All other factors being equal, which mode of vibration has the greatest velocity?

• A. Longitudinal Waves
• B. Shear Waves
• C. Surface Waves
• D. Rayleigh Waves

Answer: (A)

When a wave propagates in a solid, the compressional (longitudinal) mode moves particles in the same direction as the wave, pushing and pulling the material. The speed of this mode is governed by how stiff the material is to compression (bulk modulus) relative to its density, giving v_P = sqrt((K + 4/3 G)/ρ). This compressional stiffness is typically larger than the stiffness that governs shear motion, so the longitudinal wave travels faster than the shear wave, which depends on the shear modulus and has v_S = sqrt(G/ρ). Surface and Rayleigh waves are confined to the near-surface region and have more complex motion; their speeds are generally lower than the bulk shear speed. So, with all other factors equal, the compressional (longitudinal) wave is the fastest.