Biomedical Engineering Reference
InDepth Information
all directions in an infinite space). The main advantages of the
RayleighSommerfeld integral are: (1) this intuitive formulation
is based on the superposition of Green's functions; and (2) this
integral formula is easily evaluated with a computer program.
Disadvantages of the RayleighSommerfeld integral include rel
atively long computation times and large errors in the nearfield
region due to slow convergence.
The RayleighSommerfeld integral describing the pressure
radiated from a finite transducer aperture driven by a time
harmonic input signal is
2
1
0
−1
−−′
jk

rr

j
ωρ
π
e
∫
0
jt
ω
px yzt
(, ,,)
=
Ue
dS
,
(6 .1)
0
2

rr
− ′

S
where
p
represents the pressure, (
x ,y,z
) are the observation coor
dinates in Cartesian space,
t
is the time variable,
j
is
−
, ω is
the radian frequency of the excitation, ρ
0
is the density of the
medium,
U
0
represents the uniform normal particle velocity on
the face of the transducer,
e
j
ω
t
denotes a timeharmonic excita
tion,
S
indicates that the integration is performed over the sur
face of the radiating source,
r
represents the coordinates of the
observation point,
r
′ represents the coordinates of the source
point, 
r
−
r
′ is the distance from a source point on the radiating
aperture to the observation coordinates,
k
is the wave number,
and the integrand is the frequencydomain Green's function
for a point source surrounded by a rigid baffle. The formula in
Equation 6.1 admits direct discretization and evaluation for var
ious piston shapes, including the circular piston, the rectangular
piston, and the spherical shell.
−2
−2
−1
0
1
2
x
(units of piston radius
a
)
FIGURE 6.1
Circular piston with radius
a
centered in the
z
= 0 plane
at (
x
,
y
,
z
) = (0,0,0). The normal at the center of the circular piston is coin
cident with the
z
axis.
2/
N
are defined for numerical
integration with the midpoint rule,
N
σ
is the number of abscis
sas in the radial direction, and
N
θ
is the number of abscissas
in the θ′ direction. The double sum in Equation 6.3 evaluates
contributions from the Green's function for a point source in
an infinite baffle, where the distance 
r
−
r
′ is evaluated from
the center
r
′ of each sector on the piston face to an observation
point
r
. The contribution from the Green's function is then mul
tiplied by the area σ ′Δ σ ′Δθ ′ of each sector and superposed.
where
∆ ′ =
σ
aN
/
and
∆ ′ =
θπ
θ
σ
′
′
6.2.1.1 Circular piston
The RayleighSommerfeld integral for a circular transducer
(Figure 6.1) excited with a timeharmonic and spatially uniform
normal particle velocity
U
0
is given by
−−
′
jk

rr

j
ωρ
π
2
π
a
e
∫
∫
0
jt
ω
px yzt
(, ,,)
=
Ue
σ
′
dd
σ
′
θ
,
(6.2)
0
2

rr
−
′

0
0
1
where the source points are given in cylindrical coordinates
and the observation points are given in Cartesian coordinates
with
0.8
)( sin
2 2 2
− ′ =−′σ ′θ+− ′σ ′θ+
. In Equation
6.2, primed coordinates are source coordinates, unprimed
coordinates are observation coordinates, σ′ is the radial coor
dinate of the source point, θ′ is the angle between the source
point and the
x
axis, and 
r
−
r
′ represents the distance from
each sampled source point to an observation point. The normal
of the circular piston is coincident with the positive
z
axis, and
the center of the piston is located at the origin of the polar
coordinate system.
Pressures in the nearfield region are often numerically
evaluated with the discretized version (Zemanek 1971) of the
RayleighSommerfeld integral in Equation 6.2 via

rr

(
x
cos
y
z
0.6
0.4
0.2
0
1.5
0.75
0.75
0.5
0
0.25
−0.75
0
−1.5
z
(units of
a
2
/
λ
)
x
(units of piston radius
a
)
FIGURE 6.2
Simulated nearfield pressure obtained with the
RayleighSommerfeld integral for a circular piston with radius
a
= λ.
The RayleighSommerfeld integral achieves a maximum error of 1%
in 177.19 seconds when evaluated with
NN
N
N
j
ωρ
π
σ
′
θ
′
−−′
jk

rr

e
∑
i
∑
,
(6.3)
0
jt
ω
px yzt
(, ,,)
=
Ue
σ
′
∆σσθ
′
∆ ′
===
′
N
212
on this
0
2

rr
− ′

σ
θ
′
121point by 101point grid.
i
=
1
=
1
σ
′
θ
′