In continuum mechanics, elastic solids support both longitudinal (P-waves, compressional) and transverse (S-waves, shear) modes with speeds c_P = √((λ+2μ)/ρ) and c_S = √(μ/ρ). Under stress, the photoelastic effect induces birefringence proportional to the principal stress difference — the beautiful color fringes seen in stressed glass reveal the internal stress field.
Left: photoelastic fringe pattern (stress birefringence). Right: P-wave (compression) vs S-wave (shear).