Biomedical Engineering Reference
In-Depth Information
one-dimensional integral with respect to the angle ψ, and the
result is computed using Gauss quadrature.
Near-field pressures are computed with the FNM expression
for a circular piston, and the results are evaluated on the same
121- by 101-point grid shown in Figure 6.2. For a circular piston
with radius
a
= λ, the FNM expression in Equation 6.16 requires
only
N
= 8 Gauss abscissas to achieve errors less than or equal to
0.71% throughout the grid. This result, which is calculated with
optimized C++ code called from MATLAB
®
using the same
computer described previously, is computed in 0.0342 seconds.
This example shows that near-field pressure calculations with the
fast near-field method expression for a circular piston achieves
smaller errors in less time than the Rayleigh-Sommerfeld inte-
gral. For this combination of piston radius and computational
grid, where the errors obtained with both methods are less than
1%, the FNM is more than 5,000 times faster than the Rayleigh-
Sommerfeld integral.
triangles are subtracted, where the difference in these areas
of these triangles is once again equal to the area of the rectangu-
lar piston. The superposition of these integrals is handled seam-
lessly by Equation 6.17, where the leading terms associated with
each expression determine whether a contribution from a tri-
angle is added or subtracted at every point in space. For numeri-
cal calculations, all four of the one-dimensional integrals in
Equation 6.17 are discretized (with respect to the distance
x
′ or
y
′, as appropriate) and then the result is computed using Gauss
quadrature. In the paraxial region where |
x
| <
a
and |
y
| <
b
, the
numerical performance is further improved when each triangle
in Equation 6.17 is replaced with two right triangles and contri-
butions from eight integrals are instead evaluated.
When near-field pressures are computed for a square 2λ by 2λ
piston on the same 121- by 101-point grid shown in Figure 6.4,
the FNM expression in Equation 6.17 requires only
N
= 5 Gauss
abscissas to achieve errors less than or equal to 0.34% through-
out the entire computational grid. The FNM result for a square
piston, evaluated on the same computer running optimized
C++ code called from MATLAB
®
, is obtained in 0.1291 sec-
onds. Once again, the fast near-field method expression achieves
smaller errors in less time than the Rayleigh-Sommerfeld inte-
gral. For this square piston and computational grid, where the
errors obtained with both methods are less than 1% at all points
on the grid, the FNM expression is more than 3,000 times faster
than the Rayleigh-Sommerfeld integral.
6.2.3.2 rectangular piston
The FNM expression for the time-harmonic pressure generated by
a rectangular piston (McGough 2004) in Cartesian coordinates is
2
2
2
jt
ω
−
jk
z
+
() (
y
′ +− −
xa
)
jkz
=
ρ
cU e
yb
+
e
−
′ +−
e
∫
00
px yzt
(, ,,)
(
xa
−
)
dy
′
2
2
2
π
() (
y
x
a
)
yb
−
2
2
2
−
jk
z
+
() (
y
′ ++
−
xa
)
yb
+
jkz
e
−
′ ++
e
∫
−+
(
xa
)
dy
′
2
2
6.2.3.3 Spherical Shell
The FNM expression for the time-harmonic pressure generated
by a spherically focused transducer (McGough 2004) in cylin-
drical coordinates is
() (
y
x
a
)
yb
−
2
2
2
−
jk
z
+ ′ +−
−
() (
x
yb
)
xa
+
jkz
e
−
′ +−
e
∫
+−
(
yb
)
dx
′
2
2
() (
x
y
b
)
xa
−
2
2
2
−
jk
z
+ ′ ++
−
() (
x
yb
)
xa
+
jkz
e
−
′ ++
e
∫
(6.17)
−+
(
yb
)
dx
′
,
jt
ω
przt
(, ,)
=ρ
cU e
2
2
() (
x
y
b
)
00
xa
−
π
2
2
2
aR
( cos
r
ψ−+
R
a a
)
∫
×
where 2
a
is the piston width, 2
b
is the piston height, and (
x ,y,z
)
is the coordinate of the observation point. In Equation 6.17,
the primed coordinates are the variables of integration, and the
limits of integration represent the
x
and
y
components of the dis-
tances from the observation point to the four vertices of the rect-
angular piston. The center of the rectangular piston is located at
x
= 0,
y
= 0,
z
= 0, which is the center of the Cartesian coordinate
system. Equation 6.17 evaluates the near-field pressure by super-
posing the contributions from four triangles, where the base of
each triangle is one side of the rectangular piston. In Equation
6.17, each integral represents the contribution from a single tri-
angle. When the observation point is located within the paraxial
region where |
x
| <
a
and |
y
| <
b
, the contributions from the four
triangles are added, and the sum of the areas of the triangles
is equal to the area of the rectangular piston. When the obser-
vation point is located outside of the paraxial region such that
|
x
| >
a
and |
y
| >
b
, the contributions from the two largest tri-
angles are added and the contributions from the two smallest
π
(
Rr
22
+
2
ar
cos
ψ −− ψ+
Razar
2
2
22
cos
2
z a
2
2
)
0
ψ
22 2
2
2
22
−
jk
Rrz
+ +− ψ+
2 os
r
2
z
Ra
−
−+ +
jk R
(
ign(
z
)
rz
)
×
[
e
−
e
]
d
,
(6.18)
where
a
is the aperture radius and
R
is the radius of curvature
of the spherical shell. In cylindrical coordinates (
r,z
), the center
of the sphere is located at
r
= 0 and
z
= 0, the normal at the cen-
ter of the spherical shell is coincident with the
z
axis, and the
spherical shell intersects the
z
axis at
z
= −
R
. In Equation 6.18,
ψ is the variable of integration. As for a circular piston, when
the pressure is evaluated on the
z
axis and therefore
r
= 0, all ψ
dependence is eliminated from the integrand, and the analytical
expressions for the on-axis pressure in Equations 6.12 and 6.13
are obtained. For numerical calculations, the one-dimensional
integral in Equation 6.18 is discretized with respect to the angle
ψ and evaluated with Gauss quadrature.
