red dashed line is the chord from the origin to k₀ — its slope is the phase velocity v_p. The green dashed line is the tangent at k₀ — its slope is the group velocity v_g. In the packet panel, the green vertical line tracks the envelope centroid; the red ticks track an individual wave crest. For linear dispersion they move together; for nonlinear, they separate, and the envelope also broadens in time.

Three regimes: linear (non-dispersive, $v_p = v_g$ , packet rigid), quadratic ( $v_g = 2 v_p$ , envelope outruns carrier), and square-root ( $v_g = v_p/2$ , envelope lags). Crests appear at the back of the packet and disappear at the front when $v_p > v_g$ .

Acoustic impedance — derived, not asserted

For a plane wave in a fluid, the pressure amplitude and particle-velocity amplitude are not independent — they are locked together by the wave equation.

▶ Z = ρ₀c from linearised Euler + continuity

For a plane wave $p' = P_0\, e^{i(kx - \omega t)}$ , $u' = U_0\, e^{i(kx - \omega t)}$ , the linearised Euler equation is

\rho_0 \frac{\partial u'}{\partial t} \;=\; -\frac{\partial p'}{\partial x}.

Substituting:

\rho_0 (-i\omega) U_0 \;=\; -(ik) P_0 \quad\Longrightarrow\quad U_0 \;=\; \frac{k}{\omega \rho_0} P_0.

The acoustic dispersion is $\omega = ck$ , so $k/\omega = 1/c$ , and

U_0 \;=\; \frac{P_0}{\rho_0 c}, \qquad \frac{P_0}{U_0} \;=\; \rho_0 c \;\equiv\; Z.

$Z = \rho_0 c$ is the specific acoustic impedance of the medium, with units of $\text{Pa}\cdot\text{s}/\text{m}$ .

ρ₀1.20

c343

P₀1.00 Pa

Z = ρ₀ c411.6

Density ρ₀ = 1.20 kg/m³ Wave speed c = 343 m/s Pressure amplitude P₀ = 1.00 Pa

Substituting a plane-wave ansatz into linearised Euler (ρ₀ ∂u′/∂t = −∂p′/∂x) together with the wave dispersion ω = ck gives U₀ = P₀ / (ρ₀ c) ≡ P₀ / Z. The impedance Z = ρ₀c is a *medium property*, not a wave property: it depends on ρ₀ and c, not on the specific wave. Water has Z ≈ 1.5 × 10⁶, air ≈ 410 — a factor-of-3500 mismatch that the middle-ear ossicles are built to bridge.

$Z$ is a property of the medium, not the wave: it depends on $\rho_0$ and $c$ , not on $P_0$ or $\omega$ . For water $Z \approx 1.5\times 10^6$ ; for air $Z \approx 410$ — a factor-of-3500 mismatch, the operative problem the middle-ear ossicles are built to solve.

Mechanical impedance — the same idea, generalised

For any linear system in which a sinusoidal force $F$ produces a sinusoidal velocity $v$ , the mechanical impedance is $Z_\text{mech} = F/v$ . For a damped harmonic oscillator,

Z_\text{mech}(\omega) \;=\; b \;+\; i\!\left(\omega m \;-\; \frac{k}{\omega}\right).

b (damping)0.50

m (mass)1.00

k (stiffness)4.00

ω₀ = √(k/m)2.00

Damping b = 0.50 Mass m = 1.00 Stiffness k = 4.00

Below ω₀ the stiffness term k/ω dominates and the impedance is *capacitive* (phase −90°); above ω₀ the mass term ωm dominates and the impedance is *inductive* (+90°). At ω₀ the reactive parts cancel and only the damping b remains. The complex impedance is the operative model for the basilar membrane in [Hearing Ch 4.3](/hearing/cochlea/traveling-wave).

Three frequency regimes:

$\omega \ll \omega_0 = \sqrt{k/m}$ : stiffness-controlled. $|Z| \sim k/\omega$ , phase $\to -90°$ .
$\omega \sim \omega_0$ : damping-controlled. Imaginary parts cancel, $|Z| = b$ , phase $\to 0°$ .
$\omega \gg \omega_0$ : mass-controlled. $|Z| \sim \omega m$ , phase $\to +90°$ .

The basilar membrane has exactly this impedance form, with $k$ , $m$ , $b$ that vary along the cochlea (Hearing Ch 4.3).

Energy density, intensity, and acoustic equipartition

A plane sound wave carries energy. The instantaneous energy density splits as

\mathcal{E} \;=\; \underbrace{\tfrac12 \rho_0 |\mathbf{v}'|^2}_\text{kinetic} \;+\; \underbrace{\frac{p'^2}{2 \rho_0 c^2}}_\text{potential}.

For a plane wave the two parts are in phase: simultaneously maximum at the wave crest and zero at the node.

In a plane sound wave, kinetic energy density (red) and potential energy density (green dashed) oscillate *in phase* — they are simultaneously maximum at the wave crests and zero at the nodes. On time-average they are equal (acoustic equipartition), each contributing ½ to the total energy density. The total (black) is twice the time-average of either part — a clean instance of the chapter's energy bookkeeping.

On time-average, the two parts are equal — acoustic equipartition. The intensity (energy flux per unit area, normal to the wave) is

I \;=\; \langle p'\, v'\rangle \;=\; \frac{\langle p'^2\rangle}{\rho_0 c} \;=\; \frac{P_0^2}{2 \rho_0 c}.

Intensity is what the decibel scale references. Doubling the pressure amplitude quadruples the intensity (a 6-dB rise).

Reflection at impedance discontinuities

When a plane wave hits an interface between media of impedance $Z_1$ and $Z_2$ , part transmits and part reflects. Matching pressure and normal velocity across the boundary gives

R \;=\; \frac{Z_2 - Z_1}{Z_2 + Z_1}, \qquad T \;=\; \frac{2 Z_2}{Z_2 + Z_1}.

R (amplitude)0.600

T (amplitude)1.600

R_P (power)36.0%

T_P (power)64.0%

Z₁ = 1.00 Z₂ = 4.00

At an interface between two media with impedance Z₁ and Z₂, the reflection amplitude is R = (Z₂−Z₁)/(Z₂+Z₁). For air-to-water (Z₂/Z₁ ≈ 3500), R ≈ 1.0 and 99.9% of the power reflects — the impedance-matching problem the middle ear is built to solve. When Z₁ = Z₂ (perfect match), R = 0 and all power transmits.

The corresponding power coefficients are $R_P = R^2$ and $T_P = 1 - R^2$ — and they sum to 1 by energy conservation. For air-to-water ( $Z_1 = 410$ , $Z_2 = 1.5\times 10^6$ ), $R^2 = 0.9989$ : 99.9% of the acoustic power reflects. This is the impedance mismatch the middle ear is built to compensate.

Radiation pressure

A wave carries momentum as well as energy: momentum flux per unit area = $I/c$ . When a wave is absorbed at a surface,

P_\text{rad} \;=\; \frac{I}{c} \;\;(\text{absorbed}), \qquad \frac{2 I}{c} \;\;(\text{reflected}).

For high-intensity acoustic beams, this is the basis of acoustic levitation, acoustic tweezers, and the secondary acoustic radiation force on microbubbles in cavitation flows.

The WKB approximation — waves in slowly-varying media

When the medium’s properties change slowly over a wavelength, the wave equation can be solved by the WKB ansatz: $\psi \propto A(x)\, e^{i\phi(x) - i\omega t}$ with $A$ slowly varying and $\phi$ rapidly varying. Substituting gives to leading order the local dispersion $\phi'(x)^2 = k^2(x, \omega)$ , and to next order $A(x) \propto 1/\sqrt{k(x)}$ — the amplitude grows where the wavelength contracts rule.

Taper rate α = 0.40

The WKB amplitude rule A(x) ∝ 1/√k(x) is a statement of *energy conservation* in a slowly-varying medium: the energy flux density A²·k must be constant if no energy is being created or destroyed, so A ∝ 1/√k. As the wave enters the narrower part of the duct, k(x) rises and the amplitude grows to keep the flux conserved. This is exactly the mechanism by which the cochlear traveling wave amplifies as it approaches its characteristic-frequency place.

The amplitude-vs-wavelength rule is energy conservation: the local flux $A^2 k$ is constant if no energy is created or destroyed. This is the operative tool for the cochlear traveling-wave chapter — as the wavenumber $\kappa(x, \omega)$ rises toward the characteristic-frequency place, the amplitude grows accordingly, peaking just before the resonance terminates the wave.

⏳ The history — Rayleigh, the group velocity, and the wake of a ship

Rayleigh’s Theory of Sound (1877) was the first systematic English-language acoustics text; it remains in print and still readable. Rayleigh introduced the concept of group velocity in the context of water waves, observing that wave crests on a propagating disturbance appear at the back of a packet, march forward, and disappear at the front — a counter-intuitive behaviour that requires phase velocity to differ from group velocity.

The wake of a moving ship is the cleanest example: the V-shaped Kelvin wedge has crests travelling at one speed (the phase velocity of deep-water waves) inside an envelope travelling at another (the group velocity). For deep-water gravity waves, $v_p = 2 v_g$ , so crests in the wake travel twice as fast as the wake itself.

The WKB approximation — named for Wentzel, Kramers, and Brillouin (1926) — was developed for quantum mechanics, but its origin in classical wave physics goes back to Liouville and Green in the 1830s and Rayleigh in the 1910s. The cochlear traveling wave is one of the most successful applications of WKB to a biological system.

For the cross-book applications — acoustic impedance, middle-ear matching, cochlear WKB, plane-wave Sound book — see the key examples sub-page.