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 strip waveguides.

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 range.
Figure 1. Snapshot of grid amplitudes for a mode 0 guidefunction propagating down a reduced-velocity strip waveguide.


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 s=0.9 l.

 

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 conditions.
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, etc.


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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