# 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).

## References

\[1\] 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.

