As far as I know, vacuum is the only dispersion free medium for electromagnetic waves. This makes me wonder if there are any other dispersion free media for these waves? (Experimentally established or theoreticaly predicted) If there are none for electromagnetic waves, are there any for other kinds of waves?
[Physics] Dispersion-less media
dispersionwaves
Related Solutions
Electromagnetic field induces polarisation and magnétisation in the media, which are not an instantaneous response. This results in k-vector being frequency-dependent, hence the group velocity, $$ v_g=\frac{d\omega}{dk}=\left(\frac{dk}{d\omega}\right)^{-1} $$ is different from the phase velocity $$ v_{ph}=\frac{\omega}{k}, $$ which is what we call dispersion.
Update
Dispersion and causality section of the Wikipedia article on permittivity gives a rather good review of the relevant .EM equations
Wave equation
I think the question is vaguely posed, since the answer depends on what we define as waves and wave equations. In the question cited in the OP many answers simply assumed that waves mean electromagnetic waves and wave equations means
$$
\partial_t^2u(\mathbf{x},t)=c^2\nabla^2u(\mathbf{x},t).$$
The dispersion relation in this case is obvious:
$$\omega^2-c^2\mathbf{k}^2=0.$$
Linear equations
One could talk about waves in more general sense, as solutions to any linear equation, solvable via Fourier transform, i.e., having solutions
$$
u(\mathbf{x},t) =\int d\mathbf{k}\int d\omega \tilde{u}(\mathbf{k},\omega)e^{i(\mathbf{k}\mathbf{x}-\omega t)},$$
in which case any linear operator would suffice
$$F(\partial_t, \nabla)u(\mathbf{x},t)=0.$$
By choosing function $F(\partial_t, \nabla)$ one could get almost anything. E.g.,
$$\partial_t^4u(\mathbf{x},t)=a\nabla^8u(\mathbf{x},t) + \nabla^4u(\mathbf{x},t) + cu(\mathbf{x},t)$$
has several dispersion branches.
Among more basic equations with several branches one could cite Dirac equation and Klein-Gordon equation (the latter being simply the wave equation with a constant term added).
Non-linear equations
One could go even further and consider non-linear equations that allow running solutions of the type $$f(\mathbf{k}\mathbf{x}-\omega t),$$ such as, e.g., Korteveg-de Vries equation or Sine-Gordon equation.
Which of these equations do happen?
In university physics courses one typically deals with linear theories, because the fundamental phsyics is described (mainly?) by linear theories. In more domain-specific courses one however quickly encounters equations that have higher derivatives or non-linear terms. The domains to look for more complex equations are:
- hydrodynamics
- elasticity theory
- electrodynamics of non-linear media
- non-linear theory (which deals more specifically with the equations rather than their physical content).
Remarks
First-order equations One can have also first-order wave equations, e.g., $$\partial_t u(x,t)\pm v\partial u(x,t)=0,$$ which give actial travelling wave solutions of type $f(x-vt)$. In more dimensions: $$\partial_t u(\mathbf{x},t)-\mathbf{v}\cdot\nabla u(\mathbf{x},t)=0.$$ The nuance of these equations is taht they have a preferred direction for the wave propagation (even in 1D we have either right- or left-moving wave, depending on the sign). This is why the physical theories that are symmetric in space and/or time usually have second (or generally even) partial derivatives.
One example of an equation with such a first-order term is the Navier-Stokes equation, although it is a non-linear one (but it can be linearized to give simple wave solutions).Waves vs. running waves When dealing with general form of equation $F(\partial_t, \nabla)u(\mathbf{x},t)=0$, it is necessary to keep in mind that, although it is solvable by Fourier transform, its solutions are not necessarily running waves of the form $f(\omega t-\mathbf{k}\mathbf{x})$. Requiring that solutions have this form would restrict the type of the differential operators that can be used, e.g., excluding diffusion equation.
Schrödinger equation On the other hand, Schrödinger equation (which can be viewed as a diffusion equation with complex coefficients) is certainly considered a wave equation and its solutions are often referred to as matter waves, even though they are not running waves in the restricted sense mentioned above.
Broad/flat band limit in some solid state phsyics problems one considers a broad-band limit where all electrons are assumed to have the same wave function (or wave number), while possibly ahving different energies - this can be interpreted as a continuum of frequencies corresponding to the same wavelength. The opposite and also used is the flat-band limit, where one assumes that all the wave numbers correspond to the same energy/frequency.
Best Answer
Dispersion can be locally nought and effectively nought over quite wide bands in engineered devices.
In an optical fiber, for example, the phase delays of the modal fields depends on two things:
The dispersion of the material making up the fiber and
The frequency dependence of the modal eigenvalue equation, which depends on the refractive index profile and is there even if the fiber could be made out of dispersionless materials.
The second order term in the phase as function of wavelength Taylor series determines, to first order, the total dispersion. In optical fibers, the two mechanisms above contribute exactly mutually cancelling dispersions from effects 1 and 2 in certain wavelength bands. For example, a step index profile fiber typically has quite a wide, low dispersion band at 1300nm wavelength. This band can be shifted by engineering the refractive index profile (the so called depressed cladding index profile) into the 1550nm communications band.
Grating devices can hold a system dispersionless over stunningly wide wavelength bands. For instance, carefully designed "chirped " fiber Bragg gratings are used to exactly oppose the dispersion of optical systems over hundreds of nanometers bandwidth, and thus are useful in setting up femtosecond long pulses.