# Acoustics in the Time Domain

Introduction | Appendix |

Wave Equation | Nomenclature |

Acoustic Boundary Conditions | References |

Perfectly Matched Layer (PML) |

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#### Time Harmonic Waves

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#### Time Inharmonic Waves

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#### Monopole Sources

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#### Dipole Sources

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- Dirichlet type boundary conditions. This type of boundary condition specifies the sound pressure at the boundary, and can be modeled with a DirichletCondition.

- Neumann type boundary conditions. This type of boundary condition specifies the sound particle velocity at the boundary, and can be modeled with a NeumannValue.

- Robin type boundary conditions. This type of boundary condition specifies the relation between time and normal derivatives of the sound pressure at the boundary, and can modeled with a NeumannValue since Robin type boundary conditions are technically generalized Neumann boundary conditions.

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- For the case where both media are the same we insert in the equations and we obtain absorbing boundary conditions:

- For the case where the impedance at the boundary approaches infinity, we insert in the equations and we obtain sound hard boundary conditions:

- For the case where the impedance at the boundary approaches zero, we multiply on both sides of (46) and take the limit as approaches 0. Then the equations we obtain are consistent with the definition of sound soft boundary conditions that . Note that sound soft boundary conditions should be directly modeled with DirichletCondition.

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#### Dirichlet Model

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#### Neumann Model

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#### Attenuation using an Artificial Complex Dimension

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#### PML Coordinate Transformations on the Coupled 1st Order Wave Equation

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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_{max} | maximum value of sound pressure | [Pa] |

sound pressure amplitude function | [Pa] | |

specified boundary 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}] | |

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

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

Null | separation 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] |

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

ϕ | porosity | N/A |

V_{V} | void volume | [m^{3}] |

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

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

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

ζ | sound particle displacement | [m] |

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

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

T | temperature | [K] |

Z | specific impedance | [Null] |

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

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

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

A_{t} | amplitude of transmitted wave | [Pa] |

complex spatial variable of PML | [m] | |

f_{p} | PML model parameter | [m] |

m_{f} | slope of PML artificial imaginary part | N/A |

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

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

Null | arbitrary function | N/A |

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