For aluminum and steel, the longitudinal velocity is approximately which multiple of the shear velocity?

• A. Equal to
• B. Twice
• C. Half of
• D. Four times

Answer: (B)

In metals the compressional (longitudinal) wave travels faster than the shear wave because it uses both the bulk and shear restoring forces, while the shear wave only uses shear restoring forces. In isotropic solids the speeds are related to the elastic moduli and density: Vs = sqrt(μ/ρ) for the shear wave, and Vp = sqrt((λ+2μ)/ρ) for the longitudinal wave. The ratio Vp/Vs = sqrt((λ+2μ)/μ) = sqrt(2 + λ/μ). Poisson’s ratio ν links λ/μ to a typical metal value; for aluminum and steel, ν is about 0.33, which makes λ/μ ≈ 1.9. Plugging in gives Vp/Vs ≈ sqrt(2 + 1.9) ≈ sqrt(3.9) ≈ 2. So the longitudinal velocity is about twice the shear velocity. In practical terms, Vs is around 3.0 km/s in these metals, while Vp is around 6.0 km/s, illustrating this near-2:1 relationship.