Reflecting Disc

Reflecting Disc — equivalent dipole source

www.akutek.info

HOME

INDEX

Go to www.akutek.info main page Go to article site index page

Introduction

An interesting special case among sound reflectors is the ideal rigid, infinitely thin, disc of radius a. Pierce^a (1981), with reference to Lord Rayleigh in 1897, presented an analytical solution for the scattered pressure p_sc from a disc of radius a in the low-frequency case ka<<1, resulting from an incoming plane wave pressure p_i along the x-axis normal to the disc (fig 1). The analytical solution provides verification for physical and mathematical models suggested for prediction of reflections from reflection panels in general and the disc in particular. At the bottom of this page there are several links to some related articles.

1. Scattered pressure p_sc from a disc of radius a, resulting from an incoming plane wave pressure p_i , with wave number k, along the x-axis normal to the disc, expression by Pierce:

2. In the far field, at distance r from the disc, in direction θ , the scattered pressure from can be written

p_sc = p_i∙(ka)²/ (4p r) ∙8a/3 ∙ cos θ

Pierce identifies the dipole radiation characteristics of the scattered pressure from the disc, in contrast to scattering from a sphere, which would be a combination of a monopole component and a dipole component.

3. According to Kinsler^b and Frey, a dipole source can be expressed on the form:

p_d = rc kq /(4pr) ∙kD∙cosq ,

where D is the distance between the monopole-pair of source strengths q and –q , respectively, and rc the characteristic impedance of the medium

4. Assuming that the scattered pressure from the disc is equivalent to the radiation from a dipole described in (3), what will the source strength and the distance D of the equivalent source be? Equating the right-hand expressions in (2) and (3) reveals that the source strength must be equal to the volume velocity of the cross-section p a² of the wave incident on the disc

q = p a² ∙u_i = p a² ∙p_i/rc ,

5. ...while the dipole-distance must be

D = 8a/3p ~ 0.85∙a

Note that D is equal to the classical end-correction of a pipe having radius a, and the height of the air-mass associated with the radiation reactance of a circular piston with radius a , oscillating in an infinite baffle.

6. As noted in the introduction, the above solution (1-2) is valid in the low frequency case ka<<1.

A general solution can be derived from the Helmholtz-Kirchhoff Integral Theorem, which expresses the sound pressure in a point P in a volume surrounded by a closed surface S as follows:

where p and Ñ p are the pressure and the pressure-gradient at the closed surface S, and r is the distance from P to the infinitesimal surface element dS with a normal vector n pointing away from P.

In the far field low frequency case (kr>>1 and ka<<1), a number of simplifications can be made, since e^-ikr/r vary negligibly over the surface of interest.

7. A different approach towards insight in scattering from a disc would be to study edge-scattering in the time-domain.

$Text Box: Consider the portion dz of the disc circumference being “hit” by an impulse at normal incident, similar to the case described above (figure to the left). Sound will be scattered in any direction, as if dz is some secondary source, according to the formulas derived by Peter Svensson. Link (opens in separate, resizable window): Svensson Time-Domain Edge-Diffraction. For convenience, the formula is shown below. In the case of the disc, it is particularly interesting to trace the sound paths across the disc, i.e. scattering components of higher orders. To the right is shown the path history of the j-th incident portion pi,j , its ways across the disc, and eventually its scattered portion psc,j .$