When designing sound sources for an FDTD simulation, one normally faces two principal questions. First, an excitation signal should be chosen based on the desired temporal and spectral characteristics of the source. Next, one chooses an injection method, which most generally involves imposing or superimposing the
excitation signal on a grid node. Ideally, one would have the freedom to choose any arbitrary excitation signal and to transparently inject it into the grid without any ramifications. However, several physical and numerical systems govern the process of exciting an FDTD grid, and these impose constraints that should be taken into consideration when designing sources. In this section these constraints are identified by analysing the physical and numerical systems that govern the source.
4.3.1
Transduction Constraints
In Equation (4.1), the time derivative of pressure and space derivatives of particle velocity are related to the rate of fluid emergence, q(x, t). Considering a source positioned at a single node of an FDTD grid, in which each cell occupies a volume equal to X3, then fluid emergence in the system can be expressed by means of the source’s volume velocity ˆq(t), as follows
q(x, t) = ρ0
X3qˆ(t)δ(x−x
0) (4.7)
where x0 = (x0, y0, z0)∈
R3 denotes the source position. Equation (4.7) presents
a scaling constraint which relates the amplitude of the source to the volume it occupies. By combining equations (4.2) and (4.1), the particle velocity vector is eliminated and the wave equation is derived. Note that here equation (4.2) is used without the acoustic force term. It follows from this derivation that the source term in the wave equation becomes
ψ(x, t) = ∂q(x, t) ∂t = ρ0 X3 d dtqˆ(t)δ(x−x 0 ) (4.8)
Physically, the quantity ψ(x, t) has the dimensions of density per unit time squared (kg m−3 s−2), and can be thought of as fluid emergence due to vol- ume acceleration of the source. Thus, when designing sources for use in wave equation schemes, a differentiation constraint also applies, meaning that volume velocity should be injected as its first time derivative. Observe that the source terms in Equations (4.1) and (4.5) are supplemental to the fundamental time-
space relationships, that is, if one sets q(x, t) = 0 then the homogeneous wave equation is obtained. This implies that fluid emergence is an additive process, which numerically means that source nodes should also be evaluated with the FDTD update equations for air. Therefore, there are either two or three trans- duction constraints, depending on the chosen method. For Yee-type methods, the governing physics show that volume velocity sources should be appropriately scaled (scaling constraint), as well as superimposed on the grid (superposition constraint). In wave-equation methods the same constraints apply, but in addi- tion, source functions should be differentiated in time (differentiation constraint).
4.3.2
Mechanical Constraints
In order to generate a volume velocity at the source, some mechanical process is required. This normally corresponds to some vibrating object, for example, a pulsating sphere or a moving piston. Such mechanical systems are governed by the laws of motion, and accordingly introduce further modelling constraints. Depending on the complexity of the modelled transducer, these can range from a simple second order linear system, to a full physical representation of a loud- speaker (Small, 1972). Whilst many mechanical constraints are specific to the chosen transducer, some can be generalised to the problem of sound production. Continuous DC flow, for example, is something that transducers generally cannot produce, and as such, one would expect that
Z ∞ −∞
ψ(x, t)dt = 0 (4.9)
which naturally occurs if the differentiation constraint described in Section 4.3.1 is adhered to. However, if one decides to arbitrarily chooseψ(x, t), then failure to adhere to this constraint may have detrimental ramifications, as will be further discussed.
4.3.3
Numerical Constraints
Finite difference methods are subject to numerical dispersion, which increases as the ratio of the sample rate to the modelled frequency is decreased. This results in waves whose phase velocities are dependent on frequency and in non- isotropic schemes, also on the direction of propagation. When the grid is excited at frequencies which are prone to substantial dispersion, the resulting response is contaminated with numerical errors, which not only impair the ability to per- form visual analysis, but also introduce undesired audible artefacts. Thus, if one employs a simulation in order to visually investigate wave phenomena, then it is important that high frequencies are removed from the excitation signal, to pre- vent these from being generated in the model. This is referred to as Bandwidth Constraint.
When transient phenomena are studied, an impulsive source is required and therefore bandlimiting the excitation function presents a tradeoff between the source’s bandwidth and its time compactness. If the goal of the simulation is only to produce auralisations, then one may opt to excite the grid directly with the programme material to be auralised. A more efficient way is to model the room’s response using a unit impulse, and then employ convolution in the post- processing stage. Either way, if one wishes to eliminate the audible effects of dispersion, the result should be appropriately bandlimited.
Another numerical constraint involves the temporal length of the source func- tion. If the excitation signal is not finite in time, then one should truncate it at points such that any discontinuity errors arising from the truncation will be lower than any other numerical errors inherent in the scheme. Thus, it is useful that the excitation signal is finite by definition, which is here referred to as a Finiteness Constraint. Ideally, the value of the source and all of its derivatives up to the truncation order of the scheme would be zero at simulation onset, however this has been shown to be more prominent in higher order numerical schemes (Botts et al., 2010).