Questions: Elastic Wave Propagation in Solids

Question 1

What material property is required for a solid to support S-waves (shear waves) but not required for P-waves?

• A. High density, which resists shear deformation
• B. A non-zero shear modulus G, reflecting the material's resistance to shear strain
• C. Low bulk modulus K, allowing the material to compress easily
• D. Crystalline atomic structure rather than amorphous arrangement

Answer: (B)

P-wave (compressional) velocity depends on bulk modulus K and shear modulus G as v_P = sqrt((K + 4G/3)/ρ). S-wave velocity is v_S = sqrt(G/ρ). If G = 0 (as in a liquid, which cannot resist shear), S-wave velocity is zero — shear waves cannot propagate. P-waves can still propagate in liquids because bulk modulus K remains non-zero. This is why seismic S-waves vanish in Earth's liquid outer core.

Question 2

If the bulk modulus K of a solid increases while its density ρ stays constant, the P-wave velocity in that solid increases.

• T. True
• F. False

Answer: True

P-wave velocity is v_P = sqrt((K + 4G/3)/ρ). Increasing K with ρ fixed directly increases the numerator inside the square root, so v_P increases. Physically, a stiffer material (higher K) resists compression more strongly and returns to equilibrium faster, propagating the disturbance at higher speed. This is why seismic P-wave velocities are higher in denser, stiffer mantle rocks than in softer crustal materials.

Question 3

Why is the elastic wave equation derived by applying Newton's second law to a continuous volume element rather than to individual atoms?

Model answer: Seismic wavelengths (meters to kilometers) are vastly larger than atomic spacings (angstroms), so discrete atomic structure is irrelevant at the scales of interest. Treating the solid as a continuous elastic medium — where stress and strain are defined for infinitesimal volume elements — is an excellent approximation. Newton's second law applied to such a volume element, combined with the linear stress-strain constitutive relation (Hooke's law), yields the wave equation governing macroscopic elastic deformation.

This continuum mechanics approach is a recurring strategy in physics: when the phenomenon of interest operates on length scales much larger than the microscopic structure, you replace discrete atoms with continuous fields. The validity of this approximation breaks down only when wavelengths approach atomic dimensions (e.g., in phonon physics), which is irrelevant for seismology.