Biomedical Engineering Reference
In-Depth Information
For on-axis near-field pressures generated by a circular pis-
ton, Equation 6.2 reduces to an analytical expression without
any approximations. The on-axis time-harmonic pressure gen-
erated by a circular piston excited with uniform normal particle
velocity
U
0
is given by
Rayleigh-Sommerfeld 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
Rayleigh-Sommerfeld integral is implemented in optimized C++
code that is called from MATLAB
®
as an executable program
through the MATLAB
®
MEX-file interface. The computation
time for a 121-point by 101-point 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 far-field 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 Rayleigh-Sommerfeld 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
).
Near-field and far-field pressures are often plotted with axial
distances normalized with respect to the far-field 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 near-field 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 far-field transition dis-
tance is often misinterpreted as the distance at which the far-field
approximation in Equation 6.5 is accurate. In fact, the far-field
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 far-field 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 near-field pressure computed with the
Rayleigh-Sommerfeld 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 121-point by 101-point
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
far-field directional factor for a rectangular element, and the jinc
function in Equation 6.5 is the far-field 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 far-field 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
′
