Constant Directivity Circular-Arc Arrays of Dipole Elements
Richard Taylor and Kurtis Manke
Thompson Rivers University
31 May 2017

Abstract

We devdop the theory for a broadband constant-beamwidth transducer (CBT)formed by a
conformal circular-arc line array of dipole elements. Appropriate amplitude shading of the source
distribution leads to a far-field radiation pattern that is constant above a cutoff frequency determined

by the prescribed beam width and arc radius. We illustrate the theory with examples, including
numerical simulations of magnitude responses, full-sphere radiation patterns and directivity index.
Unlike a circular-arc array of monopole elements, a dipole CBT maintains directivity control at low

frequency. We give an example of one such array that achieves just 1 dB variation in directivity
index over all frequencies.
1

Introduction

There is considerable interest in the design of
acoustic sources that exhibit broadband constant

directivity, i.e. a radiation pattern that is indepen
dent of frequency. In sound reproduction, much

minology, we refer to such an array as a Dipole
CBT. In the following section we develop the
theory for acoustic radiation from such an axray. We then use this theory to derive condi
tions on the shading function that guarantee a
frequency-independent beam pattern; while previ

of this interest stems from extensive work by ous work in this area has considered only arrays

Toole and others (see [1] and references therein) of monopole sources, we extend our theory to
showing that constant directivity is coiTelated dipole arrays. We then present results of numeri
with subjective perception of quality in stereo cal simulations that illustrate the efficacy of our
reproduction.
new shading functions as well as some of the basic
Keele [2, 3, 4, 5] has reported extensively on properties of dipole versus monopole arrays.
a constant-beamwidth transducer (CBT)formed
by a circular-arc array with amplitude shading. 2 Circular Array of Dipoles: Theory
Keele's work is based on that of Rogers and Van Consider a time-harmonic line source in the form

Buren [6], who showed that a transducer with of a circle of radius a, in free space, as shown
very constant beam pattern can be formed by a ra in Fig. I. The source elements are taken to be
diating spherical cap with frequency-independent radially-oriented dipoles. (Such an array is said to
amplitude shading based on a Legendre function. be conformal [8], in that the element orientation
In our recent work [7] we developed the theory changes with the orientation of the array surface.)
for such arrays, and derived improved shading We adopt a coordinate system in which the circle
functions.
lies in the ^.s-plane, with its center at the ori
Building on this recent work, the present pa gin. We take the x-axis (0 = 0 = 0) to be the
per considers a variation on the CBT concept: a primary "on-axis" direction of the resulting radi
constant-directivity source based on a circular-arc ation pattern. We assume the somce distribution
array of dipole sources. In keeping with past ter is continuous and iso-phase and continuous, with
I