Since the establishment of transformation optics,^[1–6] the methodology for engineering wave functional devices via coordinate transformation has spread to various wave systems due to their common character, i.e., the form invariance of wave equations in arbitrary curvilinear coordinates and curved spacetimes. Transformation acoustics (TA) is one of these developments, which offers a standard procedure to design the material parameters so that sound waves propagate along prospective trajectories or hold proleptic properties in these materials.^[7,8] The most representative device of TA is the acoustic cloak that guides sound waves traveling around the cloaked objects without scattering and thus make itself and the hidden objects invisible from acoustic detections.^[9–13] From the standard approach of TA, acoustic cloaks in fluid are made up of anisotropic materials with tensor mass densities. Though real fluids do not possess these properties, a variety of cloaks have been produced by using artificial acoustic metamaterials.^[14–18]

On the other hand, it has been realized long ago that moving media act as curved spacetimes and can be used to simulate cosmic phenomena for both electromagnetic waves^[19–22] and sound waves.^[23–30] In the pioneering works of Unruh,^[24,25] he discovered the dynamic analogy between the massless particles in the spacetime of Schwarzschild black hole and the sound waves dominated by the radially flowing fluid. This analogy is also valid for superfluid^[31–34] in which the effects associated with acoustic black hole have been observed experimentally.^[35–37] In recent years, García-Meca et al. combined the ideas of moving media and TA, and proposed the method of analogue transformation acoustics (ATA).^[38–41] They made use of background flow, instead of anisotropic effective media, to mimic the metric of spacetime, and to fabricate the desired acoustic devices, such as spacetime cloak and time-dependent spatial compressor. Through forcing the effective metric to have a certain form, the parameters of background flow, i.e., the flow velocity, the speed of sound and the mass density, can be determined. However, García-Meca et al. never discussed the feasibility of these requisite parameters of background flow, although they are not unrestricted but submitted to the laws of fluid theoretically.

In this paper, we try to reconcile the ATA-determined parameters with the basic laws of fluid dynamics and show several constraints of the background flow. First, the governing equation of sound waves in ATA is accurate only when the background flow is irrotational, or at least the vorticity of the flow Ω should be much smaller than the frequency of sound ω, in order that the analogy between the flow fields and curved spacetime is applicable.^[30] Second, there should exist external force fields (momentum sources) and external mass sources in general to guarantee momentum and mass conservations of the background flow. Third, since the spatial distribution of pressure and mass density are fixed after ATA procedures, their relation restricts the state equation of the background fluid. As an example, we put forward a scheme for constructing spherical cloaks with background flow which complies with the realistic fluid dynamics. In addition, we investigate the propagation of sound in this cloaking system with both geometric acoustics approach and solving the wave equation analytically. Our results show that the major difference of our scheme from the standard TA cloak is the phase retardation (or lead) effect of the sound waves traveling around the cloak. For eliminating the singularities of the parameters at the cloak surface, we introduce the unideal cloaks and evaluate their cloaking abilities.

To begin with, we briefly review the framework of ATA. The motion of ideal fluid (with no viscosity and thermal conductivity) obeys Euler equation and the continuity equation which correspond to the momentum and mass conservations respectively. If the fluid is barotropic, i.e., its density ρ only depends on pressure P, and the flow velocity v is irrotational, the compressible fluid submits to the following equations^[42]

Considering sound waves traveling on a background flow with the parameters {u, ψ, ρ,P} obeying Eq. (1), the sound waves perform as the perturbations of the flow. Taking the substitutions into Eq. (1), we can obtain the governing equations of sound wave in linear approximation^[24–26]

According to the procedure of ATA, if we let the effective metric either in PG system or in Sch. system equal to the requisite form gained by TA, the expressions of u, ρ, c, can be determined. However, these quantities should also compose a real solution of fluid dynamics. By plugging this quantities back into Eq. (1), the required external sources are fixed. Besides, it is notable that the governing Eq. (4) of ATA and Eq. (7) (if used) are valid only if u and u/(c² − u²) are irrotatinal. All these matters restrict the selective freedom of the background flow.

Now we show how to construct a spherical cloak with background flow. For spherically symmetric systems, all parameters are merely functions of radius and the flow velocity u is radial. Consequently, u and u/(c² − u²) would always be integrable. It is convenient to discuss the problem in spherical coordinates {r,θ,ϕ}, thus we have (γ_{i j}) = diag(1, r², r² sin² θ) and (u_i) = (u,0,0). In the light of TA, the effective metric for a spherical cloak reads^[6]

A class of functions satisfying the conditions f(a) = 0 and f(r) → r as r → ∞ is

Fig. 1.

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Fig. 1. The profiles of (a) radial transformation f(r), (b) velocity of background flow u(r), (c) mass density ρ, (d) speed of sound c(r), (e) pressure difference ΔP(r) = P − P0 in comparison with the pressure P0 at infinite, (f) constitutive relation between ρ and ΔP, (g) external potential U(r), (h) external mass source M(r), where the red, green and blue curves correspond to the transformations given in Eq. (12) with n = 2, n = 3, and n = 4 respectively. The quantity written in the square bracket beside each axis gives the unit of the corresponding coordinate axis.

In this section, we study the propagation of sound waves with geometric acoustics approximation in the shortwave limit. Under the eikonal hypothesis, the velocity potential takes the form , and the phase Φ satisfies the eikonal equation H = h(x^μ)g^μν k_μ k_ν = 0, where k_μ = ∂_μ Φ, h(x^μ) can be an arbitrary non-degenerate function, and

For spherically symmetric metric, k_ϕ is conserved along a ray, since dk_ϕ/ds = 0. If we let k_ϕ = 0, then ϕ ≡ ϕ₀, and dk_θ/ds = 0. As a result, the angular momentum of phonon perpendicular to the orbit k_θ ≡ L is also conserved. We can further remove all the wave vector terms in the canonical equations, and obtain

Figure 2 displays a set of rays (blue curves) incident with the same direction but different impact parameters. Despite the same trajectories as in TA cloak, the wave fronts Φ = const., shown by the red curves in Fig. 2, significantly differ from those in TA cloak. In our case, the wave fronts before bypassing the cloak are asymmetric to those after bypassing the cloak, however, this symmetry exists in TA cloak.^[3,9–11] This effect presents the phase retardation of the waves when bypassing the cloak, and it is essentially induced by the background flow u. In the next section, we will further investigate this effect.

Fig. 2.

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	Fig. 2. Sound rays in the acoustic cloaking system. The blue curves denote the sound rays, and the red curves denote wave fronts, where the background flow (the direction is emanative from the center) corresponds to Eq. (12) with n = 4.

According to Eq. (6), by applying the rules

In Sch. system, the boundary condition turns into

Figure 3 shows the field distributions of both velocity potential and sound pressure for the case of idea cloaking with emanative background flow. As shown in the figures, the fields are asymmetric with respect to the z = 0 plane in contrast to the case of TA cloak. For the incident half place (z < 0), the phases of the fields are compressed heavily, while the phases are stretched in the outgoing half place (z > 0). According to the expression of the wave fronts

Fig. 3.

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	Fig. 3. Distributions of (a) velocity potential , (b) sound pressure , in x = 0 plane, of the sound wave incident from the left side, where the background flow (emanative from the center) corresponds to Eq. (12) with n = 4.

Figure 4 exhibits the distributions of phase velocity and group velocity corresponding to the sound fields given in Fig. 3. As we can see, the phase velocity and the group velocity slow down as the sound wave approaches the inner boundary r = a from the left side, whereas the velocities have extremely large values at the right side of inner boundary. For the case of convergent back ground flow, the distributions in Fig. 4 should still correspond to the wave incident from the right side. It is precisely the asymmetric distribution of phase velocity that causes the phase retardation and lead in corresponding regions.

Fig. 4.

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	Fig. 4. Distributions of (a) phase velocity vp, (b) group velocity vg, in x = 0 plane, of the sound wave incident from the left side, where the background flow (emanative from the center) corresponds to Eq. (12) with n = 4.

It is noteworthy that an alternative scheme of acoustic cloak based on velocity potential wave equation has been proposed in Ref. [41]. In spite of the remarkable differences between their results and the ordinary TA cloaks treated with sound pressure equation, their proposal is more similar to a standard TA device than ours, since their primary method is still to construct effective anisotropic media with the conventional procedure used in TA,^[1,3] but our key idea totally originates from the analogy between the moving media and curved spacetime.

To realize ideal cloaking, all parameters of the background flow should tend to infinity at the inner boundary r = a as shown in Fig. 1, however, it is impossible in practice. In this section, we would discuss the unideal case with simply cutting off the infinite parameters. We still use the radial transformation f(r) to construct the background flow, but let its zero point be a′ = (1 − δ)a in order to escape the singularity at the inner boundary, when δ → 0 the cloak approaches to an ideal one. In terms of Eq. (28), the scattering wave tends to

We have checked the angular distributions of DSCS, and TSCS changing with wave length for δ = 0.1, δ = 0.05, δ = 0.02, and δ = 0.01 in comparison with the uncloaking results, as shown in Fig. 5. The results reveal that the unideal constructions still have good property of cloaking for nearly full band of wave length as δ ≤ 0.05. However, for the case of δ = 0.1, only when λ > 2a, its invisibility effect is acceptable. In addition, the angular distributions of DSCS show that the wave scatters quite uniformly in all directions when the background flow exists, whereas there are usually large components of back scattering in the uncloaking cases.

Finally, we shall discuss the required state equation of the background fluid. Though the constitutive relation between P and ρ is determined by the coordinate transformation, we can use Virial expansion to approach our constitutive relation. The Virial expansion is a physical state equation that can be derived directly from statistical thermodynamics, which takes the form

Fig. 5.

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Fig. 5. The angular distributions of differential scattering cross sections of unideal cloaks at the wave lengths: (a) λ = 0.5a, (b) λ = a, (c) λ = 2a. (d) The total scattering cross section versus wave length. In each figure, the black curve denotes the scatter of a ball with unit radius and hard boundary, the red, blue, orange, green curves correspond to the unideal cloaks of δ = 0.1, 0.05, 0.02, 0.01 respectively.

Fig. 6.

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	Fig. 6. Constitutive relationship between pressure P and density ρ fitted by 5 order Virial expansion (blue curve). The red dotted curve denotes the required constitutive relationship corresponding to the unideal cloak of δ = 0.1 constructed by the transformation function Eq. (12) with n = 4.

To summarize, we design an acoustic cloaking scheme by using background flow based on the method of ATA, and give all of the required physical parameters of the flow complying with the dynamic laws of fluid. We reveal that it is not enough to merely give the background velocity as previous works just did,^[38–40] but some restrictions should be satisfied, and some other necessities, such as the external mass and momentum sources, should be supplied. We also provide a comprehensive investigation about the propagation of sound waves in our cloaking system. According to our results, the major distinctive effect in our system is the phase retardation and lead when sounds bypass the cloak. Furthermore, we analyze the cloaking ability of unideal constructions and use Virial equation of state to fit the required constitutive relation of the background fluid. Our research not only offers a novel path to achieve acoustic cloaking but also complements the framework of ATA.