# Is there any way a longitudinal wave can have a shear velocity component?

I am simulating the propagation of an acoustic field through elastic media. I have identified a pocket of shear stress which is travelling with a speed very close to that of the longitudinal wave speed in the medium (glass optical fibre). The shear wave speed of glass is considerably lower than the measured velocity.

My silly question is: is there any way a longitudinal wave can have a shear velocity component? I'm guessing not, so the follow up is: what kind of wave could this be? It's not in a plate and is not close to the shear velocity so that rules out Rayleigh, Love, Lamb, etc. Are there any other types of exotic wave that fit this criteria?

• Does the medium have to be homogeneous? I wonder if there should be a shear component perpendicular to a gradient in the speed of sound? Is that ruled out by your "no plate" requirement? Commented Sep 28, 2014 at 23:19
• The medium of propagation is homogeneous (glass optical fibre) however it is surrounded by water so there is a change in acoustic impedance at the fibre boundaries. Commented Sep 29, 2014 at 9:57
• The fibre is a cylinder of diameter 125um - so for some wavelengths I suppose this could be considered a plate of sorts... Commented Sep 29, 2014 at 9:58
• Intuitively it appears reasonable to me to that there should be a shear component at the interface of the two media, but I don't think that answers your question. Other than that I can't think of much else. There is still the case of waves on the surface of curved media, but that's an interface, again, because unlike spacetime an acoustic medium has to be embedded in something (i.e. there is, at least, a surrounding vacuum). Commented Sep 29, 2014 at 18:24
• What do you mean by shear velocity? I ask because electrostatic lower hybrid waves (in a plasma) are longitudinal waves (i.e., $\mathbf{k} \times \mathbf{E} = 0$) and they propagate across the magnetic field, which is considered a shear velocity. Though I assume this is not the shear velocity to which you are referring. Commented Mar 31, 2016 at 14:23

Longitudinal waves in a rod (or glass fibre) travel at a speed given by Young's modulus, and as well as longitudinal motion the rod gets thicker or thinner as the wave travels. There is therefore some transverse motion and if the longitudinal strain is $$e_{xx}$$ then the transverse strain is $$e_{xx}=e_{yy}=- \sigma e_{zz}$$ with $$\sigma$$ being Poisson's ration. There is no off diagonal stress in these coordinates, but the shear modulus is involved in determining the speed.