Predict where a wave packet contributes

Write a one-dimensional complex wave as an integral of a spectral amplitude times an oscillating exponential. The real measured field is obtained by taking the real part when appropriate. A localized initial shape requires a spread of wavenumbers with coordinated phases. A single exact Fourier component extends throughout space and cannot by itself represent a sharply bounded packet. Narrowing the spectral spread tends to broaden the envelope, although the precise width relation depends on the chosen definition of width and spectral shape.

At position x and time t, the phase is kx−ω(k)t. Its derivative with respect to k is x−t times the group velocity at k. Setting that derivative to zero selects components whose group velocity equals the observation ratio x/t. Nearby spectral contributions then change phase only quadratically at leading order and can reinforce over a comparatively wider interval. This is an interference argument in wavenumber space, not a claim that individual Fourier components are little particles following trajectories.

For large positive time along a fixed ray, expand the phase around an isolated stationary point with nonzero second derivative of the dispersion. If the amplitude varies slowly over the contributing region, replace it by its value at that point and integrate the local quadratic exponential. The contribution scales as the inverse square root of time, multiplied by the spectral amplitude and a curvature-dependent factor. The phase also receives a signed π/4 correction determined by the sign of the quadratic curvature.

Several stationary points may satisfy the same observation ray. Their complex contributions must be added before taking an intensity or absolute value, so interference can remain important. If the spectral amplitude vanishes near a stationary point, that point may contribute little or require a higher-order expansion. If second derivative vanishes, the quadratic estimate is invalid and a different asymptotic scaling may arise. Endpoint contributions and nondecaying spectra also require their own treatment rather than silent use of the isolated interior-point formula.

Consider the illustrative law ω=k² in consistently scaled units. Along a ray x/t=4, the stationary condition selects k=2. A spectrum concentrated near k=7 has little amplitude at that point, so the formula does not predict a strong signal merely because a mathematical stationary point exists somewhere. The initial spectrum and the propagation law work together. Increasing propagation time at fixed ray narrows the contributing wavenumber interval and improves a valid asymptotic approximation rather than creating spectral weight that was absent initially.

Use the spectral lab to inspect one limited ingredient: the effect of including more spatial Fourier modes. Its controls are integer mode count and coefficient-decay exponent, and it displays the finite sum Σ sin(nx−t)/nᵖ. Predict whether stronger coefficient decay suppresses fine spatial structure before testing it. Every mode has the same temporal frequency in this display. It is neither a localized dispersive packet nor a stationary-phase integrator, and cannot verify the ray condition derived above.