Assumptions and definitions

Consider a medium that is

• homogeneous, that is, its properties do not change in space
• isotropic, that is, its properties do not change with direction
• infinite, that is, without bounding surfaces
• linear elastic, that is, every deformation is directly proportional to the force that produced it
• non-dissipative, that is, the total mechanical energy is conserved during the deformation

and consider

• small deformations
• with negligible body forces

Let

• ${\displaystyle 𝝃=(u,v,w)}$ be the vector of the Cartesian components of the displacement
• ${\displaystyle \lambda ,\mu }$ be the Lamé constants. They are related to the more common elastic moduli: ${\displaystyle \mu }$ is the shear modulus G, while ${\displaystyle \lambda +2\mu =K+\frac{4}{3}\mu }$, where K is the bulk modulus
• ${\displaystyle \rho }$ be the density of the medium

Equations of motion

The equation of motion of an elementary volume of the medium is

${\displaystyle \mu {\mathrm{\nabla }}^2𝝃+(\lambda +\mu )\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝝃)=\rho \frac{{\partial }^2𝝃}{\partial t^2}}$

In Cartesian components this becomes the system

${\displaystyle \mu {\mathrm{\nabla }}^{2}u+(\lambda +\mu )\frac{\partial \mathrm{\Delta }}{\partial x}=\rho \frac{{\partial }^{2}u}{\partial t^2}}$

${\displaystyle \mu {\mathrm{\nabla }}^{2}v+(\lambda +\mu )\frac{\partial \mathrm{\Delta }}{\partial y}=\rho \frac{{\partial }^{2}v}{\partial t^2}}$

${\displaystyle \mu {\mathrm{\nabla }}^{2}w+(\lambda +\mu )\frac{\partial \mathrm{\Delta }}{\partial z}=\rho \frac{{\partial }^{2}w}{\partial t^2}}$

where we have renamed the volumetric strain

${\displaystyle \mathrm{\Delta }=\mathrm{\nabla }\cdot 𝝃=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}}$

From these equations of motion two distinct wave equations can be derived.

First wave equation

The first is a wave equation for the scalar field ${\displaystyle \mathrm{\Delta }}$

${\displaystyle {\mathrm{\nabla }}^2\mathrm{\Delta }=\frac{\rho }{\lambda +2\mu }\frac{{\partial }^2\mathrm{\Delta }}{\partial t^2}}$

and describes longitudinal (compressional) waves propagating with phase velocity

${\displaystyle v=\sqrt{\frac{\lambda +2\mu }{\rho }}}$

These waves correspond to the sound waves propagating in the medium. They are also called "primary waves", or "P waves".

Second wave equation

The second is a wave equation for the vector field of the rotation ${\displaystyle 𝝎=\frac{1}{2}\mathrm{\nabla }\times 𝝃}$

${\displaystyle {\mathrm{\nabla }}^2𝝎=\frac{\rho }{\mu }\frac{{\partial }^2𝝎}{\partial t^2}}$

and describes transverse (shear) waves propagating with phase velocity

${\displaystyle v=\sqrt{\frac{\mu }{\rho }}}$

that is, a lower velocity than that of the longitudinal waves of the first equation. For this reason these waves are also called "secondary waves", or "S waves".

Special case: one-dimensional propagation

When propagation is one-dimensional, it is enough to substitute into the wave equations the conditions

${\displaystyle \frac{\partial 𝝃}{\partial y}=\frac{\partial 𝝃}{\partial z}=0}$

The first equation easily reduces to

${\displaystyle \frac{{\partial }^2}{\partial x^2}\frac{\partial u}{\partial x}=\frac{\rho }{\lambda +2\mu }\frac{{\partial }^2}{\partial t^2}\frac{\partial u}{\partial x}}$

which shows how the volume element undergoes successive compressions and expansions in the direction of propagation of the wave.

The second equation reduces to the system

${\displaystyle \frac{{\partial }^2}{\partial x^2}\frac{\partial v}{\partial x}=\frac{\rho }{\mu }\frac{{\partial }^2}{\partial t^2}\frac{\partial v}{\partial x}}$

${\displaystyle \frac{{\partial }^2}{\partial x^2}\frac{\partial w}{\partial x}=\frac{\rho }{\mu }\frac{{\partial }^2}{\partial t^2}\frac{\partial w}{\partial x}}$

which shows how the volume element crossed by the (shear) wave undergoes distortions that do not change its volume.

Surface waves

The previous cases show that in an infinite medium only two distinct types of waves are possible: P waves and S waves. In a finite medium, that is, in the presence of an interface between two different media, the reflection and superposition of the body waves (whether P or S) at the interface can produce new types of waves.

Surface waves are so called because their amplitude decays exponentially with distance from the interface. Consequently their energy is concentrated at the surface and propagates essentially in two dimensions: their amplitude decays with the distance r from the source as ${\displaystyle \frac{1}{\sqrt{r}}}$ rather than as ${\displaystyle \frac{1}{r}}$, as happens for body waves.

Rayleigh waves

They are surface waves produced by the combination of P waves and (vertically polarised) S waves at the surface. If the medium is homogeneous their phase velocity is 92% of that of the S waves that generated them; otherwise they are dispersive. The points reached by a Rayleigh wave move along ellipses, in the retrograde sense with respect to the motion of the wave. The size of the ellipses decreases with distance from the surface.

Love waves

They are surface waves produced by the reflection of S waves at the surface in a medium in which the velocity of S waves decreases as the distance from the surface decreases. Their speed lies between that of S waves in the surface layer and that of S waves in the deeper layers, and depends on frequency (they are dispersive). The points reached by a Love wave move perpendicular to the direction of propagation of the wave and parallel to the surface.

Further reading and links

For a more elementary overview of the various types of waves and their properties, see the pages

• what is a wave
• transverse and longitudinal waves
• how is a wave described?
• speed of mechanical waves
• the sound wave
• speed of sound

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