7.4 Elastic waves: P-waves and S-waves

Give an elastic solid its equation of motion and it supports waves — two kinds, travelling at two speeds. The compressional wave carries changes of volume; the shear wave carries changes of shape. That a solid supports both, while a fluid supports only the first, is the central diagnostic of seismology and the sharpest expression of what elasticity adds over fluid mechanics.

The Navier–Cauchy equation

Apply Newton’s second law to a small element of the continuum. The net force per unit volume is the divergence of the stress tensor, so

\rho\,\ddot{\mathbf{u}} \;=\; \nabla\cdot\boldsymbol{\sigma}.

▶ Substituting Hooke's law Derivation

Insert the isotropic constitutive law $\sigma_{ij} = \lambda\,\varepsilon_{kk}\,\delta_{ij} + 2\mu\,\varepsilon_{ij}$ and use $\varepsilon_{ij} = \tfrac12(\partial_i u_j + \partial_j u_i)$ . Taking the divergence, the volumetric term contributes $\lambda\,\partial_i(\nabla\cdot\mathbf{u})$ and the shear term contributes $\mu\,[\partial_i(\nabla\cdot\mathbf{u}) + \nabla^2 u_i]$ . Collecting,

\rho\,\ddot{\mathbf{u}} \;=\; (\lambda + \mu)\,\nabla(\nabla\cdot\mathbf{u}) \;+\; \mu\,\nabla^2\mathbf{u}.

This is the Navier–Cauchy equation, the equation of motion of an isotropic elastic solid.

Two waves from the Helmholtz decomposition

Any vector field splits into a curl-free part and a divergence-free part. Applied to the displacement, this separates the Navier–Cauchy equation into two independent wave equations.

▶ Longitudinal and transverse waves Derivation

Write $\mathbf{u} = \mathbf{u}_L + \mathbf{u}_T$ with $\nabla\times\mathbf{u}_L = 0$ (longitudinal) and $\nabla\cdot\mathbf{u}_T = 0$ (transverse). For the longitudinal part $\nabla^2\mathbf{u}_L = \nabla(\nabla\cdot\mathbf{u}_L)$ , so the equation becomes $\rho\,\ddot{\mathbf{u}}_L = (\lambda + 2\mu)\nabla^2\mathbf{u}_L$ . For the transverse part $\nabla\cdot\mathbf{u}_T = 0$ , so the first term drops and $\rho\,\ddot{\mathbf{u}}_T = \mu\,\nabla^2\mathbf{u}_T$ . Each is a wave equation, with speed the square root of the stiffness over density.

The two speeds are

c_P \;=\; \sqrt{\frac{\lambda + 2\mu}{\rho}} \;=\; \sqrt{\frac{K + \tfrac43 G}{\rho}}, \qquad c_S \;=\; \sqrt{\frac{\mu}{\rho}} \;=\; \sqrt{\frac{G}{\rho}}.

where

$c_P$: P-wave (longitudinal) speed m/s
$c_S$: S-wave (shear) speed m/s
$K$: bulk modulus Pa
$G$: shear modulus Pa
$\rho$: mass density kg/m³

K (bulk)1.50

G (shear)0.60

c_P/c_S1.96

Bulk modulus K = 1.50 Shear modulus G = 0.60

The P-wave (top) is compressional: particles oscillate *along* the propagation direction; the medium alternately compresses and rarefies. The S-wave (bottom) is shear: particles oscillate *perpendicular* to the propagation direction. P-waves are always faster (involve both K and G); S-waves involve only G — they don't exist in fluids (where G = 0). The c_P / c_S ratio is always > √2 and diverges as the material approaches incompressibility.

The P-wave (primary, longitudinal) is compressional: particles oscillate along the propagation direction, and both the bulk modulus $K$ and shear modulus $G$ resist the motion. The S-wave (secondary, shear) is transverse: particles oscillate across the propagation direction, resisted by $G$ alone. Since $c_P$ carries the extra $\tfrac43 G$ under the root, the P-wave is always faster — it arrives first, which is how it earned its name.

Fluids, solids, and seismology

A fluid has $G = 0$ : no shear stiffness, hence no shear wave. It transmits only the compressional wave, at speed $\sqrt{K/\rho}$ — the ordinary speed of sound in a liquid or gas. A solid, carrying both, is distinguished precisely by its ability to transmit shear.

The consequence organises seismology. For steel ( $E\approx 200\,\text{GPa}$ , $\nu\approx 0.3$ , $\rho\approx 7800\,\text{kg/m}^3$ ), $c_P\approx 5900\,\text{m/s}$ and $c_S\approx 3200\,\text{m/s}$ . An earthquake radiates both; because they travel at different speeds, the P-wave outruns the S-wave, and the P–S arrival delay at a seismometer grows linearly with distance to the source — the basic ruler for locating an earthquake. And because the Earth’s liquid outer core kills the S-wave, the shadow it casts across the far side revealed the core to be molten.