Rayleigh Surface Waves

Rayleigh surface waves with retrograde elliptical trajectory on the free surface of a solid.

Conceptual context

When an elastic wave encounters a free surface (the interface between a solid and vacuum or air), propagation modes appear that do not exist in the bulk. Rayleigh waves are the most important surface mode: they combine longitudinal and transverse displacement in a retrograde elliptical trajectory confined to the vicinity of the surface. They are responsible for most seismic damage in shallow earthquakes and form the basis of modern material characterization techniques.

Unlike body waves (P and S), Rayleigh waves do not spread geometrically in three dimensions but in two, allowing them to transport energy over large distances with less relative attenuation. Their velocity $v_R$ is slightly less than the S-wave velocity of the same medium, and their amplitude decays exponentially with depth.

🟢 Essential level

Rayleigh waves travel along the free surface of a solid. The particles describe retrograde ellipses: at the wave crest they move backward relative to the propagation direction, combining vertical (up-down) and horizontal (forward-backward) motion. It is like watching an ocean wave roll, but in reverse.

[!NOTE] The velocity $v_R$ of these waves is always less than the S-wave velocity ($v_s$) of the same material. For most rocks and metals, $v_R$ lies between 87 and 96 percent of $v_s$.

Rayleigh waves exist only near the surface: their amplitude $A_z$ decreases rapidly with depth $z$. At a depth equal to one wavelength $\lambda_R$, the amplitude is already less than 30 percent of the surface value $A_0$. This makes them ideal for inspecting surface layers without needing access to the material interior.

During an earthquake, Rayleigh waves arrive after P and S (because they are slower), but they produce the most intense and prolonged surface motion: they cause the greatest destruction in buildings.

🔵 Formal level

Basic formulas

The Rayleigh wave velocity is obtained by solving the secular equation:

Rayleigh secular equation

Where $\eta$ is the dimensionless ratio of $v_R$ to $v_s$, and $\xi$ is the ratio of $v_s$ to the P-wave velocity. The secular equation is a sixth-degree polynomial admitting a single real root between 0 and 1 for any stable medium.

In practice, the Viktorov approximation is used, reproducing the exact root with error below 0.5 percent:

Rayleigh wave velocity (approximation)

This formula shows that $v_R$ depends only on $v_s$ and $\nu$. For $\nu$ equal to 0.25, $v_R$ is 0.92 times $v_s$; for $\nu$ approaching 0.5, it reaches 0.955 times $v_s$.

Real student questions

Why is the Rayleigh wave slower than S if it has a P component (which is faster)?

The Rayleigh wave does not inherit the velocity of its components separately. The free-surface boundary condition couples P and S waves such that the coherent combination travels slower than either constituent alone. It is an effect of confined constructive interference.

If amplitude decays with depth, why are Rayleigh waves detected thousands of km from the epicentre?

The exponential decay is with depth, not horizontal distance. Along the surface, the wave attenuates much more gently (as the inverse square root of distance) because it propagates in 2D, not 3D. That is why it dominates distant earthquake records.

What is the difference between a Rayleigh wave and an ocean wave?

Ocean waves are gravity waves in a fluid; particles move in prograde ellipses (at the crest, motion is in the propagation direction). Rayleigh waves are elastic waves in a solid and particles move in retrograde ellipses (at the crest, backward). Also, ocean waves depend on gravity, while Rayleigh waves depend only on elasticity.

Can a Rayleigh wave be used to detect surface cracks?

Yes. Since amplitude is concentrated near the surface, any surface defect (crack, corrosion, delamination) interrupts propagation and generates detectable reflections. Non-destructive SAW (Surface Acoustic Waves) testing exploits this principle in the aerospace and microelectronics industries.

Final synthesis

Rayleigh waves are surface modes combining P and S deformation under the free-surface condition. Their velocity $v_R$ depends exclusively on the Poisson ratio and $v_s$, their amplitude decays exponentially with depth, and their effective penetration $d_{pen}$ is on the order of one wavelength. These three properties make them surface diagnostic tools in seismology, geotechnics, and industrial testing.