# Acoustics in the Frequency Domain

Introduction | Perfectly Matched Layer (PML) |

Helmholtz Equation | Nomenclature |

Acoustic Boundary Conditions | References |

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"Complex Exponential Representation" |

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#### Monopole Sources

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#### Dipole Sources

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Time Domain Modeling | Frequency Domain Modeling | |

Governing PDE | Wave equation | Helmholtz equation |

Dependent variable | transient p(t,X) | stationary p(X) |

Harmonic excitation | yes | yes |

Inharmonic excitation | yes | no |

Computational cost | high | low |

Accuracy | low | high |

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#### Wave Equation: Time Domain Modeling

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#### Helmholtz Equation: Frequency Domain Modeling

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#### Accuracy Comparison

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#### Impedance Boundary Conditions in Time Harmonic Analysis

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#### Absorbing Boundary Conditions in Time Harmonic Analysis

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#### Sound Hard Boundary Conditions in Time Harmonic Analysis

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#### Sound Hard Boundary Conditions in Eigenfrequency Analysis

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#### Normal Velocity Boundary Conditions in Time Harmonic Analysis

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#### Sound Soft Boundary Conditions in Time Harmonic Analysis

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#### Sound Soft Boundary Conditions in Eigenfrequency Analysis

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#### Pressure Source Boundary Conditions in Time Harmonic Analysis

##### Dirichlet Model

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##### Neumann Model

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#### Radiation Boundary Conditions in Time Harmonic Analysis

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Symbol | Description | Unit |

ρ | density of a medium | [kg/m^{3}] |

c | speed of sound in a medium | [m/s] |

p | sound pressure | [Pa] |

p | local sound amplitude | [Pa] |

specified boundary pressure | [Pa] | |

conjugate of sound pressure | [Pa] | |

t | time | [s] |

t_{end} | simulation end time | [s] |

X | position vector | [m] |

s | direction switch | N/A |

F | optional dipole source | [N/m^{3}] |

dipole source strength | [N/m^{3}] | |

θ | dipole directivity angle | [rad] |

Q | optional monopole source | [1/s^{2}] |

monopole source strength | [1/s^{2}] | |

Null | seperation distance of dipole source | [m] |

λ | wavelength of sound | [m] |

Ω | simulation domain | [m] |

Null | wave number | [rad/m] |

Null | sound wave frequency | [Hz] |

ω | sound wave angular frequency | [rad/s] |

δ | Dirac delta function | N/A |

regularized delta function | N/A | |

Null | regularization parameter | [m] |

Null | mesh spacing | [m] |

Null | sound source location | [m] |

Null | effective bulk modulus | [Pa] |

α | attenuation factor | [m^{2}/(s·N)] |

ϕ | porosity | N/A |

V_{v} | viod volume | [m^{3}] |

V_{T} | total volume | [m^{3}] |

R_{f} | flow resistivity | [kg/(m^{3}·s)] |

β | standard deviation of a Gussian pulse | [m] |

ζ | sound particle displacement | [m] |

Null | sound particle velocity | [m/s] |

Null | specified boundary velocity | [m/s] |

T | sound wave period | [t] |

Z | characteristic impedence | [Null] |

Z_{b} | boundary impedance | [Null] |

A_{r} | amplitude of reflected wave | [Pa] |

A_{i} | amplitude of incident wave | [Pa] |

σ | absorbtion coefficient of PML | [rad/(s·m)] |

σ_{max} | maximum value of absorbtion coefficeint | [rad/(s·m)] |

1. Ihlenburg, Frank. The Medium-Frequency Range in Computational Acoustics: Practical and Numerical Aspects. Journal of Computational Acoustics, Vol.11, No. 2 175-193, 2003.

2. Heutschi, Kurt. Lecture Notes on Acoustics I. Swiss Federal Institute of Technology Zurich, 2016.

3. Johnson, Steven. Notes on Perfectly Matched Layers (PMLs). MIT, 2010.

4. Bilbao, Stefan and Hamilton, Brian. Directional Source Modeling In Wave-Based Room Acoustics Simulation. IEEE, 2017.

5. Peskin, Charles. The Immersed Boundary Method. Cambridge University, 2002.

6. Russell, Daniel, Titlow, Joseph and Bemmen, Ya-Juan. Acoustic monopoles, dipoles and quadropoles: An experiment revisited. American Journal of Physics 67, 660, 1999.

7. Vita, Micro. The Wave Equation with a Source. Oklahoma State University.

8. J. De Moerloose and M. A. Stuchly, Behavior of Berenger's ABC for evanescent waves, IEEE Microwave and Guided Wave Letters, vol. 5, no. 10, pp. 344-346, Oct. 1995.

9. J. Berenger, Evanescent waves in PML's: origin of the numerical reflection in wave-structure interaction problems, IEEE Transactions on Antennas and Propagation, vol. 47, no. 10, pp. 1497-1503, Oct. 1999.

10. J. De Moerloose, Jan & Stuchly, Maria. Reflection analysis of PML ABCs for low-frequency applications, IEEE Microwave and Guided Wave Letters, vol. 6., no. 4, pp. 177-179, Apr. 1996.

11. E. Turkel and A. Yefet. Absorbing PML boundary layers for wave-like equations, Applied Numerical Mathematics, vol. 27, pp. 533-557, 1998.

12. G. Pan, A. Abubakar and T. Habashy. An effective perfectly matched layer design for acoustic fourth-order frequency-domain finite-difference scheme, Geophysical Journal International, vol. 188, pp. 211-222, 2012.

13. T. J. Cox, P. D'Antonio. Acoustic absorbers and diffusers: Theory, design, and application, London: Spon Press, 2004.

14. E. W. Weisstein. Fast Fourier Transform, MathWorld--A Wolfram Web Resource. Retrieve from: http://mathworld.wolfram.com/FastFourierTransform.html.