Shear Waves vs. Compression Waves

I would think that exactly 1/2 the speed would be partially based upon amplitude and frequency. As to why they travel slower (I'm guessing here):

Following a sine wave centered about 0 along the x axis... for compression waves, starting at 0 degrees you have the front of the first wave. At 180 degrees you have the front for the second wave, and at 360 degrees the front for the third wave (think full wave rectifier). For shear waves you do not see the front for the first wave until 90 degrees. the second wave front hits at 270, the third is then 90 on the next wave. So the wave fronts for both types are generated from a single sine wave and are both 180 degrees apart, 90 degree phase shift from one another. However, if you look at the original sign wave, it spends less time at the zero line (just crosses over it) where the compression wave fronts are generated. The crest and valley (90 and 270) create the shear wave fronts at a point of tangency. The compression waves would then be sharper and appear to "come and go" faster than the shear waves which would be less pronounced appearing to "come and go" slower.

You could also trig out the wave and determine the length of the wave segment for each to check the motion / length of each segment and compare it to the time / movement along the x axis.

Take a peek at this and run the animation. http://einstein.byu.edu/~masong/HTMstuff/C13A4.html

If you come up with a more definitive answer, please post it. It is interesting.

This is not exactly 1/2!

Compression is governed by lengthwise stiffness, so compression wave velocity by square root of (elastic modulus / mass).

Shear is governed by shear modulus.

The relation of shear modulus to elastic modulus is near 0.3 in metals but can range from below 0.2 to 0.5 - depending on compressibility.

So shear wave velocity is governed by (shear velocity/mass).

RHABE