Question

What is the relationship between longitudinal and transverse waves?

Answer 1

S-waves are shear waves and L-waves (P-waves) are longitudinal waves. The wave speed for longitudinal waves is given by

cL=λ+2μρ−−−−−−√cL=λ+2μρ
and for shear waves, the speed is given by

cS=μρ−−√cS=μρ

.

Here λλ and μμ are the so-called Lamé parameters. These parameters can of course be expressed in terms of EE and νν, 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, cL>cscL>cs. However, I don't see how the S-wave speed can be determine from L-wave speed unless you know both EE and νν up front. I assume by L-wave, you mean, longitudinal wave.

For steel, ρ=7800ρ=7800 kg/m33, E=200E=200e9 Pa and ν=0.3ν=0.3. For these values,

λ=Eν(1+ν)(1−2ν)≈115.38 GPa and μ=E2(1+ν)≈76.92 GPa.λ=Eν(1+ν)(1−2ν)≈115.38 GPa and μ=E2(1+ν)≈76.92 GPa.

For these values,

cL≈5875 m/s and cS≈3140 m/s.cL≈5875 m/s and cS≈3140 m/s.

If you need more information on this, you could look into the book Wave motion in elastic solids by Karl Graff.

Answer 2

Hey mate here is your answer in attachment