Surface Waves: Rayleigh & Love

Surface Waves: Rayleigh & Love#

Learning Goals#

  • Derive dispersion relations for Rayleigh and Love waves

  • Interpret particle motions (retrograde Rayleigh, transverse Love)

  • Connect theory to measurements in Labs 4–6

Rayleigh Waves (vertical–radial, retrograde)#

  • Boundary conditions at the free surface lead to the Rayleigh equation: $\(\left(2 - \beta^2\right)^2 - 4\sqrt{1 - \beta^2}\,\sqrt{1 - \alpha^2 \beta^2} = 0\)\( where \)\beta = c_R / V_S\( and \)\alpha = V_P / V_S$.

  • Dispersion in layered media: \(c_R(f)\) depends on near-surface \(V_S(z)\).

Love Waves (transverse, SH)#

  • SH confined to a low-velocity layer over a faster half-space yields:

    \[\tan(k_{z1} h) = \frac{\rho_2 k_{z1}}{\rho_1 k_{z2}}\]

    leading to discrete modes and dispersion curves \(c_L(f)\).

  • Particle motion: horizontal, transverse to propagation.

Particle Motion & Energy#

  • Rayleigh: retrograde elliptical in the vertical–radial plane.

  • Love: purely transverse; no vertical component.

Practical Workflow#

  • Lab 4 (theory): derive/inspect dispersion relations and particle motion.

  • Lab 5 (practicum): measure dispersion (MFT, phase/group velocities).

  • Lab 6 (Love waves): focus on SH-guided waves; compare with Rayleigh.

  • Use these lecture notes for equations; keep notebooks focused on computation.

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