3.3 Plane waves and complex impedance

The phasor of 3.1 lived in time alone; the spiral of 3.2 added a decay rate. This third lesson takes the same machinery into space as well as time, producing plane waves of the form $e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})}$ . It also generalises the phasor algebra into the impedance picture — every linear acoustic element (a tube, a resonator, a loudspeaker, a wall) is characterised by a complex-valued impedance $Z(\omega)$ that lives in the complex plane and varies with frequency.

This is the lesson the rest of the bookshelf actually uses. Almost every Sound-book equation traffics in plane waves and impedances; this is where both are introduced.

Plane waves

A propagating plane wave is the phasor framework extended to space-and-time:

\boxed{\;p(\mathbf{r}, t) \;=\; \operatorname{Re}\!\bigl[\, P_0\, e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})}\, \bigr].\;}

The argument $\omega t - \mathbf{k} \cdot \mathbf{r}$ is a real number (since $\mathbf{k}$ and $\mathbf{r}$ are real); the phasor $P_0$ is a complex amplitude carrying both magnitude and overall phase. The wave moves in the direction $\hat{\mathbf{k}}$ at speed $v_p = \omega / |\mathbf{k}|$ — the phase velocity.

The key algebraic move: spatial derivatives become multiplication by $-i \mathbf{k}$ :

\nabla\bigl[ e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})} \bigr] \;=\; -i \mathbf{k}\, e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})}.

In particular, the Laplacian becomes multiplication by $-|\mathbf{k}|^2$ . The wave equation $\partial_t^2 p = c^2 \nabla^2 p$ collapses to the algebraic dispersion relation

\omega^2 \;=\; c^2\, |\mathbf{k}|^2 \quad\Longleftrightarrow\quad \omega = \pm c\, |\mathbf{k}|.

The wave’s frequency and wavenumber are tied by the speed of sound. This is the algebra you do dozens of times in the Sound book: substitute a plane-wave ansatz, recover the dispersion relation, read off the speed of propagation.

The same trick works for any linear PDE with constant coefficients. The Helmholtz equation, the heat equation, the Schrödinger equation — all reduce to algebraic relations among $\omega$ and $\mathbf{k}$ when you substitute the plane-wave ansatz. See Foundations 6.2 and 6.7 for the full development.

Complex impedance

For a linear element driven at a single frequency $\omega$ , the impedance is the ratio of the driving phasor to the response phasor:

Z(\omega) \;\equiv\; \frac{\tilde V}{\tilde I}, \qquad \tilde V \text{ across element}, \tilde I \text{ through it}.

In acoustics the analogous ratio is pressure-over-volume-velocity for an acoustic element, or pressure-over-particle-velocity for a wave at a point (Sound 5.4). In electrical circuits it is voltage-over-current, the standard EE definition. In mechanics it is force-over-velocity. The same algebra applies in all three settings.

Impedance is complex in general, because the response can lag or lead the drive in phase. The phasor algebra makes that explicit: $Z(\omega) = R + i X(\omega)$ , with the real part $R$ being the resistive component (in phase with the drive, dissipating energy) and the imaginary part $X(\omega)$ being the reactive component (90° out of phase, storing energy).

For a series RLC circuit:

Z(\omega) \;=\; R \;+\; i\omega L \;+\; \frac{1}{i\omega C} \;=\; R \;+\; i\!\left(\omega L - \frac{1}{\omega C}\right).

The resistor $R$ contributes a real-valued term (no phase shift). The inductor contributes $i\omega L$ (90° lead, voltage ahead of current). The capacitor contributes $1/(i\omega C) = -i/(\omega C)$ (90° lag, voltage behind current). At a special frequency $\omega_0 = 1/\sqrt{LC}$ the inductive and capacitive reactances cancel and $Z = R$ — pure resistance, no phase shift. This is resonance.

R = 1.00 L = 1.00 C = 1.00

probe frequency ω = 1.00

A series RLC circuit has impedance Z(ω) = R + i(ωL − 1/ωC). The blue curve traces Z as ω sweeps from low to high — at low ω the capacitive term 1/(ωC) dominates and Im[Z] is very negative; at high ω the inductive term ωL dominates and Im[Z] is very positive; in between, Z passes through the *resonance* point where Im[Z] = 0 and the impedance is purely resistive at value R. The right panel shows |Z(ω)| on log-log axes — the V-shape with minimum at resonance is the canonical "RLC response" you'll see on every Bode plot of a real-world filter or transducer.

Slide $R$ , $L$ , $C$ to change the circuit; slide $\omega$ to sweep the probe along the impedance curve. Two things to absorb:

The impedance traces a curve in the complex plane as $\omega$ sweeps. At low $\omega$ the capacitive term dominates and $Z$ has large negative imaginary part. At high $\omega$ the inductive term dominates and $Z$ has large positive imaginary part. In between, $Z$ passes through the resistive point $Z = R$ at resonance.
$|Z(\omega)|$ on log-log has a V-shape with minimum at resonance. The asymptotes are $|Z| \sim 1/(\omega C)$ at low $\omega$ and $|Z| \sim \omega L$ at high $\omega$ , giving slopes $-1$ and $+1$ respectively on the log-log plot.

This is the same V-shape that appears on every Bode plot of every real-world filter or transducer. The impedance picture in the complex plane (called a Nyquist plot in control theory) and the magnitude-vs-frequency plot (the Bode plot) are two views of the same complex function $Z(\omega)$ .

Acoustic impedance and the bookshelf

In acoustics, the specific acoustic impedance of a plane wave in a medium of density $\rho$ and sound speed $c$ is

Z_0 \;=\; \rho c.

For air at room temperature, $Z_0 \approx 1.2 \cdot 340 \approx 410$ Pa·s/m. For water, $Z_0 \approx 1.5 \times 10^6$ Pa·s/m — about 3600 times larger. This vast impedance mismatch between air and water is why sound transmits poorly across an air-water interface, and is the engineering problem the mammalian middle ear solves with the three-bone lever system (Hearing Ch 3).

For an acoustic element with a frequency-dependent impedance — a tube, a resonator, an open window, a room mode — the impedance $Z(\omega)$ becomes complex and traces curves in the complex plane just like the RLC circuit above. The transfer function of a damped resonator has exactly the same form as the RLC impedance, with the resonance frequency at $\omega_0$ and the bandwidth at $\Delta\omega \sim R/(L)$ or equivalently $\omega_0 / Q$ . The mathematics is universal; the physical labels change between domains.

Cautions

The same cautions from 3.1 apply, plus one more:

The plane-wave ansatz $e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})}$ uses the physics sign convention ( $\omega t$ first, $\mathbf{k} \cdot \mathbf{r}$ subtracted). The engineering convention swaps the signs: $e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}$ . Both are correct; both produce the same real signals. But signs flip throughout intermediate calculations, so always check which convention a textbook uses before applying formulas. The Sound book uses the physics convention throughout.

Closing the chapter

That closes Foundations 3. Three lessons developed the complex-exponential machinery in increasing generality: Euler’s formula on the unit circle (3.1), spirals in the complex plane for damped oscillations (3.2), and complex-valued impedance / plane waves with $\omega$ and $\mathbf{k}$ as independent variables (this lesson).

The arc: complex exponentials are the eigenfunctions of linear differential operators with constant coefficients. Differentiating them produces themselves multiplied by a constant — $\partial_t \to i\omega$ , $\nabla \to -i\mathbf{k}$ , $\partial_t^2 + \omega_0^2 \to (\lambda^2 + \omega_0^2)$ . This is why every linear ODE or PDE in this bookshelf collapses to algebra under a complex-exponential ansatz, and why phasor methods are the unified language of linear physics across acoustics, optics, electronics, quantum mechanics, and signal processing.