= Solution
\b[<Seismic surface-wave dispersion>.] <Rayleigh waves> combine vertical and longitudinal motion in the vertical plane of propagation; <Love waves> have horizontal transverse motion and can be guided by a low-shear-speed surface layer. A homogeneous elastic half-space supports a <Rayleigh wave> with a frequency-independent speed in the ideal elastic limit, but depth-dependent <Earth> structure makes the surface-wave <eigenfunctions> and their propagation speeds depend on <frequency>. This structural <dispersion> can occur even when each constituent material is itself nondispersive.
Distinguish <phase velocity> $c=\omega/k$ from <group velocity> $U=d\omega/dk$, which controls a narrow-band envelope in the appropriate weakly attenuating limit. Longer <periods> generally have deeper sensitivity, while shorter <periods> emphasize shallower structure. In a profile whose shear speed increases with depth this often makes long-period waves faster, but low-velocity zones, anisotropy and higher modes can change that trend. <Frequency> dependence need not be monotonic in every <Earth> model.
Measure phase delays or wave-packet arrival times across known source–receiver paths for many <periods>, accounting for source phase, cycle ambiguity, multipathing and mode identification. Compute predicted <dispersion> from trial elastic profiles and compare with observations. Linearization expresses the sensitivity schematically as
$$
\frac{\delta c(\omega)}{c(\omega)}=
\int\left[K_\beta(z,\omega)\frac{\delta\beta}{\beta}
+K_\alpha(z,\omega)\frac{\delta\alpha}{\alpha}
+K_\rho(z,\omega)\frac{\delta\rho}{\rho}\right]dz,
$$
where $\alpha,\beta$ are compressional and shear speeds. A regularized <inverse problem> then estimates depth structure; many crossing paths additionally provide <surface-wave tomography>. Joint <Love wave> and <Rayleigh wave> measurements help distinguish shear-wave polarizations and constrain radial <seismic anisotropy>. Finite bandwidth, depth averaging and correlated sensitivities mean that one <dispersion> curve does not uniquely determine a detailed <Earth> profile.
\b[<Seismic ray theory>, the <eikonal equation> and <Fermat's principle>.] In a smoothly varying isotropic medium, insert a high-frequency ansatz $u=A(\mathbf x)e^{i\omega[T(\mathbf x)-t]}$ into the scalar <wave equation>. The leading powers of <frequency> give
$$
\boxed{|\nabla T|^2=\frac1{c(\mathbf x)^2}.}
$$
The travel-time surfaces are <wavefronts> and rays are their normal trajectories, with <slowness> <vector> $\mathbf p=\nabla T$. The next-order equation transports <amplitude>, including geometrical spreading; attenuation is an additional material effect.
<Fermat's principle> says that the ray makes $\int ds/c$ stationary under path variations with fixed endpoints. Stationarity, rather than always the global minimum, permits multiple arrivals. Its Euler–Lagrange equation, with arclength $s$ and unit tangent $\hat{\mathbf t}$, is
$$
\frac{d}{ds}\left(\frac{\hat{\mathbf t}}c\right)
=\nabla\left(\frac1c\right).
$$
For horizontal layering, the horizontal <slowness> $p_h=\sin i/c$ is conserved, giving <Snell's law> at interfaces and smooth refraction between them. Here $i$ is measured from vertical. A turning point satisfies $p_hc(z)=1$; increasing speed with depth bends a downgoing ray back upwards. Travel-time tomography uses these ray integrals to infer spatial <slowness> variations.
The approximation requires <wavelength> to be small compared with structural variation scales. Caustics, turning-point <amplitudes>, diffraction, scattering and finite-frequency sensitivity require wave-based corrections. Thus <seismic ray theory> gives an economical geometric description and useful travel times, while surface-wave <dispersion> retains essential interference and boundary-mode physics.
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2015/paper-1-surface-waves-and-rays.png]
{title=Original illustrative Love-wave dispersion and depth sensitivity, alongside a turning ray}
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