8. Technical summary

This section indicates which numerical techniques were used to build the Wave Acoustics Simulator.

Links and suggestions for further reading are also provided.

The Wave Acoustics Simulator (W.A.S.) uses the Finite-difference time-domain (FDTD) numerical method to derive its solutions. Hence, all the simulations are natively carried out in the time domain.

The FDTD method has been applied to a number of disciplines ranging from electromagnetics Tavlov (2005), Schneider (2020) and geophysics Kim et al. (2010) to acoustics Yokota et al. (2002) and vibroacoustics Ferreira et al. (2017).

In order to allow waves to escape the computational domain with negligable reflections, a perfectly matched layer (PML) was implemented. The PML consists of a non-physical boundary that absorbs most of the incident sound waves. The PML boundaries used in W.A.S. were implemented according to Berenger (1994). The paper by Berenger is applied to electromagnetic waves, but a straight-forward adaptation from eletromagnetics FDTD to acoustics FDTD is provided in chapter 19 of Kunz (1993).

The simulation boundaries are implemented according to the method described by Suzuki et al. (2007), in which the physical properties (density and speed of sound) of the medium change in the boundaries region, to yield a particular value of sound absorption coefficient. The sound absorption coefficient can be set by the user in 'Boundaries->Set alpha'.

In FDTD, sound sources are most commonly implemented as sound pressure sources but implementations using particle velocity can also be found in the literature Ferreira et al. (2013). Currently, in W.A.S. all sound sources are implemented as pressure sources.

In terms of their implementation algorithm, sources can either be classified as hard, soft or transparent Schneider (1998). W.A.S. uses both hard sources (‘SINE’ and ‘NOISE’ types) and soft sources (‘PULSE’ type).

In order to ensure stability of the simulations, space and time resolutions must obbey the Courant condition. For the 2D case, the relationship between time and space resolutions is given by:

where c is the highest phase velocity of wave motion propagation (set to 343 m/s in W.A.S) and $\Delta h$ is the spatial resolution (size of a grid cell), which can be set by the user in ‘Settings->Resolution’. For further information about the Courant condition, the user is suggested to read Kunz (1993).

It is possible to view the root-mean-square (RMS) time average of the sound pressure field as an alternative to instantaneous sound pressure. The advantage of this type of visualization is that it makes it easier to to apreciate the steady-state sound pressure distribution, such as finding the location of nodal lines. The calculation method used to calculate the RMS time average of the sound pressure is given by the following integral, as described in Hopkins (2007):

where T is the averaging time.