Implementation of a Dipole Constant Directivity CIrcuiar-Arc
Array
Kurtis Manke and Richard Taylor

May 31,2017

Thompson Rivers University, Kamloops BC, Canada

Correspondence should be addressed to Kurtis Manke(inankekl20mytru.ca)
ABSTRACT

We briefly present the theory for a broadband constant-beamwidth transducer(CBT)formed by a conformal
circular-arc array of dipole elements previously developed in seminal works. This technical report considers a
dipole CBT prototype with cosine amplitude shading of the source distribution. We show that this leads to a
readily-ecjualizable response from about 1 OOHz to 1 OkHz with a far-field radiation pattern that remains constant
above the cutoff frequency determined by the beam-width and arc radius of the array, and below the critical
frequency determined by discrete element spacing at which spatial aliasing effects occur. Furthermore, we show
that the shape of the radiation pattern is the same as the shading ftinction, and remains constant over a broad band
of frequencies.
1

Introduction

Taylor, Manke and Keele in [1] developed the theory
for a constant directivity circular-arc (CBT) line ar

rays formed by continuous line sources of dipole el
ements. They have shown that choosing an appropri
ate frequency-independent amplitude shading function
leads to a far-field radiation pattern that is constant

above a cutofffrequency determined by the beam-width
and arc radius of the array.

In this report we examine the results of a prototype

CBT dipole array, and show that it confirms much of
the existing theory.

2 CBT Theory Review

Fig. 1: Geometry of a circular line source of dipoles.

The following are key results from[1]with most of the
steps omitted. See the original paper for more details.
We consider a time-harmonic acoustic line source in
the form of a circle of radius a, in free space, as shown

Referring to Fig. 1, the total(complex) pressure at O
in the far-field due to a line source of dipole elements,

with unit acceleration amplitude, is given by

in Fig. 1. The source elements are taken to be radiallyoriented dipoles. We adopt a coordinate system in
which the circle lies in the xz-plane, with its center at

p=

■ka cos (j>Y^a„f„{ka cos (l>)cos{nd)

(2)

n=0

the origin. We take the x-axis(0 = 0 = 0)to be the

primary "on-axis" direction of the resulting radiation
pattern. We assume the source distribution is iso-phase

where k is the wave number [2, p. 312] and with

f„(x) = 2ni"J'„ix)

and continuous, with strength that varies with polar

angle a according to a dimensionless and frequencyindependent"shading function" 5(a)(sometimes also

(3)

where J„ is a Bessel function of the first kind [3].

called the amplitude taper).

Taylor, et al. go on to derive the following properties

We assume the shading function S(a) is even,so it can
be expressed as a Fourier cosine series

from (2):

5(a) = Y^a„cos{na)
n=0

(1)

• Each circular harmonic shading mode is mapped

to a corresponding radiation mode in the far field