a) Zero-order approximation (Rayleigh's equation)

Assuming the fluid is incompressible ( ${\rho }_0=const$ ), it is easy to show that the motion of the interface is described by a second-order nonlinear differential equation [8]:

where $R(t)$ - the radius of the cavity; ${\rho }_0$ - the density of the liquid; $P(t)$ – the pressure in the liquid at the interface; $P_0$ - the hydrostatic pressure.

If it is necessary to study the sound radiation of the cavity, the incompressibility condition must be modified so that the speed of sound in the liquid is finite. In cases where the speed of expansion of the sphere is small compared to the speed of sound in a divergent medium, the density perturbations caused by the expansion of the sphere will also be small. Therefore, for such processes, the propagation of a divergent wave can be described by a solution of linear acoustics that satisfies the boundary condition of velocity continuity on the surface of a sphere. Then the pressure distribution in the liquid is given by the equation $[9,11]$:

Differential equation (1) does not allow us to take into account the energy emitted by the compression wave, and is a zero-order approximation in which all terms of the order $R^{\prime }$ /care ignored (where c is the speed of sound in a liquid). The elastic modulus is assumed to be infinitely large and, consequently, the speed of sound in water is infinitely large as well.

Considering the collapse of a spherical cavity having at the initial moment the radius $R_{max}$ and the pressure P inside, significantly less than the hydrostatic PO, and taking into account that this pressure ratio persists for most of the period of pulsations, Rayleigh obtained a simple solution to the equation (1) [14]:

Formula (6) is often called the Rayleigh-Willis formula. It is very useful for estimating the relative energy and acoustic characteristics of various types of marine seismic sources. As can be seen from the Rayleigh-Willis formula, the pulsation period increases in proportion to the cube root of the total energy and decreases in proportion to the hydrostatic pressure in the power of 5/6. For processes accompanied by the formation of spherical gas bubbles in water (explosions of condensed explosives, underwater gas explosions, electric discharge, exhaust of compressed gas), this dependence is in better agreement with the experiment, the greater the depth of immersion of the source and the less heat exchange between the gas in the bubble and the surrounding liquid. Note, however, that formula (6) does not apply if the pulsating bubble is located near the liquid boundary due to a change in the nature of the spreading flows.

b) First-order approximation: Herring and Keller-Kolodner equations

The assumption that the speed of sound in water is permanent (and in this case the elastic modulus is considered as a constant and there is a linear dependence of pressure on density) leads to a first-order approximation, which contains terms of the order $R^{\prime }/c$. Acoustic approximation allows us to take into account the energy loss due to radiation in this case.

In equation (9), Keller and Kolodner left the first two terms and omitted the second two, without examining their relative values [17]. In [21], an attempt was made to prove that the second term in equation (9) is not related to linear elastic theory and Newton's second law. Indeed, in deriving equation (9), Keller and Kolodner began their analysis with the wave equation, which applies only to a linearly compressible fluid with a constant speed of sound, when the particle velocity is sufficiently small. Then they linked the particle velocity to pressure via the Bernoulli equation, which has a much broader application than the wave equation. The authors of [21] correctly pointed out the incompatibility of the wave equation and the Bernoulli equation and the appearance of terms in equation (9) that are not related to linear elastic theory. However, in their analysis, they also omitted terms that are significant for linear theory when calculating the pressure field in the near zone.

Since the particle velocity in a linearly compressible fluid is quite small, we transform equation (9), by omitting the terms of the order $(R^{\prime }/c{)}^2$. In the end we get:

c) Second-order approximation (approximation of Kirkwood-Bethe)

The value of φ propagates at a constant speed For the case when the rate of expansion of the cavity in the liquid is high and the density perturbations caused by the expansion of the cavity are significant, the Kirkwood-Bethe approximation is applicable $[8,9]$. This method, developed in the study of underwater explosions, consists in determining the invariant of motion, which is taken as the function

where h -specific enthalpy of the liquid; u -particle velocity.

The value of $\mathrm{\Phi }$ propagates at a constant speed equal to $u+c$ (where c is the local speed of sound). Under this condition, the equation of motion of the interface takes the form:

where $H(t)$ and $c(t)$ are the specific enthalpy and speed of sound in the liquid at the interface, respectively. Both of these functions depend on the pressure $p(t)$. Equation (12) was first obtained by Gilmore [8, 9]. The relationship between pressure and enthalpy is found from the experimental dependence of pressure on density under isentropic compression, which is described by the formula [11]:

where B and n –constants that depend on the type of liquid; for water B = 2500 atm, n = 8 [20], or B = 3000 atm, n = 7 [11]. Using the equation (13),

hence

where $c_0$ -speed of sound propagation in an undisturbed liquid.

To compare equation (12) with equations (7) and (8), transform equation (16)

and determine the order of value

According to the approximate solution of equation (1) given by Rayleigh [9], the value $\frac{P-P_0}{{\rho }_0}$ is proportional to $R^2$. Therefore, the expression $\frac{P-P_0}{P_0+B}$ has the order $(R^{\prime }/c_0{)}^2$. Thus, up to terms of order $(R^{\prime }/c_0{)}^2$

Similarly, we can show that, neglecting the terms of order $(R^{\prime }/c{)}^2$, for the enthalpy H we get

Substituting the expressions (18) and (19) in equation (12) and neglecting the term $\frac{R^{\prime }}{\rho c_0^2}{P}^{\prime }$, having an order $(R^{\prime }/c_0{)}^2$, we get the Keller and Kolodner equation (8).

We now transform equation (12) by multiplying each of its terms by c:

Dividing each term of this equation by $c+R^{\prime }$ and neglecting the terms of order $(R^{\prime }/c{)}^2$, we arrive at the equation

where $y=R(H+\frac{R^{\prime 2}}{2})$;

It is easy to show that, ignoring the terms of order $(R^{\prime }/c{)}^2$, we find an expression for the pressure field from (21) that coincides with equation (11).

II. RESULTS OF NUMERICAL SOLUTION OF EQUATIONS OF THE MOTION OF SPHERICAL CAVITY. ANALYSIS OF THE ASSUMPTIONS MADE IN RELATION TO THE UNDERWATER EXHAUST OF COMPRESSED AIR

a) Comparison of results of numerical solution of equations of the motion of a spherical gas cavity in a liquid

So, to describe the motion of a pulsating spherical bubble in a liquid, we can apply three different differential equations, depending on the order to which the terms $R^1/c$ are taken into account. The assumption of incompressibility of the fluid leads to the differential equation (1), in which all terms of the order $R^1/c$ are ignored – this is an approximation of the zero order. The pressure field in a liquid is described by equation (3).

The assumption that the speed of sound in a liquid is constant leads to equation (7) or (8), which take into account the terms of the order $R^{\prime }/c$ and which coincide with each other up to the terms of the order $(R^{\prime }/c{)}^2$. This is a first-order approximation. The pressure field is given by equation (11). Finally, the Kirkwood-Bethe approximation leads to differential equation (12), which takes into account second-order terms with respect to $R^{\prime }/c$. This is a second-order approximation. The pressure field is given by equation (21).

Let's assume that a sphere with radius $R_0$ (initial volume of the sphere $V_{01}=(4/3)\pi R_0^3$ ) with gas pressure inside $P_{01}$ exceeding the hydrostatic pressure is placed in an unlimited extent of liquid with density ${\rho }_0$ and hydrostatic pressure $P_0$. At the initial moment, the shell of the sphere is instantly removed and the bubble expands with acceleration under the influence of the pressure difference in the bubble and in the surrounding liquid. In this case, an acoustic signal is emitted, the maximum pressure of which is observed at t=0. Assuming that the change in the state of the gas in the bubble occurs according to the adiabatic law, the pressure in the bubble as a function of its radius is described by the dependence

where $\gamma$ -is the adiabatic exponent (for air $\gamma =1,4$ ). Knowing the pressure in the bubble, we can solve differential equations (1), (7) (or (8)) and (12) by setting the following initial conditions:

Numerical solutions of these equations performed in MathCAD by the fourth-order Runge-Kutta-Gill method at $P_{01}=12$ MIIa, $P_0=0,2$ MIIa, $R_0=0,089$ m ( $V_{01}=3\partial m^3$ ), are shown in Fig. 1. The solutions of equations (7), (8), representing the first-order approximation, and equation (12), describing the second-order approximation, as well as the pressure in the liquid, calculated respectively from equations (11) and (21), completely coincide and are represented by a

$P_{01}=12MPa$, $P_0=0,2MPa$, $R_0=0,089m$; (a) - dependence of the bubble radius on time; (b) - pressure in the compression wave; ____ equation (1); ____ equations (7), (8) and (12)

Numerical analysis of the equations of motion of the cavity shows that at an initial pressure $P_{01}=10\mathbf{\nabla }\mathbf{\cdot }15MPa$ for volumes $V_{01}=0$, $1\mathbf{\nabla }\mathbf{\cdot }10d{m}^3$ (namely, at such pressures and volumes of operating chambers, pneumatic sources are used), approximations of the first and second orders give the same results. Since the vibrations of a free bubble represent an idealized case for a pneumatic source, when the energy losses associated with the flow of air through the exhaust windows of the source are ignored, the expansion rate of real bubbles will be at least no greater than the expansion rate calculated for free bubbles. Hence, we conclude that the rate of expansion of bubbles generated by pneumatic emitters is small and there is no need to use a second-order approximation.

The zero - order approximation - equation (1) - is the roughest approach to the problem of acoustic radiation and is mainly used to obtain semi-quantitative information about the oscillating bubble.

The first-order approximation remains, which leads to equations (7) and (8). In our opinion, the use of these equations is most advantageous, since with the same accuracy of the solution as equation (12), these equations are simpler. Since equations (7) and (8) coincide up to second-order terms with respect to $R^{\prime }/c$, any of these equations can be used to describe the motion of a bubble formed by an underwater exhaust of compressed gas. The Keller-Kolodner equation (8) is most often used in the literature devoted to the theoretical study of underwater compressed air exhaust.

The effect of surface tension and viscosity can be significant only for a very small radius of the bubble and a high speed of its surface movement, which is observed, for example, in acoustic cavitation $[2,14,19]$. For bubbles generated by pneumatic sources with operating chamber volumes from 0.1 to 10.0 $d{m}^3$, according to calculations, we obtain the following order of values: $R\approx {10}^{-1}m$, $R^{\prime }\approx 10m/s$. Thus, the order of magnitude of the stress that occurs under the action of surface tension and viscosity is: