Biomedical Engineering Reference
In-Depth Information
all directions in an infinite space). The main advantages of the
Rayleigh-Sommerfeld 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 Rayleigh-Sommerfeld integral include rel-
atively long computation times and large errors in the near-field
region due to slow convergence.
The Rayleigh-Sommerfeld 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 time-harmonic 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 frequency-domain 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 Rayleigh-Sommerfeld integral for a circular transducer
(Figure 6.1) excited with a time-harmonic 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 near-field region are often numerically
evaluated with the discretized version (Zemanek 1971) of the
Rayleigh-Sommerfeld 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 near-field pressure obtained with the
Rayleigh-Sommerfeld integral for a circular piston with radius a = λ.
The Rayleigh-Sommerfeld 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
121-point by 101-point grid.
i
=
1
=
1
σ
θ
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