speed… of what (1)

Of elastic wave…

To an elastic body, if the body is isotropic and homogeneous, and if deformation of the body is small, the theory of linear elasticity is applicable.  Assuming no external force drives deformation of the body, the equation of motion can be written in this way :

\displaystyle  \mu \triangle {\bf u} + (\lambda + \mu) \nabla \nabla \cdot {\bf u} = \rho \ddot{{\bf u}}.\ \ \ \ (1)


  • \lambda, \mu: Lamé’s parameters;
  • {\bf u}:  displacement vector at each position of the body;
  • \rho: density of material.

The equation (1) looks scary.  And it does not look the same as a simplest form of wave equations:

\displaystyle c^2 \triangle u = \ddot u.\ \ \ \ (2)

where c is the speed (the phase velocity) of the wave.

Before playing with tricks, let me go straightforward first.  Decomposing the displacement vector {\bf u} into its components [u, v, w]^T, the equation (1) is rewritten as:

\displaystyle  \begin{array}{rcl}  \mu ( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}) + (\lambda + \mu) ( \frac{\partial^2 u}{\partial x \partial x} + \frac{\partial^2 v}{\partial x \partial y} + \frac{\partial^2 w}{\partial x \partial z}) = \rho \ddot{u} & & \\ \mu ( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}) + (\lambda + \mu ) ( \frac{\partial^2 u}{\partial y \partial x} + \frac{\partial^2 v}{\partial y \partial y} + \frac{\partial^2 w}{\partial y \partial z}) = \rho \ddot{v} & & \\ \mu ( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}) + (\lambda + \mu) ( \frac{\partial^2 u}{\partial z \partial x} + \frac{\partial^2 v}{\partial z \partial y} + \frac{\partial^2 w}{\partial z \partial z}) = \rho \ddot{w} & & \end{array}

This system has got more scary.  It turns out that the equation (1) is actually a system of 3 equations.  It is hard to separate the system into 3 independent equations, since components u, v,w, are entangled with each other in the system.

Where can I find the speed c in this system? So far so bad. There are several ways to capture speed.  Let me dig a bit deeper next time.

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