### 2.7 Generalizing the physical model

There are a large number of ways in which the present particularly-simple analogue model can be generalised. Obvious issues within the current physical framework are:
• Working in truly (1+1) or (2+1) dimensional systems.
• Adding vorticity, to go beyond the irrotational constraint.

Beyond these immediate questions, we could also seek similar effects in other physical or mathematical frameworks.

#### 2.7.1 External forces

Adding external forces is (relatively) easy, an early discussion can be found in [624] and more details are available in [641]. The key point is that with an external force one can to some extent shape the background flow (see for example the discussion in [249]). However, upon linearization, the fluctuations are insensitive to any external force.

#### 2.7.2 The role of dimension

The role of spacetime dimension in these acoustic geometries is sometimes a bit surprising and potentially confusing. This is important because there is a real physical distinction, for instance, between truly (2+1)-dimensional systems and effectively (2+1)-dimensional systems in the form of (3+1)-dimensional systems with cylindrical (pancake-like) symmetry. Similarly, there is a real physical distinction between a truly (1+1)-dimensional system and a (3+1)-dimensional system with transverse (cigar-like) symmetry. We emphasise that in Cartesian coordinates the wave equation

where
holds independent of the dimensionality of spacetime. It depends only on the Euler equation, the continuity equation, a barotropic equation of state, and the assumption of irrotational flow [607, 622, 626, 624].

Introducing the inverse acoustic metric , defined by

the wave Equation (119) corresponds to the d’Alembertian wave equation in a curved space-time with contravariant metric tensor:
where is the dimension of space (not spacetime). The covariant acoustic metric is then

d = 3:
The acoustic line element for three space and one time dimension reads
d = 2:
The acoustic line element for two space and one time dimension reads
This situation would be appropriate, for instance, when dealing with surface waves or excitations confined to a particular substrate.
d = 1:
The naive form of the acoustic metric in (1+1) dimensions is ill-defined, because the conformal factor is raised to a formally infinite power. This is a side effect of the well-known conformal invariance of the Laplacian in 2 dimensions. The wave equation in terms of the densitised inverse metric continues to make good sense; it is only the step from to the effective metric that breaks down. Acoustics in intrinsically (1+1) dimensional systems does not reproduce the conformally-invariant wave equation in (1+1) dimensions.

Note that this issue only presents a difficulty for physical systems that are intrinsically one-dimensional. A three-dimensional system with plane symmetry, or a two-dimensional system with line symmetry, provides a perfectly well-behaved model for (1+1) dimensions, as in the cases and above.