GG 5210/6211 Seismology I

Problem Set #9

Due November 27, 2006

Elastic Coefficients, Constitutive Relationships and Wave Propagation

1. Using the data below, calculate the P- and S-velocities for the following materials. Do this by writing a Matlab or Excel script with the formulas for velocities from elastic coefficients. I suggest also simultaneously plotting the results as a table of coefficients below the resultant velocity for each material.

Note the large differences between solids, incompressible fluids, water, and the compressible fluids such as O₂, H₂ and N₂ (that fill up the pore spaces in gas-filled rocks).

Material	Shear Rigidity	Bulk Modulus	Density
--	MBar	MBar	gm/cc
Soft Iron	0.816	1.698	7.8
Glass	0.292	0.412	3.3
Ice	0.035	0.098	0.97
Rubber	10.89 Bars	5445 Bars	1.50
Water	-0-	0.224	1.0
H2	-0-	1.417 Bars	7.9 x 10^-5
N2	-0-	1.396 Bars	1.1 x 10^-3
Basalt	0.30	0.595	2.91
Limestone	0.23	0.704	2.60
Dunite	0.68	1.20	3.26
Westerly Granite	0.32	0.550	2.62
Olivine	0.82	12.9	3.2
Perovskite	1.53	26.6	4.1

1. Show that for an incompressible fluid (like water), that Poisson's ratio ν→0.5 and V_p/V_s = ∞.
2. How does this result effect the interpretation of wave propagation in a fluid-filled body?

2. Determine whether the Lame's constant can be negative and, if so, under what physical conditions.

3. Give a physical interpretation of the fact that Young's modulus for rubber is less than for steel.

4. Suppose there are two rival rock-type models for the composition of the lower continental crust. One has a gabbroic composition and the other claims a granulite composition. Seismological evidence shows the S-wave velocity in the lower crust to be 4.0 km sec^-1. Using the data tabulated below, which model is supported best by the seismological evidence?

Material	Shear Rigidity	Bulk Modulus	Poisson's Ratio	Density
---	Mbar	Mbar	unitless	gm/cc
Gabbro	0.44	0.89	----	2.9
Granulite	----	0.95	0.25	3.5