^{a)}

^{a)}Publication of this paper was delayed by request of author so that it would appear in the same issue with “Higher-order acoustic diffraction by edges of finite thickness” by Dezhang Chu, Timothy K. Stanton, and Allan D. Pierce.

^{1}, Dezhang Chu

^{1}and Guy V. Norton

^{2}

### Abstract

The acoustic diffraction by deformed edges of finite length is described analytically and in the frequency domain through use of an approximate line-integral formulation. The formulation is based on the diffraction per unit length of an infinitely long straight edge, which inherently limits the accuracy of the approach. The line integral is written in terms of the diffraction by a generalized edge, in that the “edge” can be a single edge or multiple closely spaced edges. Predictions based on an exact solution to the impenetrable infinite knife edge are used to estimate diffraction by the edge of a thin disk and compared with calculations based on the T-matrix approach. Predictions are then made for the more complex geometry involving an impenetrable thick disk. These latter predictions are based on an approximate formula for double-edge diffraction [Chu *et al.*, J. Acoust. Soc. Am.122, 3177 (2007)] and are compared with laboratory data involving individual elastic(aluminum) disks spanning a range of diameters and submerged in water. The results of this study show this approximate line-integral approach to be versatile and applicable over a range of conditions.

The authors are grateful to the anonymous reviewers for their thoughtful advice that led to significant improvements of this paper. The authors are also grateful to Shirley Barkley, Jayne Doucette, and Craig Johnson, all from the Woods Hole Oceanographic Institution (WHOI), Woods Hole, MA, for preparation of the manuscript to this paper, drawing certain figures, and for construction of the disks, respectively. This research was supported by the U.S. Office of Naval Research (Grant No. N00014-02-0095), WHOI, and by a grant of computer time at the U.S. Department of Defense High Performance Computing Shared Resource Center (Naval Research Laboratory, Washington, DC).

I. INTRODUCTION

II. THEORY

A. General formulation

B. Special cases

1. Straight finite-length edge(s)

2. Disk

3. First Fresnel zone and effective length

III. NUMERICAL CALCULATIONS—THIN IMPENETRABLE DISK

IV. LABORATORY EXPERIMENT—THICK ELASTIC DISK

V. RESULTS

A. Thin, impenetrable disk

1. General observations of numerical predictions

2. Comparison with deformed knife-edge model

B. Thick, elastic disk

1. General observations of laboratory data

2. Comparisons with models

VI. DISCUSSION

VII. CONCLUSIONS

### Key Topics

- Elasticity
- 15.0
- Acoustic wave diffraction
- 13.0
- Backscattering
- 13.0
- Exact solutions
- 6.0
- Diffraction theory
- 5.0

## Figures

Deformed truncated wedge of finite length.

Deformed truncated wedge of finite length.

Diffraction geometry for disk (backscattering). The source/receiver and disk are in the and planes, respectively.

Diffraction geometry for disk (backscattering). The source/receiver and disk are in the and planes, respectively.

(Color online) Diffracted echo in backscatter direction from infinitely long, impenetrable straight knife-edge (upper) compared with diffraction in backscatter direction by the leading edge of thin, impenetrable disks (lower). Since the diffracted field spreads differently for the infinitely long edge and the disks, the plots are on arbitrary scales for comparison. The frequency is for all predictions and the diameters of the disks are 8 and . The same (exact) expression for [Eq. (18)] was used to produce the upper plot as it was in the lower plots [once integrated through use of Eqs. (15) and (16) to give Eq. (19)], as well as in Fig. 5. The angle is normal incidence to flat surface of disk and half-plane associated with the infinite knife-edge, while corresponds to edge-on incidence.

(Color online) Diffracted echo in backscatter direction from infinitely long, impenetrable straight knife-edge (upper) compared with diffraction in backscatter direction by the leading edge of thin, impenetrable disks (lower). Since the diffracted field spreads differently for the infinitely long edge and the disks, the plots are on arbitrary scales for comparison. The frequency is for all predictions and the diameters of the disks are 8 and . The same (exact) expression for [Eq. (18)] was used to produce the upper plot as it was in the lower plots [once integrated through use of Eqs. (15) and (16) to give Eq. (19)], as well as in Fig. 5. The angle is normal incidence to flat surface of disk and half-plane associated with the infinite knife-edge, while corresponds to edge-on incidence.

Impulse response in backscatter direction for -diameter thin disk as calculated with the T-matrix method. Normal incidence is 0° and edge-on incidence is 90°. The calculations were over the range 2.5–87.5°. The color scale is in decibels relative to the maximum value of the entire plot. The time delay of corresponds to the center of the disk.

Impulse response in backscatter direction for -diameter thin disk as calculated with the T-matrix method. Normal incidence is 0° and edge-on incidence is 90°. The calculations were over the range 2.5–87.5°. The color scale is in decibels relative to the maximum value of the entire plot. The time delay of corresponds to the center of the disk.

(Color online) Comparisons between T-matrix and deformed-edge calculations for thin, impenetrable disks of diameters (upper) and (lower). The partial wave target strength (PWTS) of the diffraction by leading edge only is calculated in each case. As in Fig. 3, Eq. (19) was used for the deformed edge calculations, based on an exact solution to the infinite knife-edge. The leading edge echo was numerically separated from the trailing edge echo in the impulse response time series in the T-matrix calculations, although there was difficulty resolving the two echoes (hence resulting in some contamination) for angles below about 20°. All calculations involved for later comparison with the laboratory data.

(Color online) Comparisons between T-matrix and deformed-edge calculations for thin, impenetrable disks of diameters (upper) and (lower). The partial wave target strength (PWTS) of the diffraction by leading edge only is calculated in each case. As in Fig. 3, Eq. (19) was used for the deformed edge calculations, based on an exact solution to the infinite knife-edge. The leading edge echo was numerically separated from the trailing edge echo in the impulse response time series in the T-matrix calculations, although there was difficulty resolving the two echoes (hence resulting in some contamination) for angles below about 20°. All calculations involved for later comparison with the laboratory data.

Temporally compressed echo measured in backscatter direction versus orientation for -diameter aluminum disk submerged in water. The disk is thick. Normal incidence echoes (at , 180°, and 360°), leading and trailing double-edge echoes, and circumnavigated echoes are resolved. The circumnavigated waves occur at approximately after the trailing edge echoes. Other echoes arrive near the circumnavigated echoes and are out of the scope of this analysis. The color scale is in decibels relative to the maximum value of the entire plot. Apparent echoes at normal incidence arriving at negative time delays are actually processing sidelobes from the large zero-time-delay echoes. The abbreviated terminology “leading edge” and “trailing edge” correspond to the more rigorous description “leading double edge” and “trailing double edge.” From Stanton and Chu (2004).

Temporally compressed echo measured in backscatter direction versus orientation for -diameter aluminum disk submerged in water. The disk is thick. Normal incidence echoes (at , 180°, and 360°), leading and trailing double-edge echoes, and circumnavigated echoes are resolved. The circumnavigated waves occur at approximately after the trailing edge echoes. Other echoes arrive near the circumnavigated echoes and are out of the scope of this analysis. The color scale is in decibels relative to the maximum value of the entire plot. Apparent echoes at normal incidence arriving at negative time delays are actually processing sidelobes from the large zero-time-delay echoes. The abbreviated terminology “leading edge” and “trailing edge” correspond to the more rigorous description “leading double edge” and “trailing double edge.” From Stanton and Chu (2004).

Partial wave target strength (PWTS) of leading double-edge diffracted echo versus orientation angle for aluminum disks of various diameters. Predictions are given by the solid lines and laboratory data are given by the “+.” The diameters of the disks range from , each with a thickness of . The 0° angle corresponds to normal incidence to the flat surface of the disks, while 90° corresponds to edge-on incidence. The angle is illustrated in Fig. 2. The predictions are based on a formulation from Chu *et al.* (2007) that describes the diffraction by an impenetrable infinitely long, straight double edge. That formula is incorporated into the deformed edge line integral in this paper.

Partial wave target strength (PWTS) of leading double-edge diffracted echo versus orientation angle for aluminum disks of various diameters. Predictions are given by the solid lines and laboratory data are given by the “+.” The diameters of the disks range from , each with a thickness of . The 0° angle corresponds to normal incidence to the flat surface of the disks, while 90° corresponds to edge-on incidence. The angle is illustrated in Fig. 2. The predictions are based on a formulation from Chu *et al.* (2007) that describes the diffraction by an impenetrable infinitely long, straight double edge. That formula is incorporated into the deformed edge line integral in this paper.

PWTS of leading double-edge echo versus diameter of aluminum disk at and for three orientation angles. Model predictions are given by the solid lines and laboratory data are given by the “+.” The thickness of the disks is . The angle is illustrated in Fig. 2. The predictions use the same model (impenetrable deformed double edge) as in Fig. 7. The data for the 90° angle follow the trend of varying by , as predicted by the deformed edge model, over the entire range of diameters.

PWTS of leading double-edge echo versus diameter of aluminum disk at and for three orientation angles. Model predictions are given by the solid lines and laboratory data are given by the “+.” The thickness of the disks is . The angle is illustrated in Fig. 2. The predictions use the same model (impenetrable deformed double edge) as in Fig. 7. The data for the 90° angle follow the trend of varying by , as predicted by the deformed edge model, over the entire range of diameters.

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