Building Acoustics

70 Building acoustics

z

y

dS

p(R, )

x

R

r

q

a

θ

φ

φ

z

y

dS

p(R, )

x

R

r

q

a

θ

φ

φ

or (3.47) to find the sound pressure at a point with coordinates (R,φ). At points near to
the surface, where Equation (3.46) has to be used, the distance r to surface element dS
will be given by

(^) ()
1
rRq Rq=+−^22 2sincos ,φθ^2 (3.48)
where q is the distance between the surface element and the centre of the piston. The
solution of the integral is not trivial and it must generally be solved numerically except
for points on the axis of the piston. The sound pressure in this near field will also
fluctuate in a complicated manner due to the changing phase differences of the
contributions from the different parts of the surface. The main purpose here is, however,
to show the behaviour of the pressure in the far field and through this give an example of
a source exhibiting a directional pattern quite different from our simple poles, e.g. a
dipole.
Figure 3.7 Coordinate system for calculation of sound pressure from a piston in a baffle.
At large distances from the surface, where the use of the Rayleigh integral is
applicable, we will use the following approximation for r, setting
rRq≈−sin cosφ θθand dSqq=d d. (3.49)
Inserting these expressions into Equation (3.47) we get
2
00 j( ) sin cos
00
(,,) j ˆe d e d.
2
a
pR t c u tkR q q kq
R
π
φ ρ ωφθθ
π

=⋅−

∫∫ (3.50)

The solution may be expressed as

00 2 j( ) 2J^1 ( sin )

(,,) j ˆe ,

2sin

pR t ckau tkR ka

Rka

φ ρ ω φ

φ

=⋅− ⎡ ⎤

⎢ ⎥

⎣ ⎦

(3.51)

where J 1 is a Bessel function of the first order. It is the term enclosed in parenthesis that
determines the directivity distribution of the sound pressure, an example shown in Figure