Relation between transverse velocity of sound and Young's modulus in solids

Question

There is a relation which holds in solids: $$v_{t}=\sqrt{\frac{E}{\rho}}$$, where $v_t$ is the the velocity of sound (transversal elastic waves) and $E$ and $\rho$ are Young's modulus and density of the medium respectively.

How would I derive this or can someone give me a reference where the derivation is obtained? (i would guess somehow connect wave equation and Hook's law maybe?)

Answer 1

All you need is the Navier's equation of motion (you can consult it in any book of elasticity)

$$ \rho \frac{\partial^2 w}{ \partial t^2 } = \mu \nabla^2w + (\lambda + \mu)\nabla\nabla.w $$

Naturally, you can decompose a wave $ w$ in a transversal and a longitudinal part:

$$ w = w_L + w_T $$

with the following properties :

$$ \nabla \times w_L = 0$$ $$ \nabla .w_T = 0 $$

If we focused only in the transversal part, retaking the Navier equation we have:

$$ \rho \frac{\partial^2 w_T}{ \partial t^2 } = \mu \nabla^2 w_T $$

Now you can remember that the general wave equation is

$$ \frac{\partial^2 u}{ \partial t^2 } = c^2 \nabla^2 u $$

where c is the phase velocity of the wave $ u $

So, that's it

$$ v_T = \sqrt \frac \mu\rho$$

I hope you help

J.