Chapter 3 Lecture: The Seismic Wave Equation and Wave Properties

Purpose¶

In previous lectures, we described seismic waves using the displacement field:

\mathbf{u}(\mathbf{x}, t)

(1)

But what governs how this displacement evolves in space and time?

The answer is the wave equation.

In this lecture, we introduce the seismic wave equation and show how it leads naturally to two fundamental types of motion: P waves and S waves. We then develop a simple description of wave propagation using plane waves and introduce key diagnostic quantities such as velocity, wavelength, and slowness.

Learning Objectives¶

By the end of this lecture, you should be able to:

Describe the physical meaning of the seismic wave equation
Explain why elastic media support both P and S waves
Distinguish between compressional and shear motion
Define plane waves and their key properties
Understand and interpret the slowness vector
Relate mathematical descriptions of waves to observable quantities

The Wave Equation¶

A simple example of wave behavior is given by the 1D wave equation:

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}

(2)

This equation describes waves that:

propagate at a constant speed $c$
maintain their shape as they move
can travel in both directions

The key idea is that acceleration is proportional to curvature: regions where the displacement is curved experience restoring forces that drive wave motion.

The Seismic Wave Equation¶

In an elastic solid, the displacement field satisfies:

\rho \ddot{\mathbf{u}} = (\lambda + 2\mu)\nabla(\nabla \cdot \mathbf{u}) - \mu \nabla \times \nabla \times \mathbf{u}

(3)

Each term has a clear physical meaning:

$\rho \ddot{\mathbf{u}}$ : inertia (mass × acceleration)
$\nabla(\nabla \cdot \mathbf{u})$ : compressional deformation
$\nabla \times \nabla \times \mathbf{u}$ : shear deformation

Key insight:

The equation naturally separates into two types of motion:
- Compression (volume change)
- Shear (shape change)

P and S Waves¶

These two types of motion correspond to two fundamental seismic waves.

P Waves (Compressional)¶

Motion is parallel to the direction of propagation
Involve volume change (compression and dilation)

\alpha = \sqrt{\frac{\lambda + 2\mu}{\rho}}

(4)

S Waves (Shear)¶

Motion is perpendicular to the direction of propagation
Involve shear deformation only (no volume change)

\beta = \sqrt{\frac{\mu}{\rho}}

(5)

Key Observations¶

$\alpha > \beta$ : P waves travel faster than S waves
S waves depend on $\mu$ , so they do not propagate in fluids
P waves depend on both $\lambda$ and $\mu$

Plane Waves¶

To describe wave propagation, we consider a simple solution: a plane wave.

A harmonic plane wave can be written as:

\mathbf{u}(\mathbf{x}, t) = \mathbf{A} e^{i(\mathbf{k}\cdot\mathbf{x} - \omega t)}

(6)

where:

$\mathbf{A}$ : amplitude
$\mathbf{k}$ : wavenumber vector
$\omega$ : angular frequency

Interpretation¶

$\mathbf{k}$ points in the direction of propagation
$\omega$ controls how fast the wave oscillates in time
The wave varies sinusoidally in both space and time

Surfaces of constant phase are planes perpendicular to $\mathbf{k}$ .

Wave Properties¶

\omega = 2\pi f, \quad \lambda = \frac{c}{f}, \quad k = \frac{2\pi}{\lambda} = \frac{\omega}{c}

(7)

Frequency $f$ : oscillations per second
Wavelength $\lambda$ : spatial scale of the wave
Velocity $c$ : links time and space variation

Slowness¶

It is often useful to define the slowness vector:

\mathbf{s} = \frac{\mathbf{k}}{\omega}

(8)

Direction: same as propagation direction
Magnitude: $|\mathbf{s}| = \frac{1}{c}$

Interpretation¶

The dot product:

[ \mathbf{s} \cdot \mathbf{x} ]

represents travel time from the origin.

👉 This makes slowness especially useful for:

describing wave propagation
interpreting travel times
connecting to ray theory

Particle Motion¶

The direction of motion distinguishes P and S waves:

P waves: motion is parallel to propagation
S waves: motion is perpendicular to propagation

S waves can have different polarizations:

SV waves: motion in the vertical plane
SH waves: motion perpendicular to that plane

Big Picture¶

The seismic wave equation provides a unified framework for understanding wave propagation:

It describes how displacement evolves in space and time
It predicts two fundamental types of waves (P and S)
It links physical properties of Earth to observable wave behavior

Using simple solutions like plane waves, we can define key quantities—such as velocity, wavelength, and slowness—that form the foundation of seismic analysis.