= Solution
For an approximately <isotropic> <rock>, the <P wave> and <S wave> speeds are
$$
V_p=\sqrt{\frac{K+4\mu/3}{\rho}},\qquad V_s=\sqrt{\frac{\mu}{\rho}},
$$
where $K$ is <bulk modulus>, $\mu$ is <shear modulus>, and $\rho$ is <density>. Five important controls are:
* \b[Mineralogy and cementation.] Stiff, well-cemented <mineral> frameworks usually have higher <elastic modulus> values and faster <seismic velocities> than poorly consolidated <sediments>. <Density> alone is insufficient: both stiffness and <density> enter the equations.
* \b[Porosity and cracks.] Larger <porosity>, especially compliant cracks, usually lowers effective <elastic modulus> values and <seismic velocities>. Crack orientation can make the <seismic velocities> direction-dependent through <seismic anisotropy>.
* \b[Pore-fluid composition and saturation.] Replacing liquid <water> by <gas> generally lowers the effective <bulk modulus> sharply and reduces $V_p$. <Pore fluids> have negligible static <shear modulus>; $V_s$ responds chiefly through the solid framework and the changed bulk <density>, rather than through a shear restoring force in the fluid.
* \b[Effective confining pressure.] Increasing <effective stress> closes cracks and improves grain contacts, usually increasing both <seismic velocities>. Increasing <pore fluid pressure> at fixed total <stress> reduces <effective stress> and can reverse this trend.
* \b[Temperature.] At otherwise comparable conditions, increasing <temperature> generally softens <minerals> and lowers <seismic velocities>; thermal cracking or partial <melting> can lower them further. Increasing burial commonly increases both <temperature> and <pressure>, so their separate effects need not have the same sign.
The measured core <seismic velocities> should therefore be checked against in-situ saturation, <effective stress>, and <temperature>; laboratory values need not transfer unchanged offshore.
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