# Utility of displacements potentials in geophysics

In the elasticity theory, you can derive a wave equation from the fundamental equation of motion for an elastic linear homogeneous isotropic medium:

$\rho \partial^2_t \overline{u} = \mu \nabla^2 \overline{u} + (\mu+\lambda) \nabla(\nabla \cdot \overline{u})$

But in the seismology tradition, you introduce scalar and vector potentials for the P and S components of the displacements, derive wave equations for them and use them.

Now, in electrodynamics you can derive from Maxwell's equations the wave equations for Fields and for Potentials; but there you use the potentials because they compose a quadrivector. In geophysics what's the convenience of it?

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I edited your question for grammar and spelling a bit, hope you don't mind. Feel free to roll back if you disagree with the changes, but please leave the tags I added. –  j.c. Nov 27 '10 at 17:29

I believe potentials in Maxwell's equations were introduced originally to make solving equations simpler -- it wasn't until a bit later that the Lorentz symmetry was noticed. Similar potentials are introduced in 2D hydrodynamics as well, see e.g. stream function. So I would say that the reason for introducing these potentials originally was always to simplify the algebra.

Of course, whenever one has some nice trick like this, there's a deeper reason. Here the "deeper reason" for introducing various potentials has to do with the Helmholtz decomposition (even deeper, coming from the Hodge decomposition), which guarantees that vector fields can be written as a sum of a curl-free vector field and a divergence-free vector field. This decomposition "plays nicely" with wave equations in the sense that there are natural wave modes, transverse (divergence-free) and longitudinal (curl-free) which propagate independently. See the discussion of transverse and longitudinal, i.e. the application of the Helmholtz decomposition to a Fourier transform of a vector field, which obviously is highly applicable to the P (longitudinal) and S (transverse) waves in seismology.

Intuitively, the idea of the Helmholtz decomposition is a bit like breaking a vector into independent components, and then calculating what happens to each separately, except now one is working with vector fields.

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Also, adding to what j.c. states:

If you have a complicated constitutive relation between the electric field and the electric displacement field, e.g. nonlinear and in terms of a convolution (no instantanious reaction of the medium), it might be very complicated to find a representation in terms of potentials obaying a wave equation using a non-perturbative ansatz.

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