^{1,a)}, Timothy K. Stanton

^{1}and Allan D. Pierce

^{2}

### Abstract

A cw solution of acoustic diffraction by a three-sided semi-infinite barrier or a double edge, where the width of the midplanar segment is finite and cannot be ignored, involving all orders of diffraction is presented. The solution is an extension of the asymptotic formulas for the double-edge second-order diffraction via amplitude and phase matching given by Pierce [A. D. Pierce, J. Acoust. Soc. Am.55, 943–955 (1974)]. The model accounts for all orders of diffraction and is valid for all , where is the acoustic wave number and is the width of the midplanar segment and reduces to the solution of diffraction by a single knife edge as . The theory is incorporated into the deformed edge solution [Stanton *et al.*, J. Acoust. Soc. Am.122, 3167 (2007)] to model the diffraction by a disk of finite thickness, and is compared with laboratory experiments of backscattering by elastic disks of various thicknesses and by a hard strip. It is shown that the model describes the edge diffraction reasonably well in predicting the diffraction as a function of scattering angle, edge thickness, and frequency.

The authors would like to thank the reviewers for their helpful comments and suggestions, which helped to improve the quality of the paper significantly. This work was supported by the US Office of Naval Research and by the Woods Hole Oceanographic Institution.

I. INTRODUCTION

II. DEFINITION OF THE DIFFRACTION PROBLEM

III. FIRST-ORDER DIFFRACTION

A. Background and Pierce’s solution for first-order diffraction

B. Special limiting cases of first-order diffraction

IV. HIGHER-ORDER DIFFRACTION

V. RESULTS AND DISCUSSION

VI. SUMMARY AND CONCLUSION

### Key Topics

- Acoustic wave diffraction
- 21.0
- Light diffraction
- 20.0
- Exact solutions
- 18.0
- Diffraction theory
- 11.0
- Elasticity
- 11.0

## Figures

(Color online) (a) 2D cross-sectional view of diffraction by infinitely long straight truncated wedges of two thicknesses. “S” and “R” indicate the locations of the point source and receiver, respectively. (b) Three-dimensional (3D) view of diffraction by an infinitely long straight truncated wedge.

(Color online) (a) 2D cross-sectional view of diffraction by infinitely long straight truncated wedges of two thicknesses. “S” and “R” indicate the locations of the point source and receiver, respectively. (b) Three-dimensional (3D) view of diffraction by an infinitely long straight truncated wedge.

Comparison between laboratory data and knife-edge diffraction model. The “+” symbol indicates measured diffraction by the leading-edge of an aluminum disk with a diameter of and thickness of . The solid curve is the model prediction based on an infinitely long knife edge from Pierce (1974) and with the curvature of the edge accounted for using the method described in Stanton *et al.* (2007). The dashed curve is the sum of the first-order diffracted waves from the two right-angle wedges at .

Comparison between laboratory data and knife-edge diffraction model. The “+” symbol indicates measured diffraction by the leading-edge of an aluminum disk with a diameter of and thickness of . The solid curve is the model prediction based on an infinitely long knife edge from Pierce (1974) and with the curvature of the edge accounted for using the method described in Stanton *et al.* (2007). The dashed curve is the sum of the first-order diffracted waves from the two right-angle wedges at .

(Color online) Geometry illustrating different diffraction regions associated with diffraction by an infinitely long straight wedge. 2D cross-sectional view.

(Color online) Geometry illustrating different diffraction regions associated with diffraction by an infinitely long straight wedge. 2D cross-sectional view.

(Color online) Comparison between the exact solution [Eq. (2a)–(2c) normalized by ] and the diffraction amplitude defined in Eq. (5) and based on an asymptotic expansion of , i.e., Eq. (9), as a function of range to the apex. The circle on the solid curve appearing at corresponds to the position about away from the apex, or the thickness of the disk used in generating Fig. 2.

(Color online) Comparison between the exact solution [Eq. (2a)–(2c) normalized by ] and the diffraction amplitude defined in Eq. (5) and based on an asymptotic expansion of , i.e., Eq. (9), as a function of range to the apex. The circle on the solid curve appearing at corresponds to the position about away from the apex, or the thickness of the disk used in generating Fig. 2.

(Color online) Diagram of higher-order diffraction.

(Color online) Diagram of higher-order diffraction.

Geometry illustrating model angles and ranges for the diffraction by an infinitely long straight truncated wedge or a double-edge. 2D cross-sectional view.

Geometry illustrating model angles and ranges for the diffraction by an infinitely long straight truncated wedge or a double-edge. 2D cross-sectional view.

Coefficient vs angle for several cases. The dashed curve corresponds to the limit of the thickness , the dot-dashed curve is for the limit determined by Eq. (41), and the solid curve corresponds to interpolated computed from the interpolation function Eq. (32) with and the frequency of and a fit to laboratory data, as described later in this section.

Coefficient vs angle for several cases. The dashed curve corresponds to the limit of the thickness , the dot-dashed curve is for the limit determined by Eq. (41), and the solid curve corresponds to interpolated computed from the interpolation function Eq. (32) with and the frequency of and a fit to laboratory data, as described later in this section.

The ratio of width to wavelength of the truncated wedge, which is required to make the difference in edge diffraction between a knife edge and a double edge of finite thickness less than 5%, vs the scattering angle for a backscattering geometry.

The ratio of width to wavelength of the truncated wedge, which is required to make the difference in edge diffraction between a knife edge and a double edge of finite thickness less than 5%, vs the scattering angle for a backscattering geometry.

Diffraction amplitude normalized by the converged amplitude as a function of diffraction orders at two different incident angles for a monostatic scattering geometry.

Diffraction amplitude normalized by the converged amplitude as a function of diffraction orders at two different incident angles for a monostatic scattering geometry.

Comparison of laboratory data involving an aluminum disk with various diffraction models.

Comparison of laboratory data involving an aluminum disk with various diffraction models.

(Color online) Comparison of the measured partial wave target strength of an aluminum disk (diameter of ) of various thickness (, 1.0, 1.5, and ) with the model of all orders of diffractions [Eq. (28)] at different angles. For the scattering angle at 90° (bottom right), the actual angle is 89.6° to exclude the “edge-on” specular term.

(Color online) Comparison of the measured partial wave target strength of an aluminum disk (diameter of ) of various thickness (, 1.0, 1.5, and ) with the model of all orders of diffractions [Eq. (28)] at different angles. For the scattering angle at 90° (bottom right), the actual angle is 89.6° to exclude the “edge-on” specular term.

(a) Geometry and notations of the parameters for diffraction by a hard strip. The view is along the infinite length of the strip and the distance between points “” and “” is the width . (b) Comparison of data (Medwin *et al.*, 1982) with the second-order diffraction model [Eq. (47)] (solid) and Medwin’s “double” or second-order diffraction (dashed). The data are the ratio of the total field to that of one-half of the total first-order diffraction (ray paths and ). The model parameters are , , , and [Eq. (32)]. The distances from the strip to the source and the receiver are [, see (a)] and [, see (a)], respectively. The width of the strip is . The frequencies are from about . Note that the horizontal axis in Medwin *et al.* is frequency in log-scale and we use a dimensionless variable kw on a linear scale, where is the width of the strip.

(a) Geometry and notations of the parameters for diffraction by a hard strip. The view is along the infinite length of the strip and the distance between points “” and “” is the width . (b) Comparison of data (Medwin *et al.*, 1982) with the second-order diffraction model [Eq. (47)] (solid) and Medwin’s “double” or second-order diffraction (dashed). The data are the ratio of the total field to that of one-half of the total first-order diffraction (ray paths and ). The model parameters are , , , and [Eq. (32)]. The distances from the strip to the source and the receiver are [, see (a)] and [, see (a)], respectively. The width of the strip is . The frequencies are from about . Note that the horizontal axis in Medwin *et al.* is frequency in log-scale and we use a dimensionless variable kw on a linear scale, where is the width of the strip.

## Tables

Values of and diffraction in three diffraction regions shown in Fig. 3 computed using Eqs. (11)–(13), (19), and (20). In obtaining the results, we have assumed and in our computations. Note that since , , where .

Values of and diffraction in three diffraction regions shown in Fig. 3 computed using Eqs. (11)–(13), (19), and (20). In obtaining the results, we have assumed and in our computations. Note that since , , where .

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