In a homogeneous material, the velocity of shear waves compared to longitudinal waves is approximately

• A. Approximately equal to the longitudinal velocity
• B. Approximately 1/2 the longitudinal wave velocity
• C. Approximately twice the longitudinal wave velocity
• D. Approximately 3/4 the longitudinal wave velocity

Answer: (B)

In a homogeneous solid, the speeds of the two main wave types come from different elastic properties. Longitudinal (P) waves depend on both the bulk modulus and the shear modulus, while shear (S) waves depend only on the shear modulus. Since the bulk stiffness is usually larger than the shear stiffness, P-waves travel faster than S-waves. The standard formulas are Vp = sqrt((K + 4/3 μ)/ρ) for longitudinal waves and Vs = sqrt(μ/ρ) for shear waves, where K is the bulk modulus, μ is the shear modulus, and ρ is density. Taking the ratio gives Vp/Vs = sqrt(K/μ + 4/3). For typical materials, this ratio works out to about 1.7 to 2.0, meaning Vs is roughly half of Vp. For example, steel has Vp around 5900 m/s and Vs around 3100 m/s, which is close to half.