Editing Wave equation (section)

==Further generalizations ==

===Elastic waves===

The elastic wave equation (also known as the [[Linear elasticity#Elastodynamics in terms of displacements|Navier–Cauchy equation]]) in three dimensions describes the propagation of waves in an [[isotropic]] [[Homogeneity (physics)|homogeneous]] [[elastic (solid mechanics)|elastic]] medium. Most solid materials are elastic, so this equation describes such phenomena as [[seismic waves]] in the [[Earth]] and [[Ultrasound|ultrasonic]] waves used to detect flaws in materials. While linear, this equation has a more complex form than the equations given above, as it must account for both longitudinal and transverse motion:
<math display="block">
 \rho \ddot{\mathbf{u}} = \mathbf{f} + (\lambda + 2\mu) \nabla(\nabla \cdot \mathbf{u}) - \mu\nabla \times (\nabla \times \mathbf{u}),
</math>
where:
: {{mvar|λ}} and {{mvar|μ}} are the so-called [[Lamé parameters]] describing the elastic properties of the medium,
: {{mvar|ρ}} is the density, 
: {{math|'''f'''}} is the source function (driving force), 
: {{math|'''u'''}} is the displacement vector.

By using {{math|1= ∇ × (∇ × '''u''') = ∇(∇ ⋅ '''u''') − ∇ ⋅ ∇ '''u''' = ∇(∇ ⋅ '''u''') − ∆'''u'''}}, the elastic wave equation can be rewritten into the more common form of the Navier–Cauchy equation.

Note that in the elastic wave equation, both force and displacement are [[vector (geometry)|vector]] quantities. Thus, this equation is sometimes known as the vector wave equation.
As an aid to understanding, the reader will observe that if {{math|'''f'''}} and {{math|∇ ⋅ '''u'''}} are set to zero, this becomes (effectively) Maxwell's equation for the propagation of the electric field {{math|'''E'''}}, which has only transverse waves.

===Dispersion relation===

In [[Dispersion (optics)|dispersive]] wave phenomena, the speed of wave propagation varies with the wavelength of the wave, which is reflected by a [[dispersion relation]]

<math display="block">\omega = \omega(\mathbf{k}),</math>

where {{mvar|ω}} is the [[angular frequency]], and {{math|'''k'''}} is the [[wavevector]] describing [[plane-wave]] solutions. For light waves, the dispersion relation is {{math|1=''ω'' = ±''c'' {{abs|'''k'''}}}}, but in general, the constant speed {{mvar|c}} gets replaced by a variable [[phase velocity]]:

<math display="block">v_\text{p} = \frac{\omega(k)}{k}.</math>