Longitudinal and Transverse Elastic Waves

Take a homogeneous isotropic solid initially at rest, with small displacement and small strain. The divergence of the stress tensor is the internal force per volume. Newton’s law equates it to density times displacement acceleration, after any body force has been included. Insert the isotropic elastic constitutive law and the symmetric displacement-gradient strain. Because the elastic constants are spatially uniform, the derivatives collect into a vector Laplacian and the gradient of displacement divergence.

The resulting equation already shows two mechanisms. The shear modulus multiplies spatial bending of the displacement field, while the other term responds to local compression. Density is mass per volume, not a modulus, and supplies inertia. No damping, material anisotropy, or frequency-dependent constitutive response is included here. The continuum approximation also presumes wavelengths large enough compared with the material microstructure. These assumptions matter when comparing the simple prediction with ultrasound in a textured, layered, or strongly dissipative sample.

Consider a plane wave with a wavevector pointing in a fixed direction. If its displacement is parallel to that direction, displacement divergence is nonzero and both restoring terms contribute. Their sum is the longitudinal modulus, the first Lamé constant plus twice the shear modulus. For displacement perpendicular to the wavevector, divergence vanishes. Only the shear contribution remains. There are two independent transverse polarizations in three spatial dimensions, with the same speed in an isotropic solid.

Positive bulk and shear moduli ensure that the squared speeds are positive. The longitudinal modulus can also be written as the bulk modulus plus four thirds of the shear modulus; that form makes the connection with compression and distortion especially clear. Do not substitute Young’s modulus into this bulk longitudinal formula. A slender rod with free lateral surfaces can contract sideways, and its long-wavelength extensional speed instead approaches the square root of Young’s modulus divided by density. The geometry and boundary constraints determine which formula applies.

In this ideal bulk medium each polarization has a linear frequency–wavenumber relation, so its phase and group speeds coincide. A localized pulse therefore does not disperse through this particular mechanism. Boundaries can reflect waves and convert one polarization into another; a finite body can also support surface and guided modes that require additional boundary calculations. None of those effects is determined by adding two scalar sine curves. An ordinary fluid has no static shear modulus in this elastic model and consequently has no corresponding propagating bulk shear wave.

In the waves lab, set the amplitude ratio to one and predict the sum for zero phase difference and for a half-turn phase difference. The curves reinforce in the first case and cancel in the second because both displayed components have the same scalar coordinate, frequency, and wavenumber. This is an interference experiment, not a comparison of longitudinal and transverse polarizations. Two orthogonally polarized vector displacements do not cancel simply because their scalar amplitudes have opposite signs. The controls do not change density, elastic moduli, or either physical wave speed derived above.