Engineering Acoustics/Wave Motion in Elastic Solids

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

Conservation of momentum


 * $$ \tau_{ij,j} + \rho f_i = \rho{ \ddot{u_i}} ,\ (1)$$

Conservation of moment of momentum


 * $$ \tau_{ij} = \tau_{ji} ,\ (2)$$

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


 * $$ \tau_{ij,j} = \lambda \epsilon_{kk}\delta_{ij} + 2 \mu \epsilon_{ij} ,\ (3)$$

Strain-displacement relations


 * $$ \epsilon_{ij} = {1 \over 2}(u_{i,j}+u_{j,i}) ,\ (4a)$$


 * $$ \omega_{ij} = {1 \over 2}(u_{i,j}-u_{j,i}) ,\ (4b)$$

in which $$\scriptstyle\tau$$ is the stress tensor, $$\scriptstyle\rho$$ is the solid material density, and $$\scriptstyle\mathbf{u}$$ is the vector displacement. $$\scriptstyle f$$ is body force, $$\scriptstyle\lambda$$ and $$\scriptstyle\mu$$  are Lame constants. $$\scriptstyle\epsilon$$ and $$\scriptstyle\omega$$ are strain and rotation tensors.

Wave equations in infinite media
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


 * $$ ( \lambda + 2\mu )u_{j,ji} + \mu u_{i,jj}+ \rho f_i = \rho{ \ddot{u_i}} .\ (5)$$

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


 * $$ ( \lambda + 2\mu )\nabla(\nabla \cdot \mathbf{u}) + \mu\nabla^2\mathbf{u} = \rho{ \ddot{\mathbf{u}}} .\ (6)$$

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


 * $$ \mathbf{u} = \nabla \phi\ + \nabla \times\psi ,\ (7)$$

with the condition $$ \nabla \cdot \psi =0$$. The above equation is called Helmholtz (decomposition) theorem in which $$\scriptstyle\phi$$ and $$\scriptstyle\psi$$ are called scalar and vector displacement potentials. Substituting Eq. (7) in Eq. (6) yields


 * $$ [( \lambda + 2\mu )\nabla^2 \phi\ - \rho\frac{\partial^2 \phi}{\partial t^2}]+\nabla \times\ [\mu\nabla^2\psi - \rho\frac{\partial^2 \psi}{\partial t^2}] =0 .\ (8) $$

Equation (8) is satisfied if


 * $$ c_p^2 \nabla^2 \phi =\frac{\partial^2 \phi}{\partial t^2}$$    where   $$ c_p^2 =\frac{( \lambda + 2\mu )}{\rho} ,\ (9a)$$


 * $$ c_s^2 \nabla^2 \psi =\frac{\partial^2 \psi}{\partial t^2}$$    where   $$ c_s^2 =\frac{\mu }{\rho} .\ (9b)$$

Equation (9a) is a dilatational wave equation with the propagation velocity of $$\scriptstyle c_p$$. It means that dilatational disturbance, or a change in volume propagates at the velocity $$\scriptstyle c_p$$. And Eq. (9b) is a distortional wave equation; so distortional waves propagate with a velocity $$\scriptstyle c_s$$ 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).