Skip to content

Wave propagation

Seismic wave propagation

In 3D isotropic seismic wave propagation, the equations governing P-waves (compressional waves) and S-waves (shear waves) are derived from the wave equation for elastic media. These equations describe how seismic displacements propagate through an isotropic, homogeneous elastic medium.

General Wave Equation for Elastic Waves

The displacement vector \(\mathbf{u}(\mathbf{r}, t)\) satisfies:

\[ \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} = (\lambda + 2\mu) \nabla (\nabla \cdot \mathbf{u}) - \mu \nabla \times (\nabla \times \mathbf{u}), \]

where:

  • \(\rho\): density of the medium,
  • \(\lambda, \mu\): Lamé parameters,
  • \(\mathbf{u}(\mathbf{r}, t)\): displacement vector,
  • \(t\): time,
  • \(\nabla\): spatial gradient operator.

P-Wave and S-Wave Components

This general wave equation can be decomposed into P-wave and S-wave components:

P-Wave Equation (Compressional Wave)

P-waves correspond to the irrotational part of the displacement field, governed by:

\[ \nabla^2 \phi - \frac{1}{\alpha^2} \frac{\partial^2 \phi}{\partial t^2} = 0, \]

where:

  • \(\phi(\mathbf{r}, t)\) is the scalar potential such that \(\mathbf{u}_P = \nabla \phi\),
  • \(\alpha = \sqrt{\frac{\lambda + 2\mu}{\rho}}\) is the P-wave velocity.

S-Wave Equation (Shear Wave)

S-waves correspond to the solenoidal (divergence-free) part of the displacement field, governed by:

\[ \nabla^2 \mathbf{\Psi} - \frac{1}{\beta^2} \frac{\partial^2 \mathbf{\Psi}}{\partial t^2} = 0, \]

where:

  • \(\mathbf{\Psi}(\mathbf{r}, t)\) is the vector potential such that \(\mathbf{u}_S = \nabla \times \mathbf{\Psi}\),
  • \(\beta = \sqrt{\frac{\mu}{\rho}}\) is the S-wave velocity.

Full Displacement Field

The total displacement field \(\mathbf{u}(\mathbf{r}, t)\) is the sum of the P-wave and S-wave contributions:

\[ \mathbf{u} = \nabla \phi + \nabla \times \mathbf{\Psi}. \]

Seismic Wave Velocities

P-wave velocity (\(\alpha\))

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

S-wave velocity (\(\beta\))

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

These velocities depend on the elastic properties (\(\lambda, \mu\)) and density (\(\rho\)) of the medium.

Key Notes

  • Isotropy: Assumes uniform properties in all directions.
  • Homogeneity: Assumes constant \(\lambda, \mu, \rho\) in the medium.
  • For layered or anisotropic media, additional terms or adjustments are required.

Demonstration

The demonstration is done for elastic isotropic and homogeneous media.

Starting with Newton's Second Law

In its most basic form, Newton's law states:

\[ F = ma \]

where:

  • \(F\) is the force in \(N/m\) or in \(kg.m.s^{-1}\)
  • \(m\) is the mass subject to acceleration in \(kg\)
  • \(a\) is the acceleration of the mass in \(m.s^{-1}\)

In other words, the force per unit volume is linked to the density of the material:

\[ \frac{F}{V} = \frac{m}{V}.a = \rho a \]

For a continuous medium, this translates to:

\[ \text{force per unit volume} = \text{mass per unit volume} * \text{acceleration} \]

The equation for is:

\[ F_{total} = \rho \frac{\partial^2 u}{\partial t^2} \]

where:

  • \(\rho\) is the mass per unit volume or material density in \(kg/m^3\)
  • \(u\) is the displacement vector, describing motion of particles in the medium in \(m\)
  • \(\frac{\partial^2 u}{\partial t^2}\) is the particle acceleration (second derivative of the position with respect to time)

Recovering the momentum equation

In a continuous elastic medium, there are two main sources of force acting on a small element of material:

  1. Body forces (\(F\)): external forces like gravity or a seismic source acting throughout the volume of the material.
  2. Stress gradient (\(\nabla.\tau\)): internal forces due to the stress variations within the material. Stress represents how forces are distributed inside the medium and is described by a tensor \(\tau\). The gradient of the stress tensor \(\nabla.\tau\) gives the net force acting on the element due to stress imbalances.

Therefore, summing these forces give:

\[ F_{total} = \nabla.\tau+F \]

This equation is the momentum equation, and is the foundation for deriving seismic wave equations. It says that the motion of a material element is driven by internal stress variations and external body forces.

If internal forces are negligible, then \(F\) vanishes and the simplified equations is:

\[ F_{total} = \nabla.\tau \]

Using the stress-strain equation

To solve for the displacement \(u\), we need to relate the stress tensor \(\tau\) to the dusplacement field \(u\). This is done using Hooke's law for a linear, elastic and isotropic material.

\[ \tau_{ij} = \lambda \delta_{ij} (\nabla . u) + 2 \mu e_{ij} = \lambda \delta_{ij} (\nabla . u) + \mu (\partial_i u_j + \partial_j u_i) \]

where:

  1. \(\tau_{ij}\): Stress Tensor

    • Describes internal forces per unit area within the material.
    • Subscripts \(i\) and \(j\) indicate directions:
      • \(i\): Direction of the force.
      • \(j\): Orientation of the surface on which the force acts.

  2. \(\lambda\) and \(\mu\): Lamé Parameters

    • Material-specific constants that describe how a material responds to stress.
    • \(\lambda\): Bulk modulus-related term, governing volume changes.
    • \(\mu\): Shear modulus, governing shape changes (shear deformation).
    • Together, they define the elastic properties of the medium.

  3. \(\delta_{ij}\): Kronecker Delta

    • A mathematical tool that equals:
    \[ \delta_{ij} = \begin{cases} 1, & \text{if } i = j \ (\text{diagonal terms, e.g., \(\tau_{xx}\)}), \\ 0, & \text{if } i \neq j \ (\text{off-diagonal terms, e.g., \(\tau_{xy}\)}). \end{cases} \]

  4. \(e_{ij}\): Strain Tensor

    • Describes deformation of the material:
    • \(e_{ij} = \frac{1}{2} \left( \partial_i u_j + \partial_j u_i \right)\),
    • Symmetric with respect to \(i\) and \(j\),
    • Represents how displacement gradients (\(\partial_i u_j\)) cause material deformation.

  5. \(e_{kk}\): Volumetric Strain

    • Sum of the diagonal components of the strain tensor (\(e_{xx} + e_{yy} + e_{zz}\)):
    \[ e_{kk} = \nabla \cdot \mathbf{u}. \]
    • Represents changes in the material's volume (compression or dilation).

\(\lambda \delta_{ij} e_{kk}\):

  • Describes the isotropic part of stress (uniform compression or dilation).
  • Depends on the volumetric strain \(e_{kk}\).
  • Acts only on the diagonal components of \(\tau_{ij}\) (normal stresses).

\(2\mu e_{ij}\):

  • Describes the deviatoric part of stress (shape-changing or shearing deformation).
  • Depends on the strain tensor \(e_{ij}\).
  • Acts on both diagonal and off-diagonal components of \(\tau_{ij}\).

To the wave equation

By substituting this stress-strain relationship into the simplified momentum equation, we can derive the equations governing the propagation of seismic waves, ultimately leading to the separation of P-waves and S-waves.

\[ \rho \frac{\partial^2 u}{\partial t^2} = \nabla \cdot \left( \lambda \delta_{ij} (\nabla.u) + \mu(\nabla u + (\nabla u)^T) \right) \]

If the material is homogeneous (elasticity parameters does not depend on space) \(\lambda\), \(\mu\) and \(\rho\) are constants:

\[ \rho \frac{\partial^2 u}{\partial t^2} = \lambda \delta_{ij} \nabla \cdot (\nabla \cdot u) + \mu \nabla \cdot (\nabla \cdot u) + \mu \nabla \cdot (\nabla u)^T \]

Simplifications using the vector identity \(\nabla \times (\nabla \times u) = \nabla \cdot (\nabla \cdot u) - \nabla^2 u\) can be done. Also \(\nabla^2 u = \nabla \cdot (\nabla u)^T\).

\[ \rho \frac{\partial^2 u}{\partial t^2} = \lambda \delta_{ij} \nabla \cdot (\nabla \cdot u) + \mu \nabla \cdot (\nabla \cdot u) + \mu (\nabla \cdot (\nabla \cdot u) - \nabla \times (\nabla \times u)) \]

Developing:

\[ \rho \frac{\partial^2 u}{\partial t^2} = \lambda \delta_{ij} \nabla \cdot (\nabla \cdot u) + \mu \nabla \cdot (\nabla \cdot u) + \mu \nabla \cdot (\nabla \cdot u) - \mu \nabla \times (\nabla \times u) \]

And grouping terms in \(\nabla \cdot (\nabla \cdot u)\), we get:

\[ \rho \frac{\partial^2 u}{\partial t^2} = (\lambda + 2 \mu) \nabla \cdot (\nabla \cdot u) - \mu \nabla \times (\nabla \times u) \]

Decomposition in P-waves

Taking the divergence (\(\nabla \cdot\)) of the wave equation gives the equation for P-waves:

\[ \nabla \cdot \left( \rho \frac{\partial^2 u}{\partial t^2} \right) = \nabla \cdot \left[ (\lambda + 2 \mu) \nabla \cdot (\nabla \cdot u) - \mu \nabla \times (\nabla \times u) \right] \]

Since the divergence of a curl is always zero:

\[ \nabla \cdot \left( \rho \frac{\partial^2 u}{\partial t^2} \right) = \nabla \cdot \left[ (\lambda + 2 \mu) \nabla \cdot (\nabla \cdot u) \right] \]

And that the material is homogeneous:

\[ \rho \frac{\partial^2 (\nabla \cdot u)}{\partial t^2} = (\lambda + 2 \mu) \nabla \cdot \nabla \cdot (\nabla \cdot u) \]

Lets detail a bit.

The Laplacian, \(\nabla^2\), is the natural result of applying divergence twice on a scalar field because of symmetry in second-order partial derivatives (Clairaut's theorem):

\[ \nabla \cdot \nabla \cdot (\nabla \cdot u) = \nabla^2 (\nabla \cdot u) \]

Simplifying to (homogeneous medium, \(\rho\) is constant):

\[ \rho \frac{\partial^2 (\nabla \cdot u)}{\partial t^2} = (\lambda + 2 \mu) \nabla^2 (\nabla \cdot u) \]

Define \(\phi = \nabla \cdot u\). \(\phi\) is a scalar field or scalar potential representing compressional component of the wave.

\[ \rho \frac{\partial^2 \phi}{\partial t^2} = (\lambda + 2 \mu) \nabla^2 \phi \]

Or:

\[ \frac{\partial^2 \phi}{\partial t^2} = \frac{\lambda + 2 \mu}{\rho} \nabla^2 \phi \]

Using \(\frac{\lambda + 2 \mu}{\rho} = \alpha^2 = V_p^2\) (P-wave velocity) we get the common notation:

\[ \frac{\partial^2 \phi}{\partial t^2} = \alpha^2 \nabla^2 \phi \]

Decomposition in S-waves

Taking the curl (\(\nabla \times\)) of the wave equation gives the equation for S-waves:

\[ \nabla \times \left( \rho \frac{\partial^2 u}{\partial t^2} \right) = \nabla \times \left[ (\lambda + 2 \mu) \nabla (\nabla \cdot u) - \mu \nabla \times (\nabla \times u) \right] \]

The curl of a gradient is always zero:

\[ \nabla \times \left( \rho \frac{\partial^2 u}{\partial t^2} \right) = \nabla \times \left[ - \mu \nabla \times (\nabla \times u) \right] \]

Lets detail a bit.

Simplifying to (homogeneous medium, \(\rho\) and \(\mu\) do not depend on space coordinates):

\[ \rho \frac{\partial^2 (\nabla \times u)}{\partial t^2} = - \mu \nabla \times \nabla \times (\nabla \times u) \]

But we have the vector identity \(\nabla \times (\nabla \times u) = \nabla (\nabla \cdot u) - \nabla^2 u\).

Taking the curl of this vector identity gives: \(\nabla \times \nabla \times (\nabla \times u) = \nabla \times \nabla (\nabla \cdot u) - \nabla \times \nabla^2 u\).

Lets break down this expression and simplify it:

  1. The curl of a gradient is always zero: \(\nabla \times \nabla (\nabla \cdot u) = 0\)
  2. Taking the curl of the Laplacian is equivalent to taking the Laplacian of the curl: \(\nabla \times \nabla^2 u = \nabla^2 (\nabla \times u)\)

In the end: \(\nabla \times \nabla \times (\nabla \times u) = \nabla^2 (\nabla \times u)\).

\[ \rho \frac{\partial^2 (\nabla \times u)}{\partial t^2} = \mu \nabla^2 (\nabla \times u) \]

Define \(\Psi = \nabla \times u\). \(\Psi\) is a vector field or vector potential representing the shear component of the wave.

\[ \rho \frac{\partial^2 \Psi}{\partial t^2} = \mu \nabla^2 \Psi \]

Using \(\frac{\mu}{\rho} = \beta^2 = V_s^2\) (S-wave velocity) we get the common notation:

\[ \frac{\partial^2 \Psi}{\partial t^2} = \beta^2 \nabla^2 \Psi \]