Scattering by a sphere in a tube, and related problems

Abstract

Time-harmonic waves propagate along a cylindrical waveguide in which there is an obstacle. The problem is to calculate the reflection and transmission coefficients. Simple explicit approximations are found assuming that the waves are long compared to the diameter of the cross-section d. Simpler but useful approximations are found when the lateral dimensions of the obstacle are small compared to d. Results for spheres, discs, and spheroids are given.

On blockage coefficients: flow past a body in a pipe

“…Of more relevance here are effective interface conditions, connecting solutions on two sides of an interface or ‘metascreen’, which we can take to be a configuration of perforations and small obstacles nominally in the plane z=0. As noted in [13], the effect of such a screen is often modelled using certain transmission conditions across z=0, leading to a one-dimensional problem for ϕfalse(zfalse). Let us write these ‘homogenized’ conditions as false[ϕfalse]=β11falsefalse⟨ϕfalsefalse⟩+β12falsefalse⟨ϕ′falsefalse⟩1emfalse0and1emfalse[ϕ′false]=β21falsefalse⟨ϕfalsefalse⟩+β22falsefalse⟨ϕ′falsefalse⟩,where false[ ⋅ false] denotes jump and falsefalse⟨⋅falsefalse⟩ denotes average, false[ϕfalse]=ϕfalse(0+false)−ϕfalse(0−false) and falsefalse⟨ϕfalsefalse⟩=false012falsefalse{ϕfalse(0+false)+ϕfalse(0−false)falsefalse}.…”

“…The parameters βij are to be specified in terms of the geometry and the composition of the metascreen. The literature on the interface models (3.3) is extensive; see [13] for some references. Some of these models lead to a connection with blockage coefficients, as we shall see below.…”

“…(Terms that are linear in h cancel.) For a thin rigid disc in a tube, (3.5) reduces to ([13], eqn (31)), obtained there using a different method.…”

“…Most of the papers cited in this paragraph are concerned with velocity distributions around the object, not the blockage coefficient; exceptions are the papers by Smythe [7,8] (he is interested in the difference between ϕ far upstream and far downstream, which is 2UC in our notation) and Miloh [11] (he is interested in the added mass, which is related to C; see §3a). References for analogous waveguide problems can be found in [13].…”

“…Blockage coefficients are used in various physical situations, as mentioned above. Often, they are ingredients in simple approximations, such as for the reflection coefficient when long waves interact with a body in a waveguide [13]. This is one reason why simple but accurate approximations to blockage coefficients are desirable.…”