Key examples — elasticity and continuum mechanics

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

Example 1: basilar-membrane stiffness gradient and cochlear tonotopy

The basilar membrane spans the cochlea from base to apex, with a stiffness $k(x)$ that varies by ~100× over a 35 mm length. Modelled as a damped harmonic oscillator at each position $x$, the local resonant frequency $\omega_0(x) = \sqrt{k(x)/m}$ varies by $\sqrt{100} = 10$× — covering roughly two decades, almost exactly the audible range. The exact dependence on $x$ is approximately exponential (the Greenwood map), giving the tonotopic frequency-place axis of the cochlea. See Hearing Ch 4.2.

Example 2: mechanical impedance of the basilar membrane

For each point on the basilar membrane, the impedance presented to the cochlear fluid is

with $b$ the local viscous damping, $m$ the (apparent) mass, $k(x)$ the stiffness from Example 1. This is the operative complex impedance that enters the cochlear long-wave dispersion relation $\kappa^2(x, \omega) = 2i\omega\rho/(A Z_\text{BM})$. See Hearing Ch 4.3.

Example 3: speed of sound in water vs steel

For water: $K \approx 2.2\,\text{GPa}$, $G \approx 0$ (no shear elasticity), $\rho = 1000\,\text{kg/m}^3$ → $c = \sqrt{K/\rho} \approx 1480\,\text{m/s}$.

For steel: $E \approx 200\,\text{GPa}$, $\rho \approx 7800\,\text{kg/m}^3$ → $c_P \approx 5900\,\text{m/s}$ (P-wave), $c_S \approx 3200\,\text{m/s}$ (S-wave).

Steel is 15× stiffer than water in bulk modulus, but only $\sim 4\times$ as fast in sound — because steel is also $\sim 8\times$ denser. The competition between stiffness and inertia in $\sqrt{K/\rho}$ is universal across acoustic media. See Sound Ch 4.9.

Example 4: tension–wave on a string and the discrete-to-continuum bridge

The Sound book’s Ch 3 lessons derive the 1-D wave equation by taking the continuum limit of a discrete chain of masses connected by springs. The result is the string-wave equation $u_{tt} = c^2 u_{xx}$ with $c = a\sqrt{\kappa/m}$ — the elasticity-chapter formula in disguise. In string terms, $T = \kappa a$ and $\mu_\text{lin} = m/a$, recovering $c = \sqrt{T/\mu_\text{lin}}$.

Example 5: middle-ear ossicles as nearly-rigid levers

The malleus, incus, and stapes are bone — Young’s modulus $\sim 20\,\text{GPa}$, much stiffer than the surrounding tissue. To good approximation they are treated as rigid bodies in the mechanics analysis of the middle ear (Hearing Ch 3.3), and the torque-balance + lever-arm argument from the mechanics chapter suffices. Their slight elastic compliance shows up only at very high frequencies (above 4 kHz) as a high-frequency rolloff of middle-ear transmission.

Example 6: bulk modulus of water and the speed of sound

The chapter’s $c = \sqrt{K/\rho}$ for a fluid is precisely the relation between the molecular bulk modulus computed from the Lennard-Jones potential in the intermolecular-forces chapter and the macroscopic speed of sound. For water, the LJ estimate $K \sim \varepsilon/\sigma^3 \approx 2\,\text{GPa}$ matches measurement to within a factor of two. This is one of the few “first-principles” predictions in the bookshelf where molecular-scale physics directly determines an observed macroscopic acoustic quantity.

Cross-book backlinks

• Sound Ch 3.1-3.2 — chain to continuum: discrete to wave-equation.
• Sound Ch 4.9 — speed of sound: $c = \sqrt{K/\rho}$ across fluids and solids.
• Hearing Ch 3.3 — the ossicular solution: nearly-rigid bones.
• Hearing Ch 4.2 — stiffness gradient: basilar-membrane elasticity.
• Hearing Ch 4.3 — traveling wave: mechanical-impedance derivation.
• Cavitation Ch 1.1 — water’s bulk modulus: K for liquid water.