2D/3D Acoustic Propagation Modeling:
2D focusing by a cylindrical fast lens
at finite wavelengths.
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2D surface acoustic wave resonance in co-directional
Waveguides are used in a variety
of devices where it is desirable to channel or contain propagating
wave energy within spatial boundaries. Common examples include surface
acoustic wave (SAW) filters and optical fibers. 2D reduced-velocity
strips are often used as SAW waveguides. An image from a finite-difference
time-domain (FDTD) simulation of a single reduced-velocity waveguide
strip is shown below. Figure 1 illustrates a snapshot of the amplitude
field of a continuously propagating acoustic mode 0 wavefunction
as it travels along a reduced-velocity strip guide. The guide lies
parallel to the X-axis, and is centered halfway along the Y-axis
Figure 1. Snapshot of grid amplitudes
for a mode 0 guidefunction propagating down a reduced-velocity strip
An alternative way of illustrating the energy
containment achieved by the strip waveguide of Figure 1 is to plot
intensity instead of amplitude. Figure 2 illustrates normalized
intensity versus distance along the guide for the same type of simulation
shown in Figure 1.
Figure 2. Normalized intensity for a mode 0 guidefunction propagating
down a reduced-velocity strip waveguide.
Strip waveguides placed side-by-side are referred
to as co-directional guides. One interesting aspect of co-directional
guides is the potential for resonance to occur. Figure 3 illustrates
what happens when two co-directional reduced-velocity strip waveguides
are located close enough to resonate. Normalized intensity is illustrated
over a section of the grid for the case when a mode 0 guidefunction
is sourced into one of the waveguides. Oscillations in energy between
the guides are apparent.
Figure 3. Normalized intensity illustrating resonance between
co-directional reduced-velocity strip waveguides.
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3D far-field extrapolation of 4 point sources arranged
in a plane.
The acoustic emission pattern created
by an arbitrary source region can be computed from the pressure
distribution over a surface surrounding the source region. From
the pressure and its normal derivative over any surface enclosing
any source region, an integral may be calculated to obtain the pressure
field at all points in space exterior to the surface. If the source
region can be modeled, then numerical wave propagation techniques
can be used to simulate the pressure and its normal derivative over
the enclosing surface, thus providing the input to the extrapolation
integral. Such an approach is remarkably accurate and efficient
from a computational standpoint, enabling the elimination of large
free-space propagation grids. Details of this technique and demonstrations
of its accuracy are provided in Aroyan (1996).
In this example, the far-field emission pattern
for a square arrangement of 4 point sources lying in the y-z plane
(see Figure 4) is computed. The emission pattern for such a simple
source arrangement can be analytically calculated, but is computed
here to illustrate the technique. The power of this approach is
best seen in cases of complex source regions (for example, see the
acoustic emission modeling of dolphin head tissues in the Bioacoustic
Modeling section) for which no analytical solution is available.
Closely related techniques also allow one to calculate the scattered
acoustic field from complex structures (including modeled tissues).
Figure 4. Four point sources
in the y-z plane arranged in a square configuration of side length
To calculate the pattern
below, the following procedure was used:
1. Simulate the acoustic field of the 4 point sources on a small
3D finite-difference time-domain (FDTD) grid with absorbing boundary
2. Collect the pressure and its normal derivative over an extrapolation
surface (a 6-sided rectangular 'box') surrounding the 4 point sources
on the simulation grid.
3. Use the pressure and normal derivative data from step 2 to compute
a far-field form of the Helmholtz surface integral to obtain the
far-field pattern of the 4 point sources.
4. Normalize the resulting pattern to its maximum value, and display
either intensity or decibel plots of this data.
Figure 5. Polar plot of far-field
intensity for the four point sources of Figure 4 (in phase).
3D polar plots of peaked emission data are visually
striking, but other types of plots often permit clearer visualization
of emission patterns over all spherical angles. The figures below
plot the same emission data in a flattened global projection. In
this projection, center-map corresponds to the positive x-direction,
"Above" corresponds to the positive z-direction, "L"=left
corresponds to the positive y-direction, etc.
Figure 6. Global
plots of same emission data as in Figure 5. Top figure: Surface
plot of data colored by decibel level. Bottom figure: Decibel
contour plot. Center-map is positive x-direction, "Above"
is positive z-direction, "L"=left is positive y-direction,
The extrapolation integral technique is a boundary
element method. A boundary element method was also used to calculate
the far-field pattern of a truncated Gaussian optical beam in the
example "3D far-field modeling of truncated Gaussian optical
beams". These two examples differ only in the wavelength scales
being simulated and in the geometry of the extrapolation surfaces
used: the extrapolation surface (a 6-sided 'box') in the above example
completely encloses the source region, whereas the Gaussian beam
is modeled over a single flat extrapolation surface (the input aperture).
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