Acoustic scattering from a toroidal bubble

This brief communication applies a linear model developed previously¹ for acoustic scattering from a gas bubble of arbitrary shape to the specific case of a toroidal bubble. The pulsations are modeled with a harmonic oscillator equation in which the inertia coefficient is due to an expression obtained by Strasberg² for a bubble of arbitrary shape. The model applies to a bubble that is small compared with a wavelength, but large enough that radiation damping is the dominant loss mechanism. Radiation damping dominates thermal and viscous losses for spherical bubbles with radii larger than 1 cm,³ and for cylindrical bubbles of any radius.⁴ The model was verified in the previous work¹ by comparison with an analytical solution derived by Ye⁵ for prolate spheroidal bubbles. Verification for toroidal bubbles is provided in the present work by comparison with finite-element calculations.

The primary motivation for the present analysis is the use of tethered arrays of encapsulated bubbles with radii greater than about 5 cm to abate underwater sound at frequencies below about 1 kHz.⁶ A toroidal shape can facilitate tethering, and the bubble size coincides with the parameter range of the model. It is assumed that the toroidal shell encapsulating the gas is sufficiently compliant that the stiffness coefficient in the model is determined by the gas alone. However, unencapsulated toroidal bubbles are generated by various natural phenomena, such as a vortex ring formed by injection of air from a nozzle^7,8 (or from a dolphin's blowhole⁹), a freely rising gas pocket shaped initially like a spherical cap that transitions to a toroidal shape,¹⁰ or due to formation of a liquid jet that pierces through the center of an initially spherical bubble during an asymmetric inertial collapse.¹¹ 

Consider a bubble of arbitrary shape, the volume of which at time t is denoted by V(t) and the equilibrium volume by V₀. In the linear approximation, the pulsations of the bubble can be modeled using an ordinary differential equation for the excess bubble volume $v(t)=V(t)−V0$⁠. It is assumed that the bubble is driven sinusoidally by a spatially uniform acoustic pressure $p(t)=p̃ie−iωt$⁠. By combining the lumped inertance term derived by Strasberg² with the radiation damping term derived by Devin,¹² the following equation is obtained:¹ 

where ρ is the mass density of the surrounding liquid, c is the speed of sound in the liquid, γ is the ratio of specific heats of the gas, and P₀ is the ambient pressure. The quantity $γP0$ is the bulk modulus of the gas, which may be augmented to account for additional stiffness due to a membrane or thin shell encapsulating the bubble. The constant appearing in the effective mass term of Eq. (1) is given by $C=−v̇/ϕ0$⁠, where $v̇$ is the volume velocity of the bubble and $ϕ0$ is the velocity potential on the bubble surface.

The value of C is obtained by solving the equations

and

where the integral is taken over the bubble surface with n the outward unit normal vector. Strasberg pointed out that solving for $C=−v̇/ϕ0$ is mathematically equivalent to solving for the electrostatic capacitance of an ideal conductor with the same shape as the bubble. Since this is a classical problem in the field of electrostatics, it has already been solved for a variety of shapes.

In the case of time-harmonic excitation, for which the excess volume takes the form $v(t)=ṽe−iωt$⁠, the solution of Eq. (1) is

where the undamped natural frequency is given by

For a spherical bubble with equilibrium radius R₀ the capacitance is given by $Csph=4πR0$ and Eq. (6) coincides with the traditional Minnaert frequency. Since the capacitance of a conductor with a given volume is minimized when its shape is spherical, the inequality $C/Csph>1$ holds for any nonspherical shape. Therefore, the natural frequency of a nonspherical bubble always exceeds that of a spherical bubble having the same volume.

At frequencies sufficiently low that the bubble is very small compared with a wavelength, for which Eq. (1) applies, the incident wave is scattered omnidirectionally. The scattered pressure in the far field is determined in this limit by treating the bubble as a simple source with volume velocity $−iωṽ$ and using the solution in Eq. (5) to obtain

where $k=ω/c$ is the wavenumber in the liquid. The scattering cross section, defined as the ratio of the scattered power to the intensity of the incident plane wave, is given by $σs=4πr2|p̃s/p̃i|2$⁠. Based on Eq. (7), the scattering cross section is given explicitly by

which is valid for a bubble with arbitrary shape. In order to treat the case of a bubble with a specific shape, the appropriate expression for the capacitance C must be substituted in both Eq. (8) and the expression for the natural frequency ω₀ given by Eq. (6).

For a torus with major radius b and minor radius a (see Fig. 1), the equilibrium volume is $V0=(πa2)(2πb)$⁠. An overview of the electrostatic properties of a toroidal conductor is provided by Belevitch and Boersma.¹³ They present a result for the capacitance of a torus in toroidal coordinates, an equivalent form of which is

where $ϵ=b/a≥1$ is the aspect ratio of the torus (with ϵ = 1 corresponding to a “dimpled” sphere, while the torus is thinner for larger ϵ). $Pμ$ and $Qμ$ are Legendre functions of the first and second kind, respectively, of order μ. Legendre functions of half-integer order are sometimes referred to as toroidal functions because of their relationship with problems involving toroidal geometry.

Equation (9) is valid for all aspect ratios $ϵ≥1$⁠. Thomas¹⁴ developed approximate expressions for the capacitance of thin and thick tori, which in the present notation are

These two approximations may be substituted into Eq. (6) for the undamped natural frequency to obtain

The natural frequency ω₀ of a toroidal bubble is shown in Fig. 2, normalized by the natural frequency $ωsph$ of a spherical bubble with the same volume V₀. The black curve corresponds to evaluation of Eq. (6) using Eq. (9), the red curve to Eq. (12), and the blue curve to Eq. (13).

The accuracy of approximations (12) and (13) observed in Fig. 2 within their stated ranges of applicability permits insight into how the natural frequency depends on the geometric factors a, b, and V₀ individually. For example, if a bubble maintains constant equilibrium volume V₀ as the major radius b increases, the minor radius a of the circular cross section is proportional to $1/b$⁠, for which the approximations indicate that ω₀ increases monotonically with b. If instead a is held constant, the approximations indicate that ω₀ decreases monotonically as b increases.

Verification of the results for a toroidal bubble was accomplished by comparing the expression for the scattering cross section in Eq. (8), using Eq. (9) for C, with finite-element calculations for a toroidal air bubble in water.

The solid curves in Fig. 3 were obtained from Eq. (8), normalized by the geometric cross section $πR02$ of a sphere with equivalent volume, and plotted in dB versus frequency for different aspect ratios b/a. The bubbles have fixed equilibrium volume $V0=150$ cm³, for which the radius of a spherical bubble is $R0=3.3$ cm. Values for the aspect ratio of the toroidal bubble vary from $b/a=1$⁠, for which $b=a=1.94$ cm, to $b/a=10$⁠, for which b = 9.03 cm and a = 0.903 cm. The liquid has mass density ρ = 1000 kg/m³ and sound speed c = 1484 m/s, and the gas is adiabatic with ratio of specific heats $γ=1.4$⁠. The equilibrium pressure is $P0=101 325$ Pa, resulting in a natural frequency of about 100 Hz for the spherical bubble. At 200 Hz, the highest frequency in Fig. 3, and for b = 9.03 cm, the largest value of the major radius, the nominal diameter 2b of the torus is $0.0244λ$⁠, and therefore all calculations are for bubbles substantially smaller than the wavelength λ in the liquid.

The circles in Fig. 3 were obtained using comsol to perform finite-element calculations for a fluid-fluid acoustic scattering problem with the aforementioned properties of the water and air. The boundary condition in the far field was chosen to be an outwardly propagating spherical wave to simulate an unbounded liquid. The problem was rendered axisymmetric and hence two-dimensional by assuming that the incident plane wave propagates along the symmetry axis of the torus. Figure 3 reveals that the model equations for a toroidal bubble are in excellent agreement with the numerical solutions.

Broadening of the resonance peaks in Fig. 3 with increasing aspect ratio is associated with the increase in bubble surface area, and therefore increase in radiation damping due to the increase in radiation efficiency. This effect may be quantified using the damping constant δ evaluated at the natural frequency as defined by Eller.¹⁵ Its value for a bubble of arbitrary shape relative to that for a spherical bubble with equivalent volume can be expressed as¹ $δ/δsph=ω03/ωsph3$⁠, with the increase in $ω0/ωsph$ for a toroidal bubble as a function of its aspect ratio shown in Fig. 2.

This work was supported by a postdoctoral fellowship provided by Applied Research Laboratories at The University of Texas at Austin, and by the Office of Naval Research. The authors are grateful to Mark S. Wochner for providing preliminary numerical results for the resonance frequency of toroidal bubbles in the absence of losses.