Biomedical Engineering Reference
InDepth Information
For onaxis nearfield pressures generated by a circular pis
ton, Equation 6.2 reduces to an analytical expression without
any approximations. The onaxis timeharmonic pressure gen
erated by a circular piston excited with uniform normal particle
velocity
U
0
is given by
RayleighSommerfeld integral for a circular piston are evaluated
on a desktop computer running the Windows XP operating sys
tem with a 2.4GHz Intel Core2 CPU 6600 and 4GB of RAM. The
RayleighSommerfeld integral is implemented in optimized C++
code that is called from MATLAB
®
as an executable program
through the MATLAB
®
MEXfile interface. The computation
time for a 121point by 101point grid with
N
= 212 is 177.19 sec
onds (just under 3 minutes).
22
ω
00
jt
−
jkz
−
jka z
+
pzt
(,)
=
ρ
cUe
e
−
e
.
(6.4)
In Equation 6.4,
c
represents the speed of sound and
z
indicates
the distance along the positive
z
axis with
x
=
y
= 0.
In the farfield region, the approximate pressure generated by
a circular piston is given by
6.2.1.2 rectangular and Square pistons
For a sinusoidally excited rectangular aperture with uniform
normal particle velocity
U
0
, the RayleighSommerfeld integral
in Equation 6.1 becomes
θ ≈
ωρ
j
2( sin)
sin
Jka
ka
θ
0
2
jt jkR
ω−
1
pRt
(, ,)
aU e
,
(6.5)
0
2
R
θ
a
b
−−′
jk

rr

j
ωρ
π
e
∫
∫
0
jt
ω
px yzt
(, ,,)
=
Ue
dx dy
′′
,
(6.7)
0
2

rr
− ′

2 2 2
is the distance from the center of the piston
to the observation point (
x ,y,z
),
where
Rxyz
=++
−
a
−
b
θ=
tan(
−
1
xyz
2
+
2
/)
is the angle
between the element normal (which is coincident with the positive
z
axis) and the vector that connects the center of the piston to the
observation point (
x ,y,z
), and
J
1
(·) is the Bessel function of the first
kind of order one. The expression in square brackets is also known
as the “jinc” function (analogous to the “sinc” function associated
with rectangular apertures), where the factor of 2 in the numerator
normalizes the result such that the peak value of the jinc function
is equal to one when the argument
ka
sinθ = 0 (i.e., when θ = 0).
Furthermore, when jinc(
x
) = 2
J
1
(
x
)/
x
as above, the zeros of jinc(
x
)
away from the peak value at
x
= 0 are the same as the zeros of
J
1
(
x
).
Nearfield and farfield pressures are often plotted with axial
distances normalized with respect to the farfield transition dis
tance, which is defined as
for calculations in Cartesian coordinates where
rr

− ′ =

2 2 2
− ′ +−′ +
is the distance from each point on
the radiating source to the observation point (
x ,y,z
), and primed
coordinates (
x
′,
y
′) represent the coordinates of the radiating
aperture. In Equation 6.7, the width of the rectangular piston
is 2
a
, and the height is 2
b
. The normal at the center of the rect
angular piston is coincident with the positive
z
axis, and the
center of the piston is located at the origin of the Cartesian coor
dinate system. Numerical calculations of the pressure gener
ated by a rectangular source in the nearfield region replace the
double integral with a double sum, and numerical evaluations of
these integrals with the midpoint rule replace
dx
′ and
dy
′ with
∆ ′ =
(
xx
)
(
yy
)
z
2/
(where
N
x
′
is the number of
samples, or abscissas, in the
x
direction, and
N
y
′
is the number
of abscissas in the
y
direction), yielding
xaN
x
2/
and
∆ ′ =
y
bN
y
′
′
d
2
4λ
(6.6)
N
N
y
′
j
ωρ
π
x
′
rr
rr
∆∆′′
−−′
jk


e
∑
∑
0
jt
ω
px yzt
(, ,,)
=
Ue
x
′
y
,
(6.8)
with
d
= 2
a
for a circular transducer. The farfield transition dis
tance is often misinterpreted as the distance at which the farfield
approximation in Equation 6.5 is accurate. In fact, the farfield
approximation in Equation 6.5 is only accurate at distances that
satisfy
Rd 4
0
2

− ′

i
=
1
i
=
1
x
′
y
′
where 
r
−
r
′ is the distance from each sampled source point to
each observation point. In Equation 6.8, the Δ
x
′Δ
y
′ represents the
area of a small rectangular patch on the piston face. The source
coordinate
r
′ indicates the location of the center of this patch.
In the farfield region, the approximate pressure generated by
a rectangular piston is given by
2
λ
, and other methods for computing the pres
sure are required closer to the piston.
An example of the nearfield pressure computed with the
RayleighSommerfeld integral for a circular piston with radius
a
= λ is shown in Figure 6.2. The circular piston is located in
the
z
= 0 plane and centered at (
x ,y,z
) = (0,0,0) as illustrated in
Figure 6.1. The pressure is calculated on a 121point by 101point
grid with a simplified version of Equation 6.3 that exploits the
symmetry of the circular piston by computing the integral from
0 to π and then doubling the result. Subject to the restriction
NN
/(
)
j
ωρ
π
kax
R
kby
R
0
jt jkR
ω−
px yzt
(, ,,)
≈
2
abUe
sinc
sinc
,
(6.9)
0
R
2 2 2
and the sinc function is defined as
sinc(
x
) = sin(
x
)/
x
. This definition of the sinc function is normal
ized such that the peak value is equal to one at
x
= 0 and the zeros
of this function are coincident with the zeros of the sine function
elsewhere. The product of sinc functions in Equation 6.9 is the
farfield directional factor for a rectangular element, and the jinc
function in Equation 6.5 is the farfield directional factor for a
circular element. Alternate forms of the sinc and jinc functions
where
Rxyz
=++
σ θ
, the smallest value of
N
that achieves a normal
ized error less than or equal to 1% throughout the entire grid
defined for this pressure calculation is
N
= 212. he computa
tional grid extends out to the farfield transition distance
a
2
/λ
(defined in Equation 6.6) in the axial direction, and the grid
extends out to
x
= ±1.5
a
laterally. Numerical calculations of the
==
′
N
′