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

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?)

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.

• Did you understand it? ? If so, you can thank the answer and the effort – J0KerSpin Dec 5 '17 at 20:57