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For an area $A$ with periodic in-plane boundary conditions, one polarization contributes $A2\pi kdk/(2\pi {)}^2$ normal modes of the electromagnetic field. For linear dispersion relation $\omega =ck$ and effective physical volume $Ad$, this gives the density of states in the title. No extra factor two for polarization is allowed. Opposite in-plane directions are already counted in the full circle of wavevectors.

Approximating perturbation temperatures by harmonic fields decaying as $e^{\pm \alpha \xi }$, while retaining displacement through the base gradient, produces this reduced dispersion relation. The Gibbs--Thomson relation supplies $\Gamma =\gamma T_m/(\rho L)$. This short-wave reduction suppresses perturbation advection and time dependence; its constant term and a threshold at $\alpha \kappa /V=O(1)$ are not controlled next-order results of the full diffusion problem.

For fixed complex frequency $\omega$, a spatial root is a complex wavenumber $k$ satisfying the dispersion relation. Analytic continuation of these branches from a causal temporal contour is essential in the Briggs-Bers criterion; their collision alone does not decide whether a spatial pinch point occurs.

The response to a localized perturbation must be followed in space as well as time. Growth at a fixed position is absolute wave-packet instability; growth transported away while the original position decays is convective wave-packet instability. The Briggs-Bers criterion evaluates the relevant complex spatial branches of a dispersion relation, while a real-wavenumber temporal mode test alone does not generally distinguish those behaviors.

When a dispersion relation factors as $\omega P(\omega ,k)=0$, dividing by $\omega$ discards a possible time-independent Fourier mode. This can remove a necessary stationary solution from an initial value problem. For instance, a constant-coefficient temporal equation $T^{(3)}+ia{T}^{\prime \prime }+b{T}^{\prime }=0$ has a constant solution as well as the two exponential solutions from its quadratic factor. Three initial conditions generally require all three modes. Cancellation is valid only when a nonzero frequency has already been required.

For dispersion relation $\omega =Uk-\beta k/(k^2+{\alpha }^2)$ with fixed ${\alpha }^2\ge 0$, nonzero zonal wavenumber and stationary phase imply $U=\beta /(k^2+{\alpha }^2)$. Differentiating at fixed $\alpha$ yields $c_{g,x}=U+\beta (k^2-{\alpha }^2)/(k^2+{\alpha }^2{)}^2$, which reduces to the displayed positive value for $\beta >0$. Thus a stationary Rossby wave can transport energy eastward even though its crests stay fixed.

An incompressible Boussinesq fluid with uniform magnetic field, uniform vertical rotation and uniform temperature gradient supports waves combining magnetic tension, Coriolis force and buoyancy. Put $K=|\mathbf{k}|$, $h^2=K^2-k_z^2$, $a^2=(\mathbf{B}\cdot \mathbf{k}{)}^2/({\mu }_0\rho )$ and $H=g\alpha \beta$, with ${\partial }_z{T}_0=-\beta$. Eliminating pressure and the temperature and magnetic perturbations gives the displayed dispersion relation for nonzero frequencies. Stable thermal stratification has $H<0$. Clearing frequency denominators requires checking any resulting zero roots in the original equations; a stationary magnetostatic-buoyancy balance can exist independently of the propagating branches.

For a liquid film heated below and cooled above, the quasistatic temperature of a conductively cooled film makes a thicker region colder at the surface. If surface tension decreases with temperature, Marangoni stress draws fluid towards thicker regions, competing with smoothing by capillary pressure. Writing ${\gamma }_s={\gamma }_0+bh$ gives $a_c={\gamma }_0/(3\mu )$, $a_m=b/(2\mu )$. A balanced long-wave scaling gives $H_{\tau }+(H^2{H}_X{)}_X+(H^3{H}_{XXX}{)}_X=0$. The dispersion relation $s=k^2-k^4$ has unstable band $0<|k|<1$ and fastest normal mode at $|k|=1/\sqrt{2}$. This thermal mechanism does not require surfactant.

For a water film of thickness $h$, lateral freezing speed $V$, ripple wavenumber $\alpha$, and $\Gamma =\gamma /(3\rho g)$, a long-wave model gives the dispersion relationThe fastest-growing mode has ${\alpha }_m=5^{-1/6}(\Gamma h{)}^{-1/3}$ and travels upward at speed $\sqrt{5}V/6$ when the vertical coordinate points downward.

A torsional Alfvén wave couples azimuthal velocity to a toroidal magnetic field around a fixed poloidal magnetic field. Axisymmetric induction gives ${\partial }_t{B}_{\varphi }=R{\mathbf{B}}_p\cdot \nabla \Omega$ and magnetic tension gives $\rho R^2{\mu }_0{\partial }_t\Omega ={\mathbf{B}}_p\cdot \nabla (R{B}_{\varphi })$. Their combination is the displayed wave equation. Its local dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{B}}_p{)}^2/({\mu }_0\rho )$.

In a hollow rectangular electromagnetic waveguide with perfect conductor walls and axis $z$, a transverse magnetic polarization has $B_z=0$ and a nonzero longitudinal electric field $E_z$. The wall boundary conditions impose $E_z=0$ around the transverse rectangle. Separation of variables therefore gives $E_z=E_0\sin (n\pi x/a)\sin (m\pi y/b)e^{i(k_{z}z-\omega t)}$, with $n,m\ge 1$. Setting either index to zero makes the entire field vanish. Thus there is no ${\mathrm{TM}}_{n0}$ or ${\mathrm{TM}}_{0m}$ branch. The transverse eigenvalue is $k_{\perp }^2=(n\pi /a{)}^2+(m\pi /b{)}^2$, with dispersion relation ${\omega }^2/c^2=k_z^2+k_{\perp }^2$. This exclusion is specific to this geometry and these boundary conditions.

A uniform-density inviscid shear flow with $U=U_{0}z/d$ for $|z|<d$ and $U=\pm U_0$ outside has two vorticity jumps but no walls. Decaying normal modes satisfy $w^{\prime \prime }-k^{2}w=0$ in each region. Continuity of $w$ and pressure matching at a piecewise-linear shear interface gives a two-by-two determinant whose diagonal entries are $1-2kd\pm 2kdc/U_0$ and off-diagonal entries $e^{-2kd}$. Its vanishing gives the displayed dispersion relation. The Kelvin-Helmholtz instability band is $0<kd<{\alpha }_s$, where $2{\alpha }_s-1=e^{-2{\alpha }_s}$ has its unique positive root above $1/2$, approximately $0.63923227$. Short waves decouple the two edges and remain neutral; long waves approach the vortex sheet instability with $c=\pm i{U}_0$.

For an ideal incompressible flow rotating about the axis of the field $\mathbf{B}=(J/2)s\hat{\mathbf{\varphi }}$, use cylindrical Fourier modes $e^{\sigma t+im\varphi }$. The azimuthal derivative of a cylindrical vector Fourier mode reduces induction to $\sigma \mathbf{b}=imJ\mathbf{u}/2$. If the boundary eigenproblem gives $\lambda =i\delta \nu$, with real $|\delta |<1$, eliminating the perturbation gives the displayed dispersion relation. Here $J=\nabla \times \mathbf{B}\cdot \hat{\mathbf{z}}={\mu }_0{J}_{\mathrm{phys}}$ for SI current density; the formula is conditional on that boundary eigenrelation.

For the three-layer stratified piecewise-linear shear flow, set $\alpha =kh/2$. The dispersion relation is quadratic in squared nondimensional phase velocity. Its constant term is $[2\alpha -(1+J)(1+e^{-2\alpha })][2\alpha -(1+J)(1-e^{-2\alpha })]/(4{\alpha }^2)$. Throughout the displayed band this is negative, so one squared speed is negative and gives an exponentially growing normal mode.

A nonanalyticity of the density of states associated with a stationary point of a band dispersion relation. Its form depends on dimension and on whether the stationary point is an extremum or a saddle.

An interval of particle velocities on which an equilibrium phase-space distribution function is constant. Its zero derivative removes the resonant imaginary part of the linear dielectric function for waves with phase velocity inside the interval. This permits a completely undamped electron acoustic wave in suitable single-population, nonthermal electron distributions. Zero slope alone is necessary but does not guarantee that the real dispersion relation has a root.

Consider an equal-density vortex sheet halfway between rigid walls, with equal layer depths $h$, base velocities $U_1,U_2$ and no restoring interfacial force. The irrotational perturbations have vertical factors $\cosh (k(y-h))$ above and $\cosh (k(y+h))$ below. Their kinematic boundary conditions and pressure continuity giveThe common depth factor cancels, yielding the displayed dispersion relation roots and Kelvin-Helmholtz instability for nonzero shear. The real phase velocity of the pattern is $(U_1+U_2)/2$ and the growing branch has temporal growth rate $|k(U_1-U_2)|/2$. This exact depth cancellation depends on equal depths and equal mass densities; a single nearby wall or unequal layers changes the inertia factors.

For upward internal gravity waves, let $B=c_{gz}{N}^2{A}_{\zeta }^2/\hat{\omega }$ be constant. Combining $c_{gz}=(N/k)\sin \theta {\cos }^2\theta$ with the monochromatic internal-wave overturning criterion gives ${\cot }^4\theta <B{k}^3\hat{\omega }/(N^3{\sin }^3\theta )$. Near a critical level the onset balance is $\cot \theta \sim (B{k}^3\hat{\omega }/N^3{)}^{1/4}$. This is compatible with $\hat{\omega }=N\cos \theta$; it fixes the overturning point rather than redefining the dispersion relation.

For a smooth phase $\Theta$, define local wavenumber $k={\Theta }_x$ and local angular frequency $\omega =-{\Theta }_t$. Equality of mixed derivatives gives $k_t+{\omega }_x=0$. Dividing by $2\pi$ expresses conservation of signed crest count with density $k/(2\pi )$ and flux $\omega /(2\pi )$. When a homogeneous local dispersion relation gives $\omega =\omega (k)$, the equation becomes $k_t+v_g(k)k_x=0$; its characteristic curves move at the group velocity. This phase compatibility is distinct from energy or wave-action conservation.

The wave equation is a second-order linear partial differential equation describing waves with finite propagation speed. In one spatial dimension its homogeneous, constant-coefficient form isHere $x$ is position, $t$ is time, $u(x,t)$ is the disturbance, and subscripts denote partial derivatives. For a wave equation on a string, $u$ is transverse displacement; in linear acoustics, it can be a pressure perturbation. In several spatial dimensions, $u_{xx}$ becomes the Laplacian $\Delta u$. The same mathematical structure connects vibrating strings and acoustic waves, although their physical assumptions and wave speeds differ. ^[ref]

For the whole-line Cauchy problem, prescribe both displacement $u(x,0)=f(x)$ and velocity $u_t(x,0)=g(x)$. A classical solution has continuous partial derivatives through order two; $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ suffice for D'Alembert's formula to produce one. A finite string additionally needs boundary conditions. These statements concern the unforced model with constant $c$; damping, spatially varying coefficients and nonlinear terms change the solution formula.

A travelling wave $F(x-ct)$ carries its entire profile to the right at wave speed $c$; $G(x+ct)$ travels left. Their sum is again a solution by the superposition principle. A standing wave instead has a fixed mode shape whose size and sign oscillate: in $\sin (kx)\cos (ckt)$, each node of a standing wave stays stationary. The distinction is especially clear in the pulse and string diagrams below. For this ideal model, the dispersion relation is ${\omega }^2=c^2{k}^2$: each propagation branch is nondispersive, so a localized profile translates without spreading. This makes the equation a useful reference against which to understand wave dispersion and damping.

The appealing combination is local interaction with global predictability: neighbouring displacements determine acceleration through $u_{xx}$, yet D'Alembert's formula explicitly identifies exactly which initial locations can influence a later observation. Finite propagation speed turns that observation into a domain of dependence, rather than requiring knowledge of the entire initial string.