10.6 The WKB approximation

Real media are rarely uniform: the sound speed varies with depth in the ocean, the density with height in the atmosphere, the potential with position for a quantum particle. When the medium changes slowly — little over a single wavelength — a wave adapts to it locally, and the WKB approximation makes that adaptation precise. It closes the chapter by turning the uniform-medium plane wave into a wave that tracks a varying medium.

A locally-plane wave

Where the medium varies slowly, try a wave that is plane over any small region but whose wavelength and amplitude drift with position:

\psi(x,t) \;=\; A(x)\,e^{i\phi(x) - i\omega t},

with the phase $\phi(x)$ varying rapidly and the amplitude $A(x)$ slowly.

▶ Local dispersion and the amplitude rule Derivation

Substitute the ansatz into the wave equation $\psi'' + k^2(x)\psi = 0$ , where $k(x)$ is the local wavenumber the medium would support. The phase and amplitude separate by their rates of variation. At leading order the rapidly-varying terms give the local dispersion relation

\phi'(x)^2 = k^2(x) \;\;\Longrightarrow\;\; \phi(x) = \int k(x)\,dx,

so the wave simply accumulates local phase. At next order the equation relating amplitude to phase reads $2A'\phi' + A\phi'' = 0$ , which integrates to

A(x) \propto \frac{1}{\sqrt{k(x)}}.

where

$k(x)$: local wavenumber, set by the medium at $x$ rad/m
$\phi(x)$: accumulated phase rad
$A(x)$: slowly-varying amplitude varies

What the amplitude rule means

The result $A\propto 1/\sqrt{k}$ says the amplitude grows where the wavelength contracts. Its origin is energy conservation: the energy flux of the wave is proportional to $A^2 k$ (amplitude-squared times the propagation factor), and if no energy is added or removed as the wave advances, $A^2 k$ must stay constant, forcing $A\propto 1/\sqrt{k}$ . Where the medium squeezes the wavelength short, the same energy is packed into a smaller region and the amplitude rises.

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.

This single rule organises a wide range of slowly-varying-medium phenomena. It explains why ocean swells steepen and rise as they run into shoaling water, why atmospheric and internal gravity waves grow in amplitude as they climb into thinner air, and — carried into quantum mechanics, where it is the semiclassical approximation — why a particle’s wavefunction amplitude is largest where it moves slowest. The approximation holds wherever the fractional change in wavelength over one wavelength is small; it breaks down precisely where $k(x)\to 0$ , at a turning point, where the wave can propagate no further and reflects. The WKB method is the bridge between the exact plane wave of a uniform medium and the behaviour of real waves in the graded media they actually travel through.