Biomedical Engineering Reference
In-Depth Information
where the integral is evaluated in spherical coordinates and the
pressure is computed in Cartesian coordinates with rr
where the expression in square brackets is the jinc function.
For pressure evaluations on-axis where r = 0, the jinc function
is equal to one, so Equation 6.14 is exactly equal to the expres-
sion in Equation 6.13 when evaluated at the geometric focus.
Equation 6.14 provides an accurate numerical estimate of the
main lobe width for all values of a < R . Equation 6.14 also accu-
rately represents the sidelobe levels as well as the locations of
the first few nulls and sidelobe peaks when the f / # , defined for a
spherical shell as R /(2 a ), is greater than one, where much better
agreement over a wider range of r values is achieved with a larger
f / # .
An example of the pressure computed with the Rayleigh-
Sommerfeld integral for a spherically focused transducer with
aperture radius a = 20λ and radius of curvature R = 20 λ is
shown in Figure 6.6. The geometric center of the spherically focused
transducer is located at ( x ,y,z ) = (0,0,0), and the normal at the cen-
ter of the spherical shell is coincident with the z axis. Furthermore,
the spherical shell intersects the z axis at z = − R , and the circle that
defines the aperture opening is located at z
|
− ′ =
|
2 2 2
− θ ′φ+− θ ′φ+− φ In Equ-
ation 6.10, primed coordinates are source coordinates, unprimed
coordinates are observation coordinates, ( R ,θ′,φ′) is the spheri-
cal coordinate of a source point, and | r r ′| represents the dis-
tance from each sampled source point to the observation point.
The limits of integration for the φ′ coordinate are selected such
that the center of the spherical shell intersects the z axis at z = − R
(i.e., on the negative z axis), and the origins of both the source
and observer coordinate systems are located at the geometric
focus of the spherical shell. The spherical coordinate system
defined for a spherically focused transducer is shifted relative to
the coordinate systems defined in the center of the piston face for
circular and rectangular pistons in Equation 6.2 and Equation
6.7, respectively.
The discretized version of the Rayleigh-Sommerfeld integral
in Equation 6.10 is
(
xR
cos
sin) (
yR
sin
sin) (
zR
cos
).
2 2 as
illustrated in Figure 6.5. The pressure is evaluated on a 101-point by
201-point grid that extends from x = − a to x = a laterally and from z
= − a to z = a axially, where the lateral extent of the grid is equal to
the aperture diameter and the minimum axial value is determined
by the coordinate of the aperture opening. The pressure calcula-
tions are performed with a simplified version of Equation 6.11
that exploits the cylindrical symmetry of the spherically focused
transducer and the computational grid by computing the integral
from 0 to π and then doubling the result. The number of abscissas
in both directions is again restricted such that NN
N
=− −=−
Ra
a
N
φ
θ
−−′
jk
|
rr
|
j
ωρ
π
e
jt
ω
2
px yzt
(, ,,)
=
0
Ue
R
si
n
φφθ
∆∆
,
(6 .11)
0
2
|
rr
− ′
|
i
=
1
i
=
1
φ
θ
1
/ N , N φ is the number of
abscissas in the φ′ direction, and N θ is the number of abscissas
in the θ′ direction. In Equation 6.11, R 2 sinφ′Δ φ′Δθ ′ represents the
area of a small patch on the spherical shell, the source coordinate
r ′ indicates the location of the center of this patch, and r indi-
cates the location of the observation point.
The spherical shell also admits exact expressions for the on-
axis pressure as well as approximate expressions for the pressure
in the focal plane. The on-axis time-harmonic pressure gener-
ated by a spherical shell excited with uniform normal particle
velocity U 0 is given by
where ∆ ′ =
φ
sin(/)/
aR N
, ∆ ′ =
θπ θ
φ
φ θ . he
smallest value of N that achieves a normalized error less than or
==
N
R
z
2
2
22
+ −+
j
kkR z
+
jt
ω
jk
z
2
z
Ra R
pr
(
=
0
,
zt
,)
=
ρ
cU
e
e
e
(6.12)
00
1
for all points where r = 0 and z ≠ 0 (i.e., at all points on the axis
of symmetry other than the geometric focus) when pressures are
evaluated in cylindrical coordinates ( r,z ) with the radial coordinate
defined as r
0.75
0.5
= 2 2 . The expression for the on-axis pressure
in Equation 6.12 is exactly equivalent to the on-axis expression in
(O'Neil 1949). At the geometric focus where r = 0 and z = 0, the time
harmonic pressure generated by a spherical shell is
xy
0.25
0
1
0.5
jt jkR
00
ω−
2
2
0.5
pr
( ,
===ωρ
z
,)
t
j
Ue
(
R
Ra
).
(6.13)
0
0
−0.5
−0.5
An expression from (O'Neil 1949) for the approximate pressure
in the focal plane ( z = 0) is
−1
−1
x (units of aperture radius a )
z (units of aperture radius a )
2
2
=≈ωρ −−
+
RR
(
Ra
Rr
)
FIGURE 6.6 Simulated pressure obtained with the Rayleigh-
Sommerfeld integral for a spherically focused transducer with aperture
radius a = 20λ and radius of curvature R = 20 λ . The Rayleigh-
Sommerfeld integral achieves a maximum error of 1% in 88.22 seconds
when evaluated with NN
prz
(,
0, )
t
j
0
2
2
2
2
2(
Jkar Rr
karRr
/
+
)
22
jt jk
ω−
Rr
+
1
xU e
,
(6.14)
===
N
109 on this 101-point by 201-point
0
2
2
/
+
σ
θ
grid.
 
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