Acoustic waves

When pressure is the only restoring force (i.e., g=0), Eqn. 41 becomes,

and assuming a compressible gas (i.e., ), this simplifies by taking the square root of the dot product with k to,

the dispersion relation for acoustic waves. Hence acoustic waves are longitudinal (v₁ is along k), non-dispersive (phase velocity v_p = ω/k = c_s and group velocity v_g = dω/dk = c_sare equal), and isotropic (phase speed does not vary with k).