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Recover a compactly supported acoustic source $Q$ from measurements of its outgoing field, such as its far-field pattern. The Helmholtz equation forward problem is specified with the Sommerfeld radiation condition. At a single frequency, the source-to-far-field operator at fixed frequency has invisible fixed-frequency nonradiating sources, and its compactness makes recovery of the visible component unstable.

The angular spectrum is the transverse Fourier transform of a monochromatic field on a reference plane. In a uniform medium, each transverse wavevector component propagates with a longitudinal wavenumber satisfying the Helmholtz equation. Components beyond the propagating wavenumber range are evanescent waves. The outgoing angular spectrum chooses the longitudinal branch that propagates or decays away from the plane.

The angular spectrum representation expresses a monochromatic wave as a superposition of plane waves. In a homogeneous, lossless, source-free half-space with real wavenumber $k=2\pi /\lambda >0$, transverse Fourier transform components propagate independently, providing a useful solution of the Helmholtz equation.^[ref]

Use angular frequency $\omega$ and time dependence $e^{-i\omega t}$, and define $\tilde{U}(k_x,k_y;0)=\int \int U(x,y,0)e^{-i(k_{x}x+k_{y}y)}dxdy$. Then
The outgoing branch of the angular spectrum has nonnegative real and imaginary parts of $k_z$ for propagation toward $z>0$. A propagating angular-spectrum component has real $k_z$; an evanescent angular-spectrum component has $k_z=i\kappa$ and decays as $e^{-\kappa z}$. The source-plane amplitudes and their complex arguments matter, not just their squared magnitudes.^[ref]

For an original example, a transverse period $\lambda /2$ gives $k_x=2k$, $k_y=0$ and $\kappa =\sqrt{3}k$. After distance $z=\lambda /4$, its amplitude is multiplied by $e^{-\pi \sqrt{3}/2}\simeq 0.0658$. Such fine spatial detail does not propagate with unchanged amplitude to the far field.

The paraxial angular-spectrum propagator uses $k_z\simeq k-(k_x^2+k_y^2)/(2k)$. For the source-plane Gaussian beam $U_0=e^{-(x^2+y^2)/w_0^2}$, its transverse spectrum is proportional to $e^{-w_0^2(k_x^2+k_y^2)/4}$. The on-axis Gaussian integral therefore gives
The denominator produces the axial Gouy phase $-\arctan (z/z_R)$. This connects the phase shift with interference of transverse spectral components, rather than an abrupt phase jump at the focus.^[ref]

The scalar propagation formula does not authorize arbitrary independent vector components. A physical electromagnetic field must satisfy the transverse polarization constraint in an angular spectrum; in a uniform source-free medium,$\mathbf{k}\cdot \tilde{\mathbf{E}}=0$. Boundary conditions and source consistency accompany a vector reconstruction.

Bessel functions are standard solutions of the ordinary differential equationThe parameter $\nu$ is the order. A first-kind solution iswhere $\Gamma$ is the gamma function and a branch of a multivalued function is chosen for a noninteger power. For nonnegative integer $n$, $J_n$ is entire; a second-kind solution $Y_n$ is singular at zero.^[ref]

These functions arise when separation of variables is applied to the Helmholtz equation in cylindrical coordinates. A circular membrane requires a radial solution regular at the centre, selecting $J_n$; its boundary condition then selects zeros of $J_n$. For example, $J_0(z)=1-z^2/4+z^4/64-\cdots$ is the radial profile for the rotationally symmetric family.

Let ${\Gamma }_9$ have center weight $-10/3$, axial weights $2/3$ and diagonal weights $1/6$, and let $M_h$ have center weight $2/3$ and axial weights $1/12$. Taylor expansion gives $h^{-2}{\Gamma }_{9}u=\Delta u+h^2{\Delta }^{2}u/12+O(h^4)$ and $M_{h}u=u+h^2\Delta u/12+O(h^4)$. On a smooth solution of the constant-coefficient Helmholtz equation, the second-order term becomes $h^2\Delta (\Delta u+\lambda u)/12=0$. Thus the normalized PDE residual is fourth order, while the original unscaled stencil residual is sixth order. This local cancellation alone does not prove uniform global accuracy at or near a resonance.

For the cubic polytropic interior, $\nabla \rho$ vanishes at each vertex, so the locally defined potential $\Phi =C-2K\rho$ predicts zero gravitational acceleration there. But at the vertex $(0,0,0)$, the actual self-gravitational acceleration of the positive mass distribution iswhose three components are strictly positive. The fields cannot match continuously. Thus the interior Helmholtz equation and zero density on the faces do not construct an isolated static cubic star. External stresses or an external gravitational field would be needed to realize such a boundary-value construction.

The interior Helmholtz equation for a polytrope of index one also admits a positive solution vanishing on the faces of a cube:Its central value is ${\rho }_c$ and its average is $\bar{\rho }=8{\rho }_c/{\pi }^3$. Together with $\Phi =C-2K\rho$, it solves the interior hydrostatic equilibrium and Poisson equation for Newtonian gravity. A local interior solution need not match the external field of its own mass.

For the upper half-space and the sign convention $(\Delta +k^2)G_D={\delta }_{r^{\prime }}$, reflect the source $r^{\prime }=(x^{\prime },y^{\prime },z^{\prime })$ to $r^{\prime \ast }=(x^{\prime },y^{\prime },-z^{\prime })$ and use the displayed difference. This is the negative of the conventional outgoing kernel, whose source sign is $-\delta$. The method of images makes the Dirichlet boundary condition zero on the plane. With $R=|r^{\prime }-r|$ and $\cos \theta =z^{\prime }/R$, its outward normal derivative there isThe outward unit normal is $-e_z$. Differentiating $e^{ikR}/R$ gives $e^{ikR}(ik/R-1/R^2)$; the real and reflected source contributions add in the normal derivative. Green's second identity therefore expresses a solution of the Helmholtz equation as $\varphi (r^{\prime })={\int }_{z=0}\varphi (r){\partial }_n{G}_D(r,r^{\prime })dS$ when the boundary term at infinity vanishes. Suitable decay or compatible outgoing Sommerfeld radiation conditions are needed; vanishing at infinity without further control is not a general uniqueness condition for a real-frequency Helmholtz problem.

Consider the Helmholtz equation $(\Delta +k^2)u=0$ in $z>0$, with $k>0$, boundary value $u(x,0)=g(x)$, and upward outgoing or decaying propagation. Taking the tangential Fourier transform, with frequency $\xi$, givesThus the upward normal derivative is the Fourier multiplier $i\beta (\xi )$. The outward unit normal of the upper half-space points downward, so its Dirichlet-to-Neumann map has multiplier $-i\beta (\xi )$ instead. The normal orientation and the square-root branch determine both signs.

For $|\xi |<k$ the modes propagate; for $|\xi |>k$ they are evanescent waves decaying with height. At high frequency $\beta (\xi )\sim i|\xi |$, so the map grows in magnitude like one tangential derivative. Small-amplitude boundary roughness can therefore have a large derivative effect when it varies on a short spatial scale.

The quadratic wave phase follows from an ordinary distance calculation. Write ${\rho }^2=(x-\xi {)}^2+(y-\eta {)}^2$. The Taylor expansion of the distance to a screen point isIn the slowly varying propagation prefactor, $r\simeq z$ suffices. In the complex exponential $e^{ikr}$, however, a small distance error is multiplied by the large wavenumber $k$. Retaining the quadratic term while dropping the quartic term requires both $\rho /z\ll 1$ and $k{\rho }^4/(8z^3)\ll 1$ over the contributing aperture and the screen region being considered. The second condition is a wave phase accuracy requirement; the first is the geometric paraxial approximation.

Expanding the retained quadratic term separates three effects:The outside quadratic factor changes the wave phase at each screen point but has unit complex modulus, so it leaves that screen's light intensity unchanged. The quadratic factor inside the integral changes relative wave phases between contributions from different aperture points, so it changes their wave interference. The remaining linear factor supplies a Fourier transform. Fraunhofer diffraction drops the aperture quadratic factor when it is nearly constant; it does not require dropping the screen quadratic factor when predicting a later coherent field.

There is also a useful proof from the Helmholtz equation. Substituting $U=e^{ikz}u$ gives $u_{zz}+2ik{u}_z+u_{xx}+u_{yy}=0$. Neglecting $u_{zz}$ yields the parabolic wave equation. A transverse Fourier transform evolves each spatial frequency $\mathbf{\nu }$ by $\exp [-iz|\mathbf{\nu }{|}^2/(2k)]$, using the transform kernel $e^{-i\mathbf{\nu }\cdot \mathbf{x}}$. Its inverse Fourier transform is the quadratic Fresnel propagator kernel above. Thus quadratic wave phase in space and quadratic wave phase in spatial frequency are two views of the same propagation.

Two Green functions of the same Helmholtz equation with the same homogeneous Robin boundary condition satisfy the displayed reciprocity by Green's second identity. It is a bilinear identity without complex conjugation, so it concerns the transpose rather than the Hermitian adjoint. In an exterior problem both kernels use the same radiation convention.

A Herglotz wave function is a superposition of incident plane waves,It solves the Helmholtz equation everywhere and is real analytic. The finite sphere area makes the density integrable, so each spatial derivative can be taken under the integral.

The incident wave is the prescribed background solution that illuminates a scatterer. It solves the unperturbed Helmholtz equation in scalar time-harmonic models. The scattered contribution is defined by subtracting this field from the total field; reflected waves from a planar boundary are included unless a separate reference reflection is explicitly removed.

A force-free magnetic field is linear when the proportionality between its curl and itself is a spatial constant. The curl of the curl identity and zero divergence give the vector Helmholtz equation ${\nabla }^2\mathbf{B}+{\lambda }^2\mathbf{B}=0$. Here $\lambda$ has inverse-length units; specifying a constant current-to-field coefficient with another normalization changes its numerical definition.

The logarithmic wave perturbation measures relative complex amplitude and phase through $\psi ={\psi }_i{e}^{\chi }$. A continuous branch requires nonzero incident and total fields in the region considered. For the Helmholtz equation with scattering potential convention $L_0\psi =-V\psi$, its equation is $(\Delta +2\nabla \log {\psi }_i\cdot \nabla )\chi +\nabla \chi \cdot \nabla \chi =-V$. The dot product is bilinear, not a squared modulus.

Fix time dependence $e^{-i\omega t}$ and a positive real wavenumber $k$. The one-dimensional outgoing Green function for $L={\partial }_x^2+k^2$ on the real line isFor $x>x^{\prime }$, the phase $k(x-x^{\prime })-\omega t$ propagates rightward; for $x<x^{\prime }$, the phase $k(x^{\prime }-x)-\omega t$ propagates leftward. Both move away from the point source. Reversing the time convention reverses the corresponding spatial exponential.

Away from $x=x^{\prime }$, each branch solves the homogeneous Helmholtz equation. The unnormalized function $e^{ik|x-x^{\prime }|}$ is continuous and has first derivative jump $ik-(-ik)=2ik$. Division by $2ik$ therefore gives the unit Dirac delta function required by the Green function jump conditions. For a compactly supported forcing, Green-function representation gives an outgoing solution by integrating this Green kernel against the forcing.

The outgoing angular spectrum propagates prescribed scalar boundary data into a half-space. For $e^{-i\omega t}$ time dependence, $\beta (q)=\sqrt{k^2-q^2}$ has nonnegative real part on propagating modes and nonnegative imaginary part on evanescent modes. Its trace at the boundary is $g$, and each mode solves the Helmholtz equation. The spectral representation can be interpreted through distributions for nondecaying plane-wave data.

The outgoing branch of the angular spectrum selects the longitudinal wavenumber that represents forward radiation and bounded evanescent decay in a specified source-free half-space. For real $k>0$, time dependence $e^{-i\omega t}$ and propagation into $z>0$,The first root is positive real and the second positive imaginary. They accompany the factor $e^{i{k}_{z}z}$.^[ref]

The scalar Helmholtz equation permits both signs. For $q<k$, $e^{i{k}_{z}z}$ and $e^{-i{k}_{z}z}$ describe opposite longitudinal phase propagation under the stated time convention. For $q>k$, the corresponding factors are decaying and growing exponentials. A source-free upper half-space with no incident field from infinity uses the forward, bounded choice.

For an original transverse spatial frequency $q=1.25k$, the decaying root is $k_z=0.75ik$. At $z=\lambda$, amplitude changes by $e^{-1.5\pi }\simeq 0.00898$. Choosing the other sign would instead multiply it by approximately 111.3. A sign error can therefore become an enormous near-field propagation error without changing the input data.

This choice is a statement about the physical region and boundary conditions, not an instruction to delete reflected waves from every problem. An interface or cavity can require both propagation directions in a finite region. Likewise, propagation towards negative $z$ must use a convention consistent with that region. At $q=k$, the root is zero and the component is grazing; formulas that divide by $k_z$ need separate attention.

Backward reconstruction of a decayed evanescent angular-spectrum component multiplies measurement error by an increasing exponential. That is an inverse problem's conditioning issue, rather than evidence that the forward outgoing branch should grow. Stating the time convention, physical half-space and radiation condition prevents the square-root symbol from hiding these choices.

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 $r$, $G_k=e^{ikr}{e}^{-ik\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }}/(4\pi r)+O(r^{-2})$.

The parabolic wave equation is a one-way, slowly varying envelope approximation to the Helmholtz equation. For carrier wavenumber $k$ and transverse coordinate $z$, a common form is $2ik{E}_x+E_{zz}=0$.