7.5 Strings, membranes, and plates

The full elastic solid of the previous lessons has a lower-dimensional family of special cases — the string, the membrane, and the plate — that recur throughout physics and engineering. Each is a thin elastic structure, and each has its restoring force dominated by a different mechanism: tension in the string and membrane, bending stiffness in the plate. That difference decides whether their waves are dispersionless or dispersive.

The string: tension as the restoring force

A string is the simplest elastic system — a one-dimensional continuum whose only stiffness is the tension pulling it taut. Its transverse deflection $y(x,t)$ obeys the wave equation

\rho_\text{lin}\,\frac{\partial^2 y}{\partial t^2} \;=\; T\,\frac{\partial^2 y}{\partial x^2}, \qquad c \;=\; \sqrt{\frac{T}{\rho_\text{lin}}}.

where

$y(x,t)$: transverse deflection m
$T$: tension (a force, not a stress) N
$\rho_\text{lin}$: linear mass density kg/m
$c$: wave speed m/s

The restoring force is geometric: where the string curves, the tension pulling on the two ends of a small segment does not quite cancel, and the leftover points back toward the axis in proportion to the curvature $\partial^2 y/\partial x^2$ . This is the same equation the discrete chain of masses produces in the continuum limit; here it applies to a stretched string of any material. Its dispersion is linear, $\omega = ck$ : every wavelength travels at the same speed, so a plucked shape holds together as it propagates.

Tension T = 1.00 Linear density μ_lin = 1.00

A transverse pulse on a taut string propagates at c = √(T/μ_lin). Raising the tension quadruples-the-stiffness — and the speed doubles. Doubling the mass density slows the wave by √2. The wave equation u_tt = c² u_xx is exactly what you get from applying Newton's second law to an element of the string with tension restoring its curvature.

Membranes and plates: two ways to be stiff

Stretch the string idea to a two-dimensional sheet and two distinct regimes appear, depending on which restoring mechanism dominates.

Membrane — bending stiffness negligible; the restoring force is entirely the in-plane tension, as in a drumhead. The transverse displacement obeys the two-dimensional wave equation $\ddot{w} = c^2\nabla^2 w$ , with the tension-to-density speed of the string. Dispersion is linear, $\omega = ck$ .
Plate — bending stiffness dominant; the sheet resists being curved, as a sheet of metal does. The restoring force involves the fourth spatial derivative of the displacement, and the governing equation is $\ddot{w} = -\dfrac{D}{\rho_A}\nabla^4 w$ . Dispersion is quadratic, $\omega \propto k^2$ .

where

$w$: transverse displacement of the sheet m
$D$: flexural rigidity of the plate N·m
$\rho_A$: area mass density kg/m²
$k$: wavenumber rad/m

Highlight:

A membrane stretches under in-plane tension; the restoring force is tension, and the dispersion is *linear* (ω = ck): all wavelengths travel at the same speed. A plate resists bending via a stiffness D ∝ Eh³; the dispersion is *quadratic* (ω ∝ k²): short wavelengths travel faster than long ones. The basilar membrane is closer to a plate than a membrane, and its quadratic dispersion is what makes the cochlear traveling-wave place-map work.

The difference is not cosmetic. A membrane’s linear dispersion means all wavelengths travel together; a plate’s quadratic dispersion means short wavelengths travel faster than long ones, so a flexural pulse spreads and reshapes as it propagates. This dispersive spreading is why bending waves in plates and beams smear out, why a struck metal sheet rings with a shimmering, pitch-shifting timbre, and — in any graded elastic sheet whose stiffness varies with position — why different frequencies come to rest at different places. The mechanism a structure uses to resist deformation is written directly into the dispersion relation of the waves it carries.