What is the relationship between longitudinal and transverse waves? As the title says, is there any relationship between these two waves in a solid?
I'm currently working in NDT ultrasound technologies and I need to calibrate a system by determining the speed of a L-wave, and from this value determine the speed of the S-wave associated. I read a bit on different sites but figured this would be the best place to ask.
I looked for graphs showing different curves of wave-type velocity vs material type but didn't find anything really usefull.
I'm also a 2nd year physics student so bring them concepts and equations on!
 A: S-waves are shear waves and L-waves (P-waves) are longitudinal waves. The wave speed for longitudinal waves is given by 
$$ c_L = \sqrt{\frac{\lambda + 2\mu}{\rho}}$$
and for shear waves, the speed is given by
$$ c_S = \sqrt{\frac{\mu}{\rho}}$$.
Here $\lambda$ and $\mu$ are the so-called Lamé parameters. These parameters can of course be expressed in terms of $E$ and $\nu$, the Young's modulus and Poisson's ratio respectively.
For S-waves, the particle motion is transverse to the direction of wave propagation. On the other hand, for P-waves, the particle motion is parallel to the wave propagation direction.
So, as you can see from the above, $c_L > c_s$. However, I don't see how the S-wave speed can be determine from L-wave speed unless you know both $E$ and $\nu$ up front. I assume by L-wave, you mean, longitudinal wave.
For steel, $\rho = 7800$ kg/m$^3$, $E = 200$e9 Pa and $\nu = 0.3$. For these values, 
$$
\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)} \approx 115.38\text{ GPa and }
\mu = \frac{E}{2(1+\nu)} \approx 76.92 \text{ GPa}.
$$
For these values,
$$
c_L \approx 5875 \text{ m/s} \text{ and } c_S \approx 3140 \text{ m/s}.
$$
If you need more information on this, you could look into the book Wave motion in elastic solids by Karl Graff.
