Key examples — waves

Where the chapter’s machinery shows up across the bookshelf.

Example 1: air-to-water impedance mismatch

Air $Z_1 = 410$, water $Z_2 = 1.5\times 10^6$. Reflection coefficient $R = (Z_2 - Z_1)/(Z_2 + Z_1) = 0.9995$; power reflection $R_P = R^2 = 0.999$. Only 0.1% of acoustic power transmits across a flat air-water interface — that’s a 30 dB transmission loss. The middle ear corrects this with the ossicular lever ($\sim 1.3\times$) and the eardrum-to-stapes area ratio ($\sim 17\times$), giving a combined pressure-amplification factor close to the required impedance ratio. See Hearing Ch 3.1-3.3.

Example 2: cochlear traveling wave as WKB

The cochlear long-wave equation has a slowly-varying wavenumber $\kappa(x, \omega)$ set by the local basilar-membrane impedance. The WKB amplitude rule $A(x) \propto 1/\sqrt{\kappa(x)}$ predicts that the wave amplitude rises sharply as $\kappa$ grows toward the resonance — exactly what is measured in in vivo cochlear-mechanics experiments. The growth is amplified further by the active outer-hair-cell feedback (Hearing Ch 4.5), but the WKB skeleton accounts for the passive component of the place-map gain. See Hearing Ch 4.3.

Example 3: plane harmonic waves in the Sound book

Sound Ch 5.1 takes the plane-wave ansatz from this chapter and works it through the full acoustic system: pressure, particle velocity, energy density, intensity, all derived in lock-step. This is the operative parameter set for everything in chapters 5-8 of Sound.

Example 4: standing waves in the ear canal

The ear canal is effectively a closed pipe (closed at the eardrum, open at the entrance). Standing-wave resonance gives the first mode at $f = c/(4L) \approx 3.4\,\text{kHz}$ for a 25 mm canal — exactly the frequency of peak hearing sensitivity. The next mode is $3 c/(4L) \approx 10\,\text{kHz}$, which sets a secondary maximum in the HRTF (head-related transfer function). See Hearing Ch 2.4.

Example 5: radiation pressure on a microbubble

A focused MHz acoustic beam at intensity $I = 1\,\text{W/cm}^2 = 10^4\,\text{W/m}^2$ exerts radiation pressure $P_\text{rad} = I/c \approx 7\,\text{Pa}$ on an absorbing surface — small in absolute terms but enough to translate a microbubble at $\mu$m/s velocities. Acoustic levitation, acoustic tweezers, and the secondary radiation force on microbubbles in clouds all derive from this chapter’s $P_\text{rad} = I/c$ formula. See Sound Ch 5.4.

Cross-book backlinks

• Sound Ch 5.1 — plane harmonic waves: full plane-wave description.
• Sound Ch 5.2 — acoustic energy density: kinetic + potential split.
• Sound Ch 5.3 — intensity and flux: I = ⟨p’v’⟩.
• Sound Ch 7.1 — reflection coefficient: R = (Z₂−Z₁)/(Z₂+Z₁).
• Hearing Ch 2.4 — ear canal resonance: quarter-wave standing wave.
• Hearing Ch 3.1 — impedance problem: air-to-water mismatch.
• Hearing Ch 4.3 — cochlear traveling wave: WKB on a slowly-varying medium.