Monopole

Definition

The acoustic monopole (point source) with amplitude $a$ at position $\bm r_0$ is defined by the pressure field [BSUA70]

\[p_\mathrm{M}(\bm r) = \cfrac{a}{4 \pi} \cfrac{\mathrm{e}^{-\mathrm{j} k R}}{R} \,, \qquad R = |\bm r - \bm r_0| \,,\]

that is, by the free-space Green's function of the Helmholtz equation scaled by the amplitude. The field is singular at the position of the monopole.

API

The API provides the following constructor with default values:

ex = monopole(;
        position  = error("missing argument `position`"),
        frequency = error("missing argument `frequency`"),
        embedding = Medium(ε0, μ0),
        amplitude = 1.0
)

The constructors of the acoustic excitations are collected in the Acoustic submodule, so that their names do not clash with the electromagnetic ones.

Radiated Field

The pressure of the monopole itself (without scatterer) is as given above. Since it depends on the position solely via the distance $R$ to the monopole, its gradient is radial with respect to the monopole,

\[\nabla p_\mathrm{M} = \cfrac{\partial p_\mathrm{M}}{\partial R} \hat{\bm R} = -\cfrac{a}{4 \pi} (1 + \mathrm{j} k R) \cfrac{\mathrm{e}^{-\mathrm{j} k R}}{R^2} \hat{\bm R} \,, \qquad \hat{\bm R} = \cfrac{\bm r - \bm r_0}{|\bm r - \bm r_0|} \,,\]

so that the two Cauchy data (the surface traces) on a surface with outward unit normal $\hat{\bm n}$ read

\[\gamma_0 p_\mathrm{M} = p_\mathrm{M} \qquad \text{and} \qquad \gamma_1 p_\mathrm{M} = \hat{\bm n} \cdot \nabla p_\mathrm{M} = -\cfrac{a}{4 \pi} (1 + \mathrm{j} k R) \cfrac{\mathrm{e}^{-\mathrm{j} k R}}{R^2} (\hat{\bm R} \cdot \hat{\bm n}) \,.\]

For $k R \gg 1$ the Neumann trace approaches $-\mathrm{j} k (\hat{\bm R} \cdot \hat{\bm n}) p_\mathrm{M}$, the local plane-wave result.

Unlike a plane wave, a monopole does possess a far field. With $|\bm r - \bm r_0| = r - \hat{\bm r} \cdot \bm r_0 + \mathcal{O}(1/r)$ it is

\[p_{\mathrm{M},\infty}(\hat{\bm r}) = \lim_{r \rightarrow \infty} r \, \mathrm{e}^{\mathrm{j} k r} p_\mathrm{M}(\bm r) = \cfrac{a}{4 \pi} \mathrm{e}^{\mathrm{j} k \hat{\bm r} \cdot \bm r_0} \,,\]

that is, the factor $\mathrm{e}^{-\mathrm{j} k r} / r$ is omitted, as for the electromagnetic excitations. Its magnitude is the same in all directions, as it has to be for a point source: the position of the monopole enters the phase alone.

API

The general API is employed:

p  = field(ex, Pressure(point_cart))

FF = field(ex, FarField(point_cart))

γ₀ = field(ex, PressureTrace(point_cart))

γ₁ = field(ex, PressureNormalGradient(point_cart))

Without further specification the outward normal $\hat{\bm n} = \hat{\bm r}$ of a sphere centered in the origin is assumed at every location. One normal vector per location may be provided instead, which is required, e.g., for the facets of a surface mesh:

γ₁ = field(ex, PressureNormalGradient(point_cart, normals))

See PressureNormalGradient for details.

Scattered Field

The scattered field computation follows [BSUA70]. For the sound-hard and the sound-soft sphere it is obtained by a series in the spherical wave functions, see Acoustic Spheres; for the spheroids and the disc by a series in the spheroidal wave functions, see Spheroidal Solution.

API

The general API is employed:

p  = scatteredfield(sp, ex, Pressure(point_cart))

FF = scatteredfield(sp, ex, FarField(point_cart))

γ₀ = scatteredfield(sp, ex, PressureTrace(point_cart))

γ₁ = scatteredfield(sp, ex, PressureNormalGradient(point_cart))

where sp is a HardSphere, a SoftSphere, a Spheroid, or a Disc. The traces and the far field are described under Quantities.

Total Field

API

The general API is employed:

p  = field(sp, ex, Pressure(point_cart))

FF = field(sp, ex, FarField(point_cart))

γ₀ = field(sp, ex, PressureTrace(point_cart))

γ₁ = field(sp, ex, PressureNormalGradient(point_cart))