If someone may, I expect a mathematical comparison of the governing equations "Maxwell Equations" for wave propagation and "Conservation of momenta & equilibrium equations" for stress and deformation. I am confused if a stress wave exists? If yes, how similar it is to EM waves?


2 Answers 2


The displacements field equations can be derived for a cubic small element $\Delta x \Delta y \Delta z$ of a body, by taking the components of the net force on it:

$$F_{x+\Delta x} - F_x = ma_x$$ $$F_{y+\Delta y} - F_y = ma_y$$ $$F_{z+\Delta z} - F_z = ma_z$$

$$(\sigma_{xx(x + \Delta x)} - \sigma_{xx(x)})\Delta y \Delta z + (\sigma_{yx(y + \Delta y)} - \sigma_{yx(y)})\Delta x \Delta y + (\sigma_{zx(z + \Delta z)} - \sigma_{zx(z)})\Delta x \Delta z = ma_x$$

Dividing by $\Delta x \Delta y \Delta z$ and taking the limit when deltas go to zero:

$$\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{yx}}{\partial y} + \frac{\partial \sigma_{zx}}{\partial z} = \rho a_x = \rho \frac{\partial^2 u_x}{\partial t^2}$$ where $\rho$ is the density and $u_x(x,y,z)$ is the displacement at the point to the $x$ direction. The same for the $y$ and $z$.

For the element doesn't rotate, $\sigma_{ij} = \sigma_{ji}$. Considering the material linear elastic, there are linear relations between stresses and deformations:

$$\sigma_{ij} = E_{ijkl}\epsilon_{kl}$$

where E is the elasticity matrix.

As $$\epsilon_{kl} = \frac{1}{2}\left(\frac {\partial u_k}{\partial x_l} + \frac {\partial u_l}{\partial x_k}\right)$$

We end up with three differential equations of second order, to find the 3 components of the displacement field $u(x,y,z)$. In the left side there are second derivatives with respect to x, y and z. At the right side second derivatives with respect to time. There is some similarty with the wave equation for EM, but more parameters and variables. The solutions depends on boundary conditions.


I am reaching way back in my memory, but I recall that in seismology there are at least two distinct types of mechanical wave propagation. Both are wave phenomena (and are thus modeled as solutions of wave equations), but one is more similar to electromagnetic waves than the other is.

  • Shear waves or S-waves are transverse waves, in that the displacement occurs perpendicular to the direction of propagation, as in electromagnetic waves.

    EDIT February 2021: But there is a crucial difference between S-waves and electromagnetic waves: electromagnetic waves involve two distinct fields (yes, I know that E and B fields are part of a unified field, but the propagation really depends on their interaction). Further, S-waves require a physical medium to carry them. Electromagnetic waves require no aether.

  • P-waves, on the other hand, are pressure waves, and the compression and dilation occur along the direction of propagation.

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