## Acoustic Scattering Problems

For our first application we consider the problem of time-harmonic acoustic scattering from a bounded obstacle. Here, an obstacle is excited by an incident pressure wave and we seek the far-field pattern of the response. In principle, the incident wave could be generated by modeling an actual source, but under the assumption that the source is far from the obstacle, we approximate the incident wave by a plane wave

Here, k = ω/c is the wavenumber, depending on the angular frequency ω and speed of sound c, and ê is the direction of the source. The incident wave pinc and the scattered wave p are complex-valued, with the associated timedependent quantities given by

P inc(x, t) = Re( pinc(x)eiωt) P(x, t) = Re( p(x)eiωt)

The obstacle is assumed to occupy a bounded region int ⊂ IR3 with Lipschitz boundary . The scattered pressure p satisfies the (homogeneous) Helmholtz equation outside the obstacle

−p − k2 p = 0 in = IR3 \ int (10.1)

along with the Neumann boundary condition (for the case of a rigid scatterer)

∂p ∂n

= g = −∂p inc

∂n on (10.2)

and the Sommerfeld radiation condition

∂p ∂r

+ ikp =: w = o(r−1) as r → ∞ (10.3)

In the region exterior to a sphere large enough to enclose the obstacle, the solution to Equations (10.1-10.3) can be represented by the Atkinson-Wilcox expansion

p(r) = e −ikr

r

∞∑

un(ψ, θ ) rn

= e −ikr

r P(r)

where r, ψ, θ are the standard spherical coordinates (see Appendix A). The main idea behind infinite elements is to remove the exponential phase factor e−ikr/r and focus on the approximation of the remainder P . Our derivation closely follows [62].