Geometric Optics and WKB

Geometric optics and WKB replace a rapidly oscillating wave by a local phase, rays, and slowly transported amplitudes. Their control parameter compares the wavelength with every scale sampled by the solution; large frequency by itself is not enough. The general wave hierarchy belongs to Waves, Optics, and Radiation.

Helpful background. Differential equations for relativity supplies the principal-symbol and transport-equation language used in the WKB hierarchy.

The short-wavelength hierarchy

Let $L$ be no larger than the shortest background, envelope, wavefront, boundary, or mode-splitting scale, and define

$$\epsilon=\frac{\lambda}{L}\ll1.$$

For a declared linear wave operator, use the local ansatz

$$\Psi=e^{iS/\epsilon} \left(A_0+\epsilon A_1+\epsilon^2A_2+\cdots\right), \qquad k_\mu=\nabla_\mu S.$$

Successive powers of $\epsilon$ have distinct jobs:

Order	Equation	Meaning
Leading	$p(x,k)=0$ for the principal symbol $p$	Eikonal or local dispersion relation
Leading characteristics	$dx^\mu/ds=\partial p/\partial k_\mu$, $dk_\mu/ds=-\partial p/\partial x^\mu$	Rays in phase space
Next	Transport equation for $A_0$ and any polarization	Flux dilution, focusing, and internal-state transport
Later	Equations for $A_1,A_2,\ldots$	Finite-wavelength corrections and a truncation residual

This phase, ray, and transport ordering is the Liouville–Green/WKB hierarchy; its usefulness depends on an error estimate over the stated interval, not on the formal series alone Bender and Orszag 1999, Ch. 10, printed pp. 484–543.

For a massless wave whose principal symbol is $p=\tfrac12g^{\mu\nu}k_\mu k_\nu$, the leading equations become

$$k^\mu k_\mu=0, \qquad k^\nu\nabla_\nu k_\mu=0.$$

For the standard lossless scalar normalization, the next equation is

$$\nabla_\mu\left(A_0^2k^\mu\right)=0.$$

Vacuum electromagnetism adds a transverse screen-polarization class that is parallel transported modulo gauge. The curved-spacetime derivations are on The Eikonal Expansion, Light Rays as Null Geodesics, and Amplitude and Polarization Transport.

What the scale ratio does—and does not—control

The ratio $\lambda/L$ must stay small along the entire ray segment and across the beam width used by the observable. A locally $O(\epsilon)$ correction can accumulate over a sufficiently long propagation distance, so the interval and norm belong to the approximation statement. One should record the wave operator, branch of the dispersion relation, initial or boundary data, polarization basis, variables held fixed, truncation order, and residual.

WKB is not the same as a far-zone expansion: distance from a source does not make the wavelength short relative to the medium or curvature. It is also not automatically an adiabatic approximation in time, although the two can share a small parameter in a particular problem. In a dispersive medium, the relevant principal symbol is the constitutive dispersion relation; metric-null rays and vacuum polarization transport cannot be imported unchanged.

Where WKB fails and how to proceed

Failure surface	Diagnostic	Appropriate method
Turning point	A local wave number vanishes or two characteristic branches coalesce	Airy-type or other uniform approximation and connection formula
Caustic	The ray-bundle Jacobian vanishes and one-phase transport predicts divergent amplitude	Multiphase or uniform diffraction treatment
Sharp boundary or aperture	Coefficients or data vary on the scale $\lambda$	Boundary-layer, diffraction, or direct wave solution
Mode degeneracy or conversion	Eigenvalue separation competes with transport coupling	Coupled-mode or matrix WKB analysis
Dispersion or absorption	Frequency-dependent or complex principal symbol	Re-derived rays, transport, and flux for the material model
Stokes transition	A subdominant exponential changes across a complex-sector boundary	Branch, contour, and Stokes multiplier tracked explicitly

Ordinary local WKB is nonuniform at a turning point. Uniform turning-point approximations and their error control are organized in NIST DLMF 1.2.7, § 2.8, while remainder and Stokes questions are treated in § 2.11. The construction and matching logic are developed in Bender and Orszag 1999, Ch. 10, printed pp. 484–543.

Three quick checks discriminate a controlled result. Constant coefficients must recover plane waves with constant collimated amplitude. A lossless spherical ray tube in flat space must give $A_0\propto r^{-1}$. Substituting the truncated field into the original wave equation must leave a residual of the claimed order in the declared norm. For practical matching, multiple scales, and uniform repairs, use Asymptotics and Matched Expansions.

References

• Bender, Carl M., and Steven A. Orszag. Advanced Mathematical Methods for Scientists and Engineers I. Springer, 1999, Ch. 10, printed pp. 484–543. DOI.
• National Institute of Standards and Technology. Digital Library of Mathematical Functions, Release 1.2.7, June 15, 2026, §§ 2.8 and 2.11. § 2.8. § 2.11.