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The acoustic far field is the radiation region where the observer is many reduced wavelengths from the source and far compared with its geometric extent. Three-dimensional outgoing acoustic density perturbations spread as $R^{-1}$; two-dimensional harmonic fields spread as $R^{-1/2}$. The condition is independent of the acoustic compact-source approximation $k_0\ell \ll 1$. Differentiating a retarded phase produces the radiating part of a Green function integral; derivatives of the spreading factor produce near-field terms.

For $\mathrm{Re}p>0$, put $\lambda =p^{1/3}$ using the principal cube root and $r_{1,2}=\lambda e^{\pm i\pi /3}$. The decaying Green function for ${\partial }_s^3+p$ on the real line isThe identities among the three characteristic roots show that $R_p$ and $R_p^{\prime }$ are continuous, while $R_p^{\prime \prime }$ has jump one. Its distributional derivative therefore satisfies $({\partial }_s^3+p)R_p={\delta }_0$. This is the kernel of the resolvent operator $(p+{\partial }_s^3{)}^{-1}$, obtained by the time Laplace transform of the Airy equation.

For $L=D_t^2+2k{D}_t+k^2+{\omega }^2$, the causal Green function satisfies $LG=\delta (t)$. It is continuous at zero, with a unit jump in its first derivative. Its smooth positive-time kernel $g$ obeys $g(0)=0,g^{\prime }(0)=1$. The zero-frequency limit is $H(t)t{e}^{-kt}$. Shifting $G$ gives the zero-initial-data response to a delayed Dirac delta function.

For $x,\xi >0$, the Green function satisfyingas $x\to \infty$ is

With normal modes $e^{ikx-i\omega t}$, a finite upper bound on $\mathrm{Im}\omega$ allows the inverse temporal Laplace transform contour to start above all temporal singularities. An unbounded growth rate at high real wavenumber prevents the standard causal Green function construction used in the Briggs-Bers criterion. For $\mathrm{Im}\omega =\alpha k^2-\beta k^4-\gamma$ with nonzero real $\beta$, the bound holds exactly when $\beta >0$.

In three dimensions,Consequently the decaying solution of $-{\nabla }^{2}u=f$ in free space is

For $t>0$ this Green function is the response to a point impulse on the infinite line, using the continuous square root of $1+i{c}_d$. The Fourier transform multiplies the initial transform by $e^{\mu t-ikUt-(1+i{c}_d)k^{2}t}$; its Gaussian inverse gives the displayed expression. At fixed $x$, its exponential rate is $\mu -U^2/[4(1+c_d^2)]$, whereas along $x=Ut$ it is $\mu$. Thus a temporally growing flow with $0<\mu <U^2/[4(1+c_d^2)]$ is convectively rather than absolutely unstable.

For the linear dispersive Stokes equation on $x>0$, with sufficiently smooth decaying initial data and $\mathrm{Re}p>0$, define $V(x,p)={\int }_0^{\infty }{R}_p(x-y)u_0(y)dy$ and let $G(p)$ be the time Laplace transform of the prescribed derivative $u_x(0,t)$. Spatial decay and the one-dimensional decaying homogeneous space giveThe Neumann boundary condition follows by differentiating at zero. The initial forcing follows from the Stokes resolvent kernel with advection. The Bromwich inversion formula gives an integral representation with no unknown boundary traces. Truncating or extending the prescribed boundary signal beyond a target time does not affect earlier values, by the Laplace transform time-shift rule and causality of this Green function.

An integral kernel is a function or distribution specifying an integral operator through $(Tf)(x)=\int K(x,y)f(y)dy$. A Green function is an integral kernel that inverts a differential operator under specified boundary conditions.

For $\mathcal{L}=D^3+a_2{D}^2+a_{1}D+a_0$ with smooth coefficients and leading coefficient one, a Green function solving $\mathcal{L}G={\delta }_{\xi }$ is continuous together with its first derivative at an interior source, and its second derivative jumps by one. Distributional differentiation shows why: a jump in $G$ would create a delta second derivative, a jump in $G^{\prime }$ a delta first derivative, and a jump in $G^{\prime \prime }$ the required delta. For a causal initial-value Green function, take the left branch zero and choose the right branch with initial values $0,0,1$ at the source.

The outgoing Green function satisfies $(\Delta +k^2)G_k=-{\delta }_{{\mathbf{r}}^{\prime }}$ and the Sommerfeld radiation condition. For time dependence $e^{-i\omega t}$, its radial phase travels outward. Away from its source it solves the homogeneous Helmholtz equation. At large $r$, $G_k=e^{ikr}{e}^{-ik\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }}/(4\pi r)+O(r^{-2})$.