SOUND RADIATION BY CYLINDRICAL HYDROACOUSTIC TRANSDUCER IN THE PRESENCE OF A SCREEN WITH ELECTRICALLY CONTROLLED ACOUSTIC POWER

INTORDUCTION Hydroacoustic antennas with acoustic screens have been widely used in the construction of hydroacoustic stations (GAS) for various purposes [1 − 6]. They can be implemented in the form of either systems of shielded transducers [7 − 9] or systems with acoustic baffl es as separate elements of system designs [1 − 3, 6]. Today, the trend of multifunctionality of modern GAS is becoming more and more popular, when several modes of operation are implemented at one station. Universatility necessitates the control of the parameters of the elements that are part of the GAS − transducers [9 − 11], acoustic screens [9], confi gurations and sizes of sonar antennas [12]. These parameters include directional properties, resonant frequencies, impedances of electrical excitation and radiation, and so on. Methods of managing these parameters are conventionally divided into passive and active. Passive methods include the use of acoustic screens with different physical properties and with different confi gurations. But their main disadvantage is the diffi culty or inability to quickly control the parameters of the antennas and their elements in the process of GAS. The peculiarity of piezoceramic hydroacoustic transducers is that they are electromechanical devices [12 − 14]. This allows you to quickly control their parameters from the electrical side [9, 10] during the operation of the GAS. The acoustic baffl e can also be made in the form of an electromechanical device and its parameters could be quickly controlled during the operation of the station from the electrical side. This is due to its impedance properties depending on the frequency range used. In the range of resonant frequencies, such a screen has the properties of an acoustically soft body, at a distance from mechanical resonances − the properties of an acoustically rigid body, and in the intermediate frequency ranges − the properties of a body with impedance properties [9]. It can be assumed that by electrically exciting the screen, which is made in the form of an electromechanical device made of piezoceramics, the radiation effi ciency of GAS can be increased [9, 10]. The purpose of this work is to obtain analytical equations that will quantify the ability to quickly control the parameters of the sonar transducer, located near the acoustic baffl e with electrically controlled parameters.

a connection of electric, mechanical and acoustic fi elds, and during its formation, due to the repeated exchange of radiated and refl ected waves between the transducer and the screen, their acoustic interaction.
In mathematical terms, these physical factors are taken into account by a joint solution of the system of differential equations in the coordinates of Figs. 1: -Helmholtz equation describing the motion of elastic media inside and outside the antenna elements:  -forced electrostatics equations for piezoceramic: Here ∆ is the Laplace operator; Ф iS − speed potential of the S-th element of the antenna inside Ф iS = Ф 1S and outside Ф iS = Ф of it; iS k − wave number inside Acoustic conditions include Sommerfeld conditions and the conditions that there is no features in the internal areas of the transducer and screen.
Electrical boundary conditions: -electric fi eld strength in the piezoceramic shells of the converter (S = 1) and the screen (S = 2) is described in the next form: -radial components of electric induction for the j-th prism in cylindrical piezoceramic shells with circumferential polarization of antenna elements by expressions is determined in next way:

OBTAINING THE CALCULATING RELATIONS
To solve the problem under consideration, lets divide the entire multiconnected areas of existence of the physical fi elds of the antenna into a number of partial domains ( Fig. 1), defi ning their boundaries in the form: -for outer area where ( ) S S r -antenna acoustic fi eld speed potential. Then the expression for the external loads of the antenna elements will look like: The solution of the "end-to-end" problem can be carried out by the method of related fi elds in multiconnected domains [2,9]. Lets present the mechanical and acoustic fi elds of the antenna elements by decomposition into Fourier series by angular and wave functions of a circular cylinder: In these equations, the traditional notations of cylindrical functions are used. To determine the unknown coeffi cients of decompositions (8) and (9), we use relations (1) − (8), pre-expressing the acoustic fi eld of the antenna in its own coordinates of each of its elements. To do this, we need to use addition theorems for cylindrical wave functions [3]: where qS r and qS  are the coordinates of the origin S O of the coordinate system of the S-th element of the antenna in the coordinate system of the q-th element.
Algebraization of functional equations (1) − (6) using relations (9) − (12) and properties of completeness and orthogonality of angular functions on the interval [0, 2π π], allows to obtain an infi nite system of linear algebraic equations for determining unknown coeffi cients of decomposition nS u , nS w , nS A nS B and as: where , Two expressions are used to calculate the acoustic fi eld outside the antenna. The expression for calculating the fi eld in the near zone of the antenna in the coordinates of the S-th element of the antenna has the form: The expression for calculating the fi eld in the far zone of the antenna is derived from relations (5) and (8)  The acoustic fi elds inside the antenna elements are determined by expression (9).
The electric fi elds of the transducer and the acoustic baffl e of the sonar antenna are determined by the following relations.
The expression for the electric excitation current of the antenna elements per unit of their height is determined from the ratio The input electrical resistance ( S Z ) of the S-th element of the antenna is initiated by Ohm's law.

ANALYSIS OF THE OBTAINED RESULTS
The obtained analytical equations allow making a more rational choice of the construction of the sonar antenna during its design or modernization in the future. As is known [2], the maximum energy effi ciency of a sonar station is limited by the mechanical, electrical and cavitation resistance of their sonar antennas. Thus, the mechanical stability of the active elements of antennas limited by the breaking forces that occur in them in operation process and depend on the maximum values of the vibrational velocities of the elements [13,15]. This necessitates fi nding numerical values of the amplitude-frequency dependences of the resonant oscillations of the antenna elements.
The electrical strength of the antenna elements, using the existing schemes of organization of their excitation, does not belong to the critical parameters of the antennas. However, knowledge of electric fi elds, in particular, the input electrical resistances of antenna elements, is essential to ensure creation electronic generators that excite these elements. Knowledge of the magnitude of electric excitation currents allows making a rational choice of materials of communication lines. Note that until recently, strict methods for calculating the electric fi elds of GAS did not exist [1,5,6]. Finally, the ability to quantify the acoustic fi elds in the near zone of sonar antennas avoids cavitation limitations of GAS, by rationally choosing acoustic pressure levels that do not exceed the threshold of the cavitation-working environment of the station.
Based on the above, we perform a qualitative analysis of the properties of the sonar antenna under consideration, based on the obtained analytical ratios. Such an analysis, although complicated by the presence in their composition of an infi nite system of linear algebraic equations (13), is possible because its matrix elements and right-hand sides have a clear physical meaning. First, we note that the system of equations (13) demonstrates the interaction between electric, mechanical and acoustic fi elds. In this case, it follows from the third equation of the system that the electric energy pumped with the adopted scheme of parallel inclusion of piezoceramic prisms in the shells of the antenna elements only in the zero mode of oscillations of the shells. The parameters of the mechanical fi elds of both shells, as follows from all equations, are determined by both internal and external acoustic fi elds, which in turn depend on the densities and speeds of sound of internal and external elastic media.
A special role in the formation of all three physical fi elds taken by the multiple exchange of emitted and refl ected sound waves, which is described in the second and third equations of system (13) by the double sums of q and m. Their presence indicates the interaction of waves, fi rst, formed by the s-th and q-th elements of the antenna, and, secondly, the m-th and n-th orders. The nature and magnitude of their impact on all fi elds determined by multiplier Техніка та озброєння Військово-Морських Сил ISSN 2414-0651 (друк) remain in the mechanical fi elds of the antenna elements, and the elements themselves behave as single-mode transducers. As follows from the second and third equations of the system (13), the decrease in the wavelength qS kr accompanied by an increase in the multiplier (1)  and, as a consequence, the effect of double the sum on the mechanical fi elds of the antenna elements. This effect manifested in the generation in the mechanical fi eld of modes of oscillations following zero. Thus, the antenna elements under consideration converted into multimode oscillatory structures. Since the right-hand side in the third equation of system (13) has not changed, the energy that is "pumped" into the antenna elements only at the zero mode of their oscillations now distributed between all modes. This leads to changes in the resonant frequencies of the oscillatory systems of the elements, the amplitudes of their resonant oscillations and the bands of their resonant frequencies. Naturally, the acoustic fi elds, the electric excitation currents and the input electrical impedances of the antenna elements also change. Quantitatively, these changes can be obtained as a result of numerical experiments.

SUMMARY
Combining many modes of operation in one GAS requires a review of approaches to the station design and organization of its work. One of them is the transition to dynamic methods of controlling the parameters of the sonar antenna of these stations. The simplest technically this can be realized using the electrical side of sonar antenna transducers. Certain possibilities in the dynamic control of antenna parameters manifested in the replacement of passive acoustic baffl es with active piezoceramic shells.
For hydroacoustic antennas formed from a piezoceramic transducer and a baffl e, analytical relations are obtained by the method of connected fi elds in multiconnected areas, which allow quantify the possibilities of dynamic control of the parameters of the hydroacoustic antenna.
A qualitative analysis of these possibilities based on the analysis of the physical meanings of the components of the obtained analytical relations is carried out.
It is shown that in such a statement there are many physical factors and their mathematical equivalents for controlling the dynamic properties of sonar antennas.