Just as in electromagnetic theory, it is sometimes useful to write the displacement in terms of scalar and vector potentials: (The vector potential A

Just as in electromagnetic theory, it is sometimes useful to write the displacement ξ in terms of scalar and vector potentials:

(The vector potential A is, as usual, only defined up to a gauge transformation, A → A + ∇φ, where φ is an arbitrary scalar field.) By inserting Eq. (12.13) into the general elastodynamic wave equation (12.4b), show that the scalar and vector potentials satisfy the following wave equations in a homogeneous solid:

Thus, the scalar potential ψ generates longitudinal waves, while the vector potential A generates transverse waves.

Equation (12.4b)

= V +Vx A. (12.13)