Engineering Acoustics/Wave Motion in Elastic Solids

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).

References edit

[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.