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This question is pretty much the same as What ties high frequency electromagnetic waves to short wavelength and converse? but much more technical, and seeking a more technical answer than any provided. This question may bleed in the territory of math stack exchange. Can migrate if others think thta would be best.

It is known that wave equations permit solutions of the form

$$ f(\boldsymbol{x}, t) = e^{i(\boldsymbol{k}\cdot \boldsymbol{x} - \omega t)} $$

However, this function only solves the wave equation if $|\boldsymbol{k}|$ and $\omega$ are related by a specific function

$$ \omega = D(|\boldsymbol{k}|) $$

For simple waves we have that $D(|\boldsymbol{k}|)$ is linear with $D(|\boldsymbol{k}|) = v|\boldsymbol{k}|$. $v$ can be interpreted as the velocity of the wave.

For more complex media or wave equations the dispersion function $D(|\boldsymbol{k}|)$ may be non-linear, in which case we say the medium exhibits dispersion.

However, in any case, we see that $D(|\boldsymbol{k}|)$ is a single valued function of $|\boldsymbol{k}|$. If I'm being imaginative I could imagine a medium that allows multiple values of $\omega$ for a single value of $|\boldsymbol{k}|$.

In fact, I think such media are possible. Examples that come to mind are

  • different phonon modes supported in solid state materials
  • different transverse modes supported in multi-mode optical fibers (though I'm not sure if this counts because I think if you consider the total magnitude of $|\boldsymbol{k}| = \sqrt{|\boldsymbol{k}_{\perp}|^2 + k_{||}^2}$ the dispersion relation is still single valued?)

So at best we can have a small number of discrete number of temporal frequencies for a given spatial frequency.

My questions are as follows.

  • Under what generic conditions do we find multiple spatial frequencies for a single temporal frequency?
  • Please provide more examples of multi-valued dispersion relations
  • Are there any examples of wave equations or media that support a continuum of temporal frequencies $\omega$ for a single value of $|\boldsymbol{k}|$
  • And if no to the previous question, why is this impossible?
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    $\begingroup$ Let $\mathbf{k}\in\Bbb R^d$ (in real life, $d=3$). Each of these dispersion relations specifies a locus in $\Bbb R^{d+1}$ for $(\omega,\,\mathbf{k}$). Your question is effectively why we get a $d$-dimensional surface, rather than a region of nonzero $(d+1)$-dimensional volume. $\endgroup$
    – J.G.
    Oct 21, 2021 at 15:14

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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.

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  • $\begingroup$ Thank you for these clarifications. I'll focus on the linear case so that plane waves are non-interacting. This seems like an ok definition of "waves". So if we have the linear differential equation $F(\partial_t^2, \nabla^2)u(\boldsymbol{x}, t) = 0$ where $F(t, x)$ is a order-$n$ polynomial, this gives rise to the dispersion relation $F(\omega^2, |\boldsymbol{k}|^2) = 0$. $F(\omega^2, |\boldsymbol{k}|^2)$ is then some surface in $\omega^2$, $|\boldsymbol{k}|^2$ space, and where this surface intersects 0 defines the dispersion relations? Is this all correct? $\endgroup$
    – Jagerber48
    Oct 22, 2021 at 1:03
  • $\begingroup$ Also do I have it right that $\partial_t$ and $\nabla$ should always appear squared in the differential equation we are analyzing? $\endgroup$
    – Jagerber48
    Oct 22, 2021 at 1:03
  • $\begingroup$ If there are both even and odd factors of $\partial_t$ and/or $\nabla$ then factors of $i$ would appear in the differential equation. I guess this is an indication that plane waves decay in time/space. If we reject these as being waves then yes, we do need all even factors probably.. $\endgroup$
    – Jagerber48
    Oct 22, 2021 at 1:13
  • $\begingroup$ @Jagerber48 I expanded the answer, to answer yoru questions above and to add mor einformation. I hope it helps. $\endgroup$ Oct 22, 2021 at 7:58

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