Engineering Acoustics/Wave Motion in Elastic Solids

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Wave types[edit]

In an infinite medium, two different basic wave types, dilatational and distortional, can propagate in different propagation velocities. Dilatational waves cause a change in the volume of the medium in which it is propagating but no rotation; while distortional waves involve rotation but no volume changes. Having displacement field, strain and stress fields can be determined as consequences.

                                                       Figure 1: Dilatational wave

                                                       Figure 1: Distortional wave

Elasticity equations[edit]

Elasticity equations for homogeneous isotropic elastic solids which are used to derive wave equations in Cartesian tensor notation are

Conservation of momentum

Conservation of moment of momentum

Constitutive equations (which relate states of deformation with states of traction)

Strain-displacement relations

in which is the stress tensor, is the solid material density, and is the vector displacement. is body force, and are Lame constants. and are strain and rotation tensors.

Wave equations in infinite media[edit]

Substituting Eq. (4) in Eq. (3), and the result into Eq. (1) gives Navier’s equation (governing equations in terms of displacement) for the media

The displacement equation of motion for a homogeneous isotropic solid in the absence of body forces may be expressed as

Displacement can advantageously be expressed as sum of the gradient of a scalar potential and the curl of a vector potential

with the condition . The above equation is called Helmholtz (decomposition) theorem in which and are called scalar and vector displacement potentials. Substituting Eq. (7) in Eq. (6) yields

Equation (8) is satisfied if



Equation (9a) is a dilatational wave equation with the propagation velocity of . It means that dilatational disturbance, or a change in volume propagates at the velocity . And Eq. (9b) is a distortional wave equation; so distortional waves propagate with a velocity in the medium. Distortional waves are also known as rotational, shear or transverse waves.

It is seen that these wave equations are simpler than the general equation of motion. Therefore, potentials can be found from Eq. (9) and the boundary and initial conditions, and then the solution for displacement will be concluded from Eq. (7).


[1] Wave Motion in Elastic Solids; Karl F. Graff, Ohio State University Press, 1975.

[2] The Diffraction of Elastic Waves and Dynamic Stress Concentration; Chao-chow Mow, Yih-Hsing Pao, 1971.