In the physics of continuous media, spatial dispersion is usually described as a phenomenon where material parameters such as permittivity or conductivity have dependence on wavevector.
Normally such a dependence is assumed to be absent for simplicity, however spatial dispersion exists to varying degrees in all materials.
The underlying physical reason for the wavevector dependence is often that the material has some spatial structure smaller than the wavelength of any signals (such as light or sound) being considered.
Since these small spatial structures cannot be resolved by the waves, only indirect effects (e.g. wavevector dependence) remain detectable.
Temporal dispersion represents memory effects in systems, commonly seen in optics and electronics.
Spatial dispersion on the other hand represents spreading effects and is usually significant only at microscopic length scales.
This usually arises due to a spreading of effects by the hidden microscopic degrees of freedom.
like so: which yields a remarkably simple relationship between the two plane waves' complex amplitudes: where the function
In electromagnetism, spatial dispersion plays a role in a few material effects such as optical activity and doppler broadening.
Spatial dispersion also plays an important role in the understanding of electromagnetic metamaterials.
[2] The constitutive relation for the polarization vector can be written as: i.e., the permittivity is a wavevector- and frequency-dependent tensor.
Considering Maxwell's equations, one can find the plane wave normal modes inside such crystals.
Nearby crystal surfaces and boundaries, it is no longer valid to describe system response in terms of wavevectors.
In materials that have no relevant crystalline structure, spatial dispersion can be important.
The non-isotropic permittivity for nonzero wavevector leads to effects such as optical activity in solutions of chiral molecules.
In isotropic materials without optical activity, the permittivity tensor can be broken down to transverse and longitudinal components, referring to the response to electric fields either perpendicular or parallel to the wavevector.
[1] For frequencies nearby an absorption line (e.g., an exciton), spatial dispersion can play an important role.
This is typically represented as a spatially dispersive loss in the plasma's permittivity.
The values of the permeability and permittivity are different in this alternative representation, however this leads to no observable differences in real quantities such as electric field, magnetic flux density, magnetic moments, and current.
As a result, it is most common at optical frequencies to set μ to the vacuum permeability μ0 and only consider a dispersive permittivity ε.
[3] In acoustics, especially in solids, spatial dispersion can be significant for wavelengths comparable to the lattice spacing, which typically occurs at very high frequencies (gigahertz and above).
In solids, the difference in propagation for transverse acoustic modes and longitudinal acoustic modes of sound is due to a spatial dispersion in the elasticity tensor which relates stress and strain.
For polar vibrations (optical phonons), the distinction between longitudinal and transverse modes can be seen as a spatial dispersion in the restoring forces, from the "hidden" non-mechanical degree of freedom that is the electromagnetic field.