The elastic wave equation or momentum equation is given by:
$$\rho \frac {\partial^2 u_i} {\partial t^2} = \ \frac {\partial \sigma_{ij}}{\partial x_j}$$
This equation has remarkable similarities with the 1D and 2D wave equations which are given by
$$\frac {\partial^2 u} {\partial x^2} = \frac 1 {c^2} \frac {\partial^2 u} {\partial t^2}$$
and
$$\frac {\partial^2 u} {\partial x^2}+\frac {\partial^2 u} {\partial y^2} = \frac 1 {c^2} \frac {\partial^2 u} {\partial t^2}$$
For instance, expanding one term of the momentum equation we obtain the following:
$$\rho \frac {\partial ^2 u_x} {\partial t^2} = \frac {\partial \sigma_{xx}}{\partial x}+\frac {\partial \sigma_{xy}}{\partial y}+\frac {\partial \sigma_{xz}}{\partial z}$$
Substituting in the values of $\sigma$ based on the constitutive elastic matrix and the definitions for strain we obtain:
$$\rho \frac {\partial ^2 u_x} {\partial t^2} = \frac{E}{(1+\nu)(1-2 \nu)} \left( \frac \partial {\partial x}\left[ (1-\nu)\frac {\partial u_x}{\partial x} + \nu \frac {\partial u_y} {\partial y} + \nu \frac {\partial u_z}{\partial z}\right] + \frac {\partial} {\partial y} \left[\frac 1 2 (1-2 \nu) (\frac {\partial u_x}{\partial y} + \frac {\partial u_y} {\partial x})\right]+\frac {\partial} {\partial z} \left[\frac 1 2 (1-2 \nu) (\frac {\partial u_x}{\partial z} + \frac {\partial u_z} {\partial x})\right] \right)$$
Which bears the following similarities with the 1D and 2D wave equations:
- The left hand side is the second partial derivative with respect to time.
- The right hand side contains several spatial derivatives taken twice.
- dividing both sides by $\rho$ we see emerge a coefficient not unlike the $1/c^2$ term in the 1D and 2D wave equations.
QUESTIONS
I have 2 questions:
- Does there exist a relationship between the 1D/2D wave equation and the elastic wave equation? Can this relationship be demonstrated?
- What analytical methods can handle a PDE with 4 independent variables and 3 dependent solutions (such as the momentum equation)?