### Abstract

A classical problem in acoustic (and electromagnetic) scattering concerns the evaluation of the Green's function for the Helmholtz equation subject to impedance boundary conditions on a half-space. The two principal approaches used for representing this Green's function are the Sommerfeld integral and the (closely related) method of complex images. The former is extremely efficient when the source is at some distance from the half-space boundary, but involves an unwieldy range of integration as the source gets closer and closer. Complex image-based methods, on the other hand, can be quite efficient when the source is close to the boundary, but they do not easily permit the use of the superposition principle since the selection of complex image locations depends on both the source and the target. We have developed a new, hybrid representation which uses a finite number of real images (dependent only on the source location) coupled with a rapidly converging Sommerfeld-like integral. While our method applies in both two and three dimensions, we restrict the detailed analysis and numerical experiments here to the two-dimensional case.

Original language | English (US) |
---|---|

Pages (from-to) | 1-13 |

Number of pages | 13 |

Journal | Wave Motion |

Volume | 51 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2014 |

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### Keywords

- Green's function
- Half-space
- Helmholtz
- Impedance
- Layered media
- Sommerfeld integral

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

**On the efficient representation of the half-space impedance Green's function for the Helmholtz equation.** / O'Neil, Michael; Greengard, Leslie; Pataki, Andras.

Research output: Contribution to journal › Article

*Wave Motion*, vol. 51, no. 1, pp. 1-13. https://doi.org/10.1016/j.wavemoti.2013.04.012

}

TY - JOUR

T1 - On the efficient representation of the half-space impedance Green's function for the Helmholtz equation

AU - O'Neil, Michael

AU - Greengard, Leslie

AU - Pataki, Andras

PY - 2014/1

Y1 - 2014/1

N2 - A classical problem in acoustic (and electromagnetic) scattering concerns the evaluation of the Green's function for the Helmholtz equation subject to impedance boundary conditions on a half-space. The two principal approaches used for representing this Green's function are the Sommerfeld integral and the (closely related) method of complex images. The former is extremely efficient when the source is at some distance from the half-space boundary, but involves an unwieldy range of integration as the source gets closer and closer. Complex image-based methods, on the other hand, can be quite efficient when the source is close to the boundary, but they do not easily permit the use of the superposition principle since the selection of complex image locations depends on both the source and the target. We have developed a new, hybrid representation which uses a finite number of real images (dependent only on the source location) coupled with a rapidly converging Sommerfeld-like integral. While our method applies in both two and three dimensions, we restrict the detailed analysis and numerical experiments here to the two-dimensional case.

AB - A classical problem in acoustic (and electromagnetic) scattering concerns the evaluation of the Green's function for the Helmholtz equation subject to impedance boundary conditions on a half-space. The two principal approaches used for representing this Green's function are the Sommerfeld integral and the (closely related) method of complex images. The former is extremely efficient when the source is at some distance from the half-space boundary, but involves an unwieldy range of integration as the source gets closer and closer. Complex image-based methods, on the other hand, can be quite efficient when the source is close to the boundary, but they do not easily permit the use of the superposition principle since the selection of complex image locations depends on both the source and the target. We have developed a new, hybrid representation which uses a finite number of real images (dependent only on the source location) coupled with a rapidly converging Sommerfeld-like integral. While our method applies in both two and three dimensions, we restrict the detailed analysis and numerical experiments here to the two-dimensional case.

KW - Green's function

KW - Half-space

KW - Helmholtz

KW - Impedance

KW - Layered media

KW - Sommerfeld integral

UR - http://www.scopus.com/inward/record.url?scp=84887627482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887627482&partnerID=8YFLogxK

U2 - 10.1016/j.wavemoti.2013.04.012

DO - 10.1016/j.wavemoti.2013.04.012

M3 - Article

AN - SCOPUS:84887627482

VL - 51

SP - 1

EP - 13

JO - Wave Motion

JF - Wave Motion

SN - 0165-2125

IS - 1

ER -