^{1}, James F. Kelly

^{1}and Robert J. McGough

^{1,a)}

### Abstract

Analytical expressions are demonstrated for fast calculations of time-harmonic and transient near-field pressures generated by triangular pistons. These fast expressions remove singularities from the impulse response, thereby reducing the computation time and the peak numerical error with a general formula that describes the near-field pressure produced by any triangular piston geometry. The time-domain expressions are further accelerated by a time-space decomposition approach that analytically separates the spatial and temporal components of the numerically computed transient pressure. Applied to a Hanning-weighted input pulse, time-space decomposition converts each spatio-temporal integral into six spatial integral evaluations at each field point. Time-harmonic and transient calculations are evaluated for an equilateral triangle with sides equal to four wavelengths, and the resulting errors are compared to pressures obtained with exact and approximate implementations of the impulse response method. The results show that the fast near-field method achieves smaller maximum errors and is consistently faster than the impulse response and methods that approximate the impulse response.

This work was supported in part by NIH Grant R01 CA093669.

I. INTRODUCTION

II. TIME-HARMONIC AND TRANSIENT NEAR-FIELD PRESSURE CALCULATIONS FOR TRIANGULAR SOURCES

A. Impulse response calculations for a triangular source

1. Time-harmonic impulse response calcuations

2. Transient impulse response calculations

3. FIELD II

4. Smoothed impulse response

B. The fast near-field method for a triangular source

1. Transient FNM calculations

2. Time-space decomposition

C. Superposition calculations with impulse response and FNM expressions

D. Transient input waveform

E. Error calculations

III. RESULTS

A. Time-harmonic near-field pressure calculations

1. Reference pressure distribution

2. FNM and impulse response calculations

3. FIELD II calculations

4. Smoothed impulse response calculations

B. Transient near-field pressure calculations

1. Reference pressure distribution

2. FNM and impulse response calculations

3. FIELD II calculations

4. Smoothed impulse response calculations

IV. DISCUSSION

A. Time and error calculations

B. Advantages of the FNM for time-harmonic and transient calculations

C. FIELD II

D. Smoothed impulse response

V. CONCLUSION

### Key Topics

- Spatial analysis
- 9.0
- Fourier transforms
- 7.0
- Transducers
- 7.0
- Computer simulation
- 5.0
- Reference fields
- 5.0

## Figures

Triangular source geometries defined for near-field pressure calculations. The near-field pressure is evaluated above the vertex (indicated in bold), and the shape of the triangle (right, acute, or obtuse) is defined by the angle . The height of each triangle is indicated by , and the bases of the individual right triangles are indicated by , , and . The acute triangle in (b) is represented by the sum of two right triangles, and the obtuse triangle in (c) is defined as the difference between two right triangles.

Triangular source geometries defined for near-field pressure calculations. The near-field pressure is evaluated above the vertex (indicated in bold), and the shape of the triangle (right, acute, or obtuse) is defined by the angle . The height of each triangle is indicated by , and the bases of the individual right triangles are indicated by , , and . The acute triangle in (b) is represented by the sum of two right triangles, and the obtuse triangle in (c) is defined as the difference between two right triangles.

Superposition operations that calculate near-field pressures generated by the equilateral triangular source ABC, where each side is 4 wavelengths long. The vertex (indicated in bold) is the projection of the observation point onto the source plane, which partitions the radiating source into three triangles with sides . (a) The field point is located inside of the equilateral triangular source, and the total field is obtained by adding the contributions from the three triangles that share a vertex at . (b) The field point is located outside of the equilateral triangular source, and the total pressure is obtained by adding and subtracting the contributions from the three triangles that share a vertex at .

Superposition operations that calculate near-field pressures generated by the equilateral triangular source ABC, where each side is 4 wavelengths long. The vertex (indicated in bold) is the projection of the observation point onto the source plane, which partitions the radiating source into three triangles with sides . (a) The field point is located inside of the equilateral triangular source, and the total field is obtained by adding the contributions from the three triangles that share a vertex at . (b) The field point is located outside of the equilateral triangular source, and the total pressure is obtained by adding and subtracting the contributions from the three triangles that share a vertex at .

Simulated time-harmonic pressure field in the plane for an equilateral triangular source with sides equal to 4 wavelengths. The reference field is generated by the impulse response method computed with -point Gauss quadrature.

Simulated time-harmonic pressure field in the plane for an equilateral triangular source with sides equal to 4 wavelengths. The reference field is generated by the impulse response method computed with -point Gauss quadrature.

Peak normalized error for calculations of near-field pressures generated by the triangular source in Fig. 2 plotted as a function of the computation time. The results show that the FNM consistently achieves smaller errors in less time than exact and approximate impulse response calculations for time-harmonic excitations.

Peak normalized error for calculations of near-field pressures generated by the triangular source in Fig. 2 plotted as a function of the computation time. The results show that the FNM consistently achieves smaller errors in less time than exact and approximate impulse response calculations for time-harmonic excitations.

Simulated transient pressure field in the plane for a equilateral triangular source with sides equal to 4 wavelengths. For this calculation, the excitation is the Hanning-weighted pulse in Eq. (16), and the transient pressure is evaluated at 85 time points in an -point grid. The result is plotted at after the initiation of the input pulse.

Simulated transient pressure field in the plane for a equilateral triangular source with sides equal to 4 wavelengths. For this calculation, the excitation is the Hanning-weighted pulse in Eq. (16), and the transient pressure is evaluated at 85 time points in an -point grid. The result is plotted at after the initiation of the input pulse.

The peak normalized error plotted as a function of the computation time for the FNM/time-space decomposition method, the impulse response method, and methods that approximate the impulse response. These errors and times are evaluated for transient near-field calculations of an equilateral triangular source with sides equal to 4 wavelengths. The excitation for these calculations is a Hanning-weighted pulse with a center frequency of .

The peak normalized error plotted as a function of the computation time for the FNM/time-space decomposition method, the impulse response method, and methods that approximate the impulse response. These errors and times are evaluated for transient near-field calculations of an equilateral triangular source with sides equal to 4 wavelengths. The excitation for these calculations is a Hanning-weighted pulse with a center frequency of .

## Tables

Basis functions for time-space decomposition with a Hanning-weighted pulse.

Basis functions for time-space decomposition with a Hanning-weighted pulse.

Number of Gauss abscissas, computation times, and time ratios that describe the reduction in the computation time achieved with the fast near-field method relative to the impulse response and methods that approximate the impulse response for peak errors of 10% and 1%. The FNM and exact impulse response results are evaluated for time-harmonic calculations on a -point grid located in the plane, and the FIELD II and smoothed impulse response results are evaluated on an -point grid in the plane that is slightly offset from the transducer face.

Number of Gauss abscissas, computation times, and time ratios that describe the reduction in the computation time achieved with the fast near-field method relative to the impulse response and methods that approximate the impulse response for peak errors of 10% and 1%. The FNM and exact impulse response results are evaluated for time-harmonic calculations on a -point grid located in the plane, and the FIELD II and smoothed impulse response results are evaluated on an -point grid in the plane that is slightly offset from the transducer face.

Comparisons of computation times, input parameters, and time ratios that describe the reduction in the computation time achieved with the FNM and time-space decomposition relative to the exact and approximate impulse response for specified maximum errors of 10% and 1%. For FNM, impulse response, and FIELD II calculations with “usetriangles,” these transient results are evaluated in an -spatial point computed at 85 time points, and for the smoothed impulse response, the results are valued at the same temporal points in a restricted -point spatial grid.

Comparisons of computation times, input parameters, and time ratios that describe the reduction in the computation time achieved with the FNM and time-space decomposition relative to the exact and approximate impulse response for specified maximum errors of 10% and 1%. For FNM, impulse response, and FIELD II calculations with “usetriangles,” these transient results are evaluated in an -spatial point computed at 85 time points, and for the smoothed impulse response, the results are valued at the same temporal points in a restricted -point spatial grid.

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