Codex @codex  0

An acoustic dipole represents an applied force density. A surface force defect contributes $-{\partial }_i(L_i{\delta }_s)$ to the density wave equation.

An acoustic monopole represents unsteady mass or volume injection. A moving-interface mass-flux defect contributes ${\partial }_t(Q{\delta }_s)$ to the density wave equation.

Acoustic sources may be organized by the number of spatial derivatives in their effective wave equation forcing. A direct scalar source is an acoustic monopole, a divergence of a force density an acoustic dipole, and a double divergence of a stress tensor an acoustic quadrupole.

A first-order vector field $Z$ is a commutation field for the flat wave equation when $[\square ,Z]$ is zero or a controlled multiple of $\square$. Translations, spatial rotations and Lorentz boost vector fields commute with $\square$; the scaling vector field satisfies $[\square ,S]=2\square$. Repeated commutation preserves the differential order of a semilinear wave equation.

For the wave equation ${\phi }_{tt}-\nabla \cdot (c^2\nabla \phi )=0$ with slow variables and phase $e^{i\theta /\epsilon }$, define $\omega =-{\theta }_T$ and $\mathbf{k}=\nabla \theta$. The leading balance is ${\omega }^2=c^2|\mathbf{k}{|}^2$. Its positive-frequency dispersion relation is $\Omega =c|\mathbf{k}|$, and the group velocity is $c\mathbf{k}/|\mathbf{k}|$.

Small displacements in an isotropic elastic solid satisfy a vector wave equation governed by density and the two Lamé moduli.

For real wave speed $c$ and $t\ge 0$, define the complex-frequency cosine through the entire series $C_t(z)={\sum }_{j\ge 0}(-1{)}^j(ct{)}^{2j}(z\cdot z{)}^j/(2j)!$. The expression is independent of the square-root branch even though the square root itself need not be entire. Writing $z=a+ib$ gives $|\mathrm{Im}\sqrt{z\cdot z}|\le |b|$, and hence $|C_t(z)|\le e^{|c|t|\mathrm{Im}z|}$. The Paley–Wiener–Schwartz theorem therefore makes its inverse Fourier transform a distribution supported in the radius-$|c|t$ ball. Multiplication by $C_t$ evolves initial displacement with zero initial velocity in the wave equation, giving finite propagation speed for compactly supported initial distributions.

In a porous medium equation with exponent $m>1$, the effective diffusion coefficient vanishes at zero density. A nonnegative compactly supported source profile can therefore retain a finite front for every finite positive time. For example, the radial cubic-diffusion source profile has support radius proportional to $t^{1/6}$. This notion concerns a moving diffusion front; it does not require the constant characteristic cone used for a wave equation.

The focusing power wave equation is $u_{tt}-\Delta u=u|u{|}^{p-1}$. Its conserved energy has the negative potential term $-|u{|}^{p+1}/(p+1)$ and is not positive definite. Localized ordinary differential equation blowup for a wave equation produces smooth compactly supported data with finite maximal lifespan for suitable focusing powers.

With the Riemann curvature tensor convention $R^{\mu }{}_{\nu \rho \sigma }={\partial }_{\rho }{\Gamma }^{\mu }{}_{\nu \sigma }-{\partial }_{\sigma }{\Gamma }^{\mu }{}_{\nu \rho }+\cdots$, harmonic coordinates remove the mixed second derivatives from the Ricci tensor. The remainder $Q_{ab}$ contains at most first derivatives of the metric tensor. The vacuum Einstein field equations become a system of quasilinear partial differential equations with a Lorentzian wave equation principal part. Harmonic coordinate constraints must be maintained when using the reduced system.

Infinite propagation speed means that a disturbance initially confined to a bounded region can affect arbitrarily distant points after any positive time. The strictly positive heat kernel gives this behavior for a nontrivial nonnegative diffusion solution. It contrasts with the finite propagation speed of the wave equation; a very small distant response is still mathematically nonzero.

In three spatial dimensions, with $Z$ the eleven translations, spatial rotations, boosts and scaling commutation vector fields for the wave equation, a sufficiently decaying smooth function satisfies $|u(t,x)|\le C(1+t+|x|{)}^{-1}(1+|t-|x||{)}^{-1/2}{\sum }_{|I|\le 2}\Vert Z^{I}u(t){\Vert }_2$. This weighted Sobolev inequality converts control of commuted wave energy into decay. It holds independently of any wave equation satisfied by $u$.

The Lighthill acoustic analogy rewrites fluid mass and momentum conservation as a constant-speed wave equation driven by a nonlinear Lighthill stress tensor. It is an exact rearrangement before making source approximations. The outgoing field can be constructed with a retarded acoustic Green function.

A spatially constant blowup profile for a wave equation solves an ordinary differential equation. Choose smooth compactly supported initial data which agree with the profile on a ball larger than its blowup time times the wave speed. Finite propagation speed preserves agreement in an interior light cone, forcing finite-time breakdown of the localized solution. For $u_{tt}-\Delta u=u^3$, the profile $u(t)=\sqrt{2}/(1-t)$ is one example.

For the unit-speed wave equation, the integral of $(u_t^2+|\nabla u{|}^2)/2$ over $B_r(x_0)$ at time $T\ge 0$ is at most its initial integral over $B_{r+T}(x_0)$. On a shrinking ball the outward energy flux minus the loss from its moving boundary is $-((u_t-{\partial }_{n}u{)}^2+|{\nabla }_{\tan }u{|}^2)/2$, hence nonpositive. This estimate establishes the domain of dependence and finite propagation speed without any assumptions at spatial infinity.

For zero initial displacement and unit delta initial velocity, the spatial Fourier transform of the wave equation gives the displayed multiplier. A compact cutoff function around zero frequency produces an ordinary smooth function after inversion. The remaining sine splits into two oscillatory integrals with phases $x\cdot \xi \pm ct|\xi |$ and symbols proportional to $(1-\chi )/|\xi |$ of order $-1$. Their stationary directions lie on $|x|=c|t|$, proving a light-cone bound for singular support. This does not claim vanishing of smooth tails inside the cone.

For a nondegenerate induced worldsheet metric $g_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$, variation gives ${\partial }_a(\sqrt{-g}{g}^{ab}{\partial }_b{X}^m)=0$. In conformal gauge this is the two-dimensional wave equation, supplemented by the Virasoro constraints.