Which statement about the relationship between shear and longitudinal wave velocities in solids is most accurate?

• A. Shear velocity is equal to longitudinal velocity
• B. Shear velocity is about half of the longitudinal velocity
• C. Shear velocity is about twice the longitudinal velocity
• D. Shear velocity is much greater than the longitudinal velocity

Answer: (B)

In solids, the compressional (P) wave travels faster than the shear (S) wave because their speeds depend on different elastic properties. The P-wave speed is determined by both the bulk modulus and the shear modulus, while the S-wave speed depends only on the shear modulus. Mathematically, c_p = sqrt((K + 4/3 G)/ρ) and c_s = sqrt(G/ρ). Since the bulk modulus K is generally larger than G, c_p is greater than c_s, with the typical ratio around 1.7. That means the shear wave velocity is about 0.6 of the longitudinal (P-wave) velocity—roughly half. So the statement that shear velocity is about half of the longitudinal velocity aligns with the common behavior of wave speeds in solids. The other ideas—that they’re equal, that shear is much greater, or that shear is twice the longitudinal velocity—don’t reflect how the moduli and density set these speeds.