Question

Derive an expression for the velocity of the transverse wave in the [100] direction in a
cubic crystal.

Answer 1

Answer on Question #51561, Physics, Solid State Physics

Derive an expression for the velocity of the transverse wave in the [100] direction in a cubic crystal.

Solution

For cubic crystals, the equation of motion can be written as:

$$\rho \ddot{u}_x = c_{11} \frac{\partial^2 u_x}{\partial x^2} + c_{44} \left( \frac{\partial^2 u_y}{\partial y^2} + \frac{\partial^2 u_x}{\partial z^2} \right) + (c_{12} + c_{44}) \left( \frac{\partial^2 u_y}{\partial x \partial y} + \frac{\partial^2 u_z}{\partial x \partial z} \right)$$

where $\rho$ is the mass density and $u_x$ is the $x$ component of the displacement $\vec{u}$. The corresponding equations of motion along $y$ and $z$ can be found by cyclic permutation.

The transverse wave in the [100] direction in a cubic crystal

$$v_T [100] = \sqrt{c_{44} / \rho}$$

Answer: $v_T [100] = \sqrt{c_{44} / \rho}$.

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