FORMULATION OF THE PROBLEM

This article describes a solution to the problem of irradiation by oscillating elastic cylindrical shell submersed into a fluid.

The sound field irradiated by the shell is characterized by irradiation impedance relating the sound pressure of the sound wave with its oscillating speed (displacement). Studies of irradiation impedances of cylindrical shells began long ago [1]. They are described in numerous publications [1–18]. However, these studies did not result in an acceptable exact solution. In this case an exact solution is considered as analytical, complete, evidence-based solution. The problems related to irradiation by cylindrical shells in a fluid have important theoretical and applied significance in hydroacoustics and are relevant.

Researchers [2] believe that this problem, being classic for mathematical physics, has a simple solution only for infinite circular cylinders. It was mentioned in [3] that the exact equation for irradiation impedance of a finite cylinder is a complex function; it cannot be expressed in a simple analytical form.

An analysis of works devoted to solution of this problem demonstrated that the described solutions are significantly different and have no single theoretical basis. The reasons for the differences are stipulated by the different types of shells (finite, constrained, infinite), irradiation by the shells in near and far fields, and determination of the axial wave number of the irradiated sound field. We will refer to the finite cylindrical shell (cylinder) as a shell with free boundary end conditions. A shell constrained by other end conditions (Navier or others) is referred to as constrained.

ANALYTICAL OVERVIEW

Let us discuss the results of an analytical overview of the main best-known works. Aiming at simplification of the equations, in most of them integration and summation of the harmonics by the circumferential angle φ are omitted. Accounting for the pressure distribution over the angle φ does not influence the results and can be carried out by known techniques.

Skuchik Equation

The best-known equation of irradiation impedance of an oscillating cylinder is given in [4]:

$$\frac{p}{w} = {{Z}_{1}} = \frac{{\rho {{\omega }^{2}}H_{n}^{{(2)}}\left( {a\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)}}{{\sqrt {{{k}^{2}} - k_{z}^{2}} H_{n}^{{(2)}}{\kern 1pt} '{\kern 1pt} \left( {a\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)}}.$$
(1)

Equation (1) is given without derivation. Neither the method of determination of kz nor the type of shell is given. Instead of derivation, reference is given to the equation for an infinite shell, where kz = 2πm/λ [1]. This reference makes it possible to believe that Eq. (1) is also for an infinite shell [3, 6, 8].

Junger Equation

The equation of a sound field irradiated by an infinite shell with preset displacement on its surface is given in [1]:

$$w(\varphi ,z) = \sum\limits_{n = 0}^\infty {\sum\limits_{m = 0}^\infty {{{\alpha }_{{nm}}}\cos \left( {\frac{{2\pi m}}{\lambda }z} \right)\cos n\varphi } } ,$$
(2)

where λ is the length of a travelling wave of deformation on the shell and n, m are the circumferential and longitudinal harmonics. The parameter 2πm/λ (2) will be referred to as the wave number of shell deformation (deformation wave number). The irradiated sound pressure is

$$p = - i\rho \omega \sum\limits_{n - 0}^\infty {\sum\limits_{m = 0}^\infty {\alpha _{{nm}}^{'}{{Z}_{2}}\cos \left( {\frac{{2\pi m}}{\lambda }z} \right)\cos n\varphi ,} } $$
(3)
$${{Z}_{2}} = \frac{{H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} \lambda }} \right. \kern-0em} \lambda }} \right)}}^{2}}} } \right)}}{{\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} \lambda }} \right. \kern-0em} \lambda }} \right)}}^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} \lambda }} \right. \kern-0em} \lambda }} \right)}}^{2}}} } \right)}}.$$
(4)

It is remarkable that Eq. (3) is also given without derivation with reference to [7], which does not contain at all equations similar to Eqs. (3) and (4), only the plane problem is considered: kz = 0.

Shenderov Equation

The equation of sound pressure of a field irradiated by a finite cylindrical shell with the oscillating displacement w(z) arbitrarily distributed over the shell length is proposed in [5]:

$$p{\kern 1pt} \text{*}{\kern 1pt} (r,z) = \frac{{\rho {{\omega }^{2}}}}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {\frac{{B(\gamma )H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right){{e}^{{i\gamma z}}}}}{{\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}d\gamma } .$$
(5)

Muzychenko–Rybak Equation

The irradiation impedance of a constrained cylindrical shell is given in [8]. In this work the oscillating speed of cylindrical shell is written as

$$V(\xi ) = {{V}_{0}}{{e}^{{iq\xi }}},$$

where q = 2πm/L is the deformation wave number determining the form of shell oscillations; L is the shell length; and ξ is the axial coordinate. The speed using the δ function is written as

$$V = \frac{{{{V}_{0}}}}{{2\pi }}\int\limits_{ - \infty }^\infty {{{e}^{{i\gamma (z - \xi )}}}d\gamma } .$$

Without derivation or a reference, using an “as known” formulation, the field of acoustic pressure is written as follows:

$$p = \frac{{{{V}_{0}}}}{{2\pi }}\int\limits_{ - \infty }^\infty {{{Z}_{1}}\;{{e}^{{i(z - \xi )\gamma }}}d\gamma } ,$$
(8)

where Z1 is presented as Eq. (1) referring to the impedance of an infinite region. The impedance of a constrained shell is given as follows:

$$\begin{gathered} {{Z}_{3}} = \frac{1}{\pi }\int\limits_{ - \infty }^\infty {{{Z}_{1}}\frac{{{{{\sin }}^{2}}\left[ {\left( {q - \gamma } \right)\left( {{L \mathord{\left/ {\vphantom {L 2}} \right. \kern-0em} 2}} \right)} \right]}}{{{{{\left( {q - \gamma } \right)}}^{2}}\left( {{L \mathord{\left/ {\vphantom {L 2}} \right. \kern-0em} 2}} \right)}}} d\gamma , \\ {{Z}_{3}} = \frac{2}{{\pi L}}\int\limits_{ - \infty }^\infty {i\rho \omega a\frac{{H_{n}^{{(1)}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}{{a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(1)'}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}} \frac{{{{{\sin }}^{2}}[(q - \gamma )L{\text{/}}2]}}{{{{{(q - \gamma )}}^{2}}}}d\gamma . \\ \end{gathered} $$
(9)

If we consider that the last term in Eq. (9) is the δ function, then, using Eq. (9) as applied to w(x), the impedance Z3 can be written as follows:

$${{Z}_{3}} = \frac{{\rho {{\omega }^{2}}H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{{({{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} L}} \right. \kern-0em} L})}}^{2}}} } \right)}}{{\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} L}} \right. \kern-0em} L}} \right)}}^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{{({{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} L}} \right. \kern-0em} L})}}^{2}}} } \right)}}.$$
(10)

The authors of [8] mentioned that while the shell length approaches infinity the impedance of the constrained shell approaches the impedance of an infinite shell, which according to Eqs. (1) and (3) is an erroneous statement, since at L → ∞ the parameter 2πm/L → 0.

Lyamshev Equation

Work [9] devoted to sound diffraction on a constrained cylindrical shell describes the equation of sound pressure referring to a field irradiated by a cylindrical shell. The equation is derived using the Green function. The part of the diffracted field related to the sound pressure in an irradiated wave caused by shell oscillations is as follows:

$$p = - \frac{{i\rho \omega }}{{2\pi }}\int\limits_{ - \infty }^\infty {\int\limits_0^L {\frac{{{{\varepsilon }_{n}}w(z)H_{n}^{{(1)}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right){{e}^{{i\gamma z}}}}}{{\sqrt {{{k}^{2}} - {{\gamma }^{2}}} {{H}_{n}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}} } dz{\kern 1pt} d\gamma ,$$
(11)

where the parameter γ is not disclosed.

Skenk Equation [10]

In [10] the sound pressure is obtained in the following form:

$${{p}_{s}} = \frac{{{{\omega }^{2}}\rho _{0}^{{}}a}}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {{{B}_{n}}(\gamma )} \frac{{{{H}_{n}}\left( {a\sqrt {{{k}^{2}} - {{\chi }^{2}}} } \right)}}{{a\sqrt {{{k}^{2}} - {{\chi }^{2}}} H_{n}^{'}\left( {a\sqrt {{{k}^{2}} - {{\chi }^{2}}} } \right)}}{{e}^{{ - i\chi z}}}d\chi ,$$
(12)

where χ is the required root of the dispersion equation of an infinite cylindrical shell.

Averbuch Equation

In [2], while calculating the irradiation of a finite cylindrical shell in the far field, the author stated that the Fourier transform of sound pressure p* and the Fourier transform of shell displacement w* are interrelated as follows:

$$p\text{*} = \frac{{H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{{\left( \xi \right)}}^{2}}} } \right)w{\kern 1pt} \text{*}}}{{\sqrt {{{k}^{2}} - {{{\left( \xi \right)}}^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{{\left( \xi \right)}}^{2}}} } \right)}},$$
(13)

where ξ = 2π/λ = –k cos θ and θ is the spherical coordinate of the observation point.

Equation (13) can be rewritten as p* = Z2w*, where Z2 is the irradiation impedance of infinite shell (4) [1, 2], with the following substitution: 2π/λ = –k cos θ. The origin of Eq. (13) is not substantiated; it is erroneous, since Z2 is only a part of the impedance. The main remark is in the absence of substantiation of the substitution 2π/λ = –kcosθ. Most likely, due to these reasons Eq. (13) was not used by other authors [3, 6, 10, 13].

It follows from the overview presented that there exist numerous solutions, differing in form and content and having no common theoretical basis. Despite having similar elements (Hankel functions), the solutions are fundamentally different. Equation (5) contains two exponents: exp(iγz) and exp(–iγz). There is only one exponent in Eq. (11). Equations (5) and (11) are similar, but are fundamentally different from Eqs. (1) and (10). Essentially the same equations of wave numbers kz = 2πm/λ and q = 2πm/L are used in the impedances of infinite (4) and constrained (10) shells, though their authors believe that the impedances of constrained and infinite shells are different. The diversity and lack of substantiation of some equations require further research.

The aim of this work is to obtain an exact solution to the problem of irradiation by an oscillating cylindrical shell in a fluid, as well as to reveal the falsity and inconsistency of a series of the known solutions [3, 4, 8–13].

THEORETICAL STUDIES

Let us derive the equation of sound pressure of the field irradiated by a cylindrical shell in a fluid and prove the falsity of a series of the known solutions.

We consider the irradiation by the side surface of a cylindrical shell with radius a and length L, placed into the cylindrical coordinates r, z, φ. The longitudinal axis of the shell is combined with the axis z. As in most of the previous works, the irradiation by edge caps and screens is not taken into account; it should be analyzed separately.

The necessity to derive Eq. (1) is stipulated as follows: (1) such a derivation was found neither in [4] nor in other publications; (2) the parameter kz should be specified; (3) the derivation is used below for analysis of the erroneous solutions.

The wave equation in cylindrical coordinates r, z, φ describing the irradiated sound field is as follows [4]:

$$\left( {\frac{1}{R}\frac{{{{\partial }^{2}}R}}{{\partial {{r}^{2}}}} + \frac{1}{{rR}}\frac{{\partial R}}{{\partial r}} + \frac{1}{{{{r}^{2}}\Psi }}\frac{{{{\partial }^{2}}\Psi }}{{\partial {{\varphi }^{2}}}}} \right) + \left( {\frac{1}{Z}\frac{{{{\partial }^{2}}Z}}{{\partial {{z}^{2}}}}} \right) = \frac{1}{{{{c}^{2}}}}\frac{{{{\partial }^{2}}T}}{{T\partial {{t}^{2}}}}.$$
(14)

The solution to Eq. (14) is written as Φ = R(r)Z(z)Ψ(φ)T(t).

The solution with relation to the pressure p of the sound field can be simplified as follows [5]:

$$p(r,z) = {{A}_{n}}H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - k_{z}^{2}} } \right){{e}^{{i{{k}_{z}}z}}},$$
(15)

where An is the required coefficient and \(H_{n}^{{(2)}}\)(rkr) is the Hankel function of the second kind. The simplification is that in Eq. (15) the summation over circumferential harmonics exp(inφ) and the dependence of the parameters as a function of time exp(iωt) are omitted. The coefficient An is determined from the boundary conditions on the shell surface:

$$w(z) = {{\left. {\frac{1}{{{{\omega }^{2}}\rho }}\frac{{\partial p}}{{\partial r}}} \right|}_{{r = a}}},$$
(16)

where w(z) is the radial oscillating displacement of the shell, ρ is the fluid density, ω = 2πf is the angular frequency, and a is the shell radius. From Eq. (15) with account for Eq. (16), we obtain the following:

$$\sqrt {{{k}^{2}} - k_{z}^{2}} {{A}_{n}}H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - k_{z}^{2}} } \right){{e}^{{i{{k}_{z}}z}}} = \rho {{\omega }^{2}}w(z),$$
$${{A}_{n}} = \frac{{\rho {{\omega }^{2}}w(z)}}{{{{e}^{{i{{k}_{z}}z}}}\sqrt {{{k}^{2}} - k_{z}^{2}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)}}.$$

Substituting An into Eq. (15), we obtain Eq. (1) in the following form:

$$p(r,z) = \frac{{\rho {{\omega }^{2}}H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)}}{{\sqrt {{{k}^{2}} - k_{z}^{2}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)}}.$$
(17)

Then, in Eq. (17) we determine the parameter kz, which is the main object of study. The solution to Eq. (14) is obtained on the basis of the condition [4]

$$\frac{1}{{{{c}^{2}}}}\frac{{{{\partial }^{2}}T}}{{T\partial {{t}^{2}}}} = - {{k}^{2}},\quad \left( {\frac{1}{Z}\frac{{{{\partial }^{2}}Z}}{{\partial {{z}^{2}}}}} \right) = - k_{z}^{2},$$
$$\left( {\frac{1}{R}\frac{{{{\partial }^{2}}R}}{{\partial {{r}^{2}}}} + \frac{1}{{rR}}\frac{{\partial R}}{{\partial r}} + \frac{1}{{{{r}^{2}}\Psi }}\frac{{{{\partial }^{2}}\Psi }}{{\partial {{\varphi }^{2}}}}} \right) = - k_{r}^{2},$$

where k, kz, kr are the arbitrary constants, which in accordance with Eq. (14) are determined as follows:

$$k_{r}^{2} + k_{z}^{2} = {{k}^{2}}.$$
(18)

Let us clarify the physical sense of these variables and determine their mathematical expression on the basis of solutions to Eqs. (14) and (18). Taking the solutions to Eq. (14) in the form of T = exp(iωt) and Z = exp(ikzz), we obtain the following: k = ω/c is the wave number of a sound wave in a fluid, kz is the wave number of a sound wave propagating in the direction of the coordinate axis z. If k, kz are the wave numbers, then kr is also the wave number of the wave propagating in the direction of the axis r. Therefore, the cylindrical wave in the coordinates r, z propagates along the axis r with the wave number kr and along the axis z with the wave number kz. Cylindrical wave moves simultaneously in the radial and axial directions with the wave numbers kr and kz, respectively. The resultant motion of the wave with the wave number k takes place at the angle θ to the axis z. The wave numbers k, kz, kr are interrelated by Eq. (18), and it follows from its solution that the required wave numbers are determined as follows:

$${{k}_{r}} = k\sin \theta ,\quad {{k}_{z}} = k\cos \theta ,$$
(19)

where θ is the angle between the vector of the wave number k and the axis z based on the axis z.

Substituting kz (19) into Eq. (17), we obtain the equation in the final form:

$$p(r,z) = \frac{{\rho {{\omega }^{2}}H_{n}^{{(2)}}(rk\sin \theta )}}{{k\sin \theta H_{n}^{{(2)'}}(ak\sin \theta )}}w(z).$$
(20)

It may seem that the difference between Eq. (20) and Eq. (1) is insignificant. In fact this difference is fundamental. Equation (1) was completed neither in terms of physical explanation of the wave number kz, nor in terms of its analytical determination. Due to the uncertainty of kz, Eq. (1) cannot be applied for practical use. The ambiguity in the understanding by the author of [4] of the number kz is confirmed by the fact that the author, referring to Eq. (3), assumed the possibility of kz = 2π/λ. Moreover, the author assumed k2 – \(k_{z}^{2}\) < 0, that is, kz > k, and in this case it was proposed to transfer to the Macdonald function.

The unexplained value of kz in Eq. (1) forced different authors to present kz at their own discretion. For instance, they presented kz in terms of the deformation wave length of the infinite shell kz = 2π/λ (4) [1, 2], or in terms of the oscillation form of constrained shell kz = 2πm/L (10) [8], or in complex form kz = k sin(α1 + iα2) [5]. Many authors performed integration over dkz upon variation of kz in the range of (–∞, ∞) (5), (11) [5, 6, 9], although the number kz does not exist in these limits. The parameter kz was not reasonably presented as kz = k cos θ in any of the works, including [1–13], in the equation of irradiation impedance of the cylindrical shell (1).

Equation (20) does not contain any signs of shell types (finite or infinite). It is important to understand that in Eq. (20) the parameters k, r, θ and, hence, kz = k cos θ are not calculated but are preset on the basis of the physical conditions of the problem to be solved. These parameters characterize the sound field irrespectively of the specific source of irradiation. The variable r can be preset in the limits of the near field starting from r = a. In some cases the angle θ is known. For instance, in the case of a pulsating shell the angle θ = 90° and kz = 0. In the case of wave propagation along the axis z, the angle θ = 0° and kz = k [4, 5].

It follows from the obtained solution that, since the wave equation and its solution do not depend on the shell type, then Eq. (20) is invariant to the shell type and characterizes only the properties of the irradiated sound wave crated by the shell displacement w(z).

The displacement w(z) in Eq. (20) can be determined from the solution to the equations of forced oscillations of the shell or presented by the Fourier series or the Fourier integral.

Equation (20) determines the physical sense and the value of kz eliminates the uncompletedness and uncertainty of Eq. (1). Equation (20) is new, reasonable, and correct. It is universal and can be applied for the solution to any problem of the hydroacoustics of any cylindrical shell. This equation is required for the solution of many problems: dispersion equations and diffraction and irradiation in far field. It can be considered as an element of the theoretical background of hydroacoustics of cylindrical shells.

Proving the Falsity of the Shenderov Equation (5) and Others Like It

Let us analyze the derivation of Eq. (5) in [5]. In deriving it, let us use our own notation assigned with accounting for time function eiωt and substitution of the speed \({v}\) for the displacement w = \({v}\)/iω and γ = kz. The main derivation stages in [5] are as follows. The solution to the Helmholtz equation is as follows:

$$p{\kern 1pt} \text{*}{\kern 1pt} (r,z) = \int\limits_{ - \infty }^\infty {{{A}_{n}}H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right){{e}^{{i\gamma z}}}d\gamma } .$$
(21)

The coefficient An is determined using the boundary condition:

$${{\left. {\frac{{\partial p{\kern 1pt} \text{*}}}{{\partial r}}} \right|}_{{r = a}}} = \rho {{\omega }^{2}}w(z).$$
(22)

On the basis of Eq. (21) and boundary condition (22), the equation for the coefficient An is obtained:

$$\int\limits_{ - \infty }^\infty {{{A}_{n}}\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)}}} \left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right){{e}^{{i\gamma z}}}d\gamma = \rho {{\omega }^{2}}w(z).$$
(23)

In order to solve Eq. (23) with respect to the coefficient An, the displacement w(z) is presented in the form of the Fourier integral with respect to the parameters γ and z:

$$\begin{gathered} w(z) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {B(\gamma ){{e}^{{i\gamma z}}}d\gamma } , \\ B(\gamma ) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {w(z){{e}^{{ - i\gamma z}}}dz} . \\ \end{gathered} $$
(24)

As a consequence of substitution of the displacement w(z) (24) into Eq. (23), the following equation is obtained:

$$\int\limits_{ - \infty }^\infty {{{A}_{n}}\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)} {{e}^{{i\gamma z}}}d\gamma = \frac{{\rho {{\omega }^{2}}}}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {{{B}_{n}}(\gamma ){{e}^{{i\gamma z}}}d\gamma } ,$$
(25)

and from the solution of Eq. (25), the coefficient An is determined:

$${{A}_{n}} = \frac{{\rho {{\omega }^{2}}{{B}_{n}}(\gamma )}}{{\sqrt {2\pi } \sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}.$$
(26)

As a consequence of substitution of An (26) into Eq. (21), Eq. (5) was obtained.

Shenderov Error

In [5] the solution of the Helmholtz equation is in the form of integrals of both parts of Eq. (15)

$$p{\kern 1pt} \text{*}{\kern 1pt} (r,z) = \int\limits_{ - \infty }^\infty {pd\gamma = \int\limits_{ - \infty }^\infty {{{A}_{n}}H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right){{e}^{{i\gamma z}}}d\gamma } } .$$
(27)

Thus, the pressure p is substituted for the integral of pressure p* (21); however, the integral of pressure p* is used as pressure p in the boundary condition:

$${{\left. {\frac{{\partial p{\kern 1pt} \text{*}}}{{\partial r}}} \right|}_{{r = a}}} = \rho {{\omega }^{2}}w(z).$$
(28)

In [5] the displacement w(z) is presented by the spectral transform as the Fourier integral with respect to the parameters γ and z. Real displacements w(z) are comprised of constituents of the type w(z) = exp[iqz], where, for instance, q = 2πm/L. Then, the displacement w(z) = exp[iqz] can be presented by the Fourier integral with respect to the parameters q and z:

$$\begin{gathered} w(z) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {B(q){{e}^{{iqz}}}dq} , \\ B(q) = \frac{1}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {w(z){{e}^{{ - iqz}}}dz} . \\ \end{gathered} $$
(29)

In order to determine the coefficient An, let us substitute the displacement w(z) (29) into Eq. (27):

$$\int\limits_{ - \infty }^\infty {{{A}_{n}}\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)} {{e}^{{i\gamma z}}}d\gamma = \frac{{\rho {{\omega }^{2}}}}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {{{B}_{n}}(q){{e}^{{iqz}}}dq} .$$
(30)

In this case it is impossible to determine the coefficient An from Eq. (30). Due to this reason, in [5] in the displacement function w(z) it is unreasonably assigned that q = γ, and as a consequence, Eq. (5) is obtained. Assigning 2πm/L = γ is erroneous. It means establishing an interrelation between the shell length L and the wave number γ. The wave equation and its solution on the left-hand side of Eq. (30), which contain γ, cannot depend on the shell length.

The parameter q is related with the shell vibration and is determined by the form (mode) of shell oscillations. The parameter γ, the axial wave number of a sound wave in a fluid, is determined by the speed of sound in the fluid. Mutual substitution of these parameters has no physical sense; since they are not physically homogeneous, they are characterized by different physical nature (though the dimensionality is the same).

Equation (5) is fundamentally different from Eq. (20). The main difference is that in Eq. (5) the Hankel function is under the sign of an improper integral. Equation (5) is a consequence of two erroneus actions: (1) the boundary condition for the integral of pressure p* is written incorrectly; (2) the parameter q is equated to the wave number of a sound wave in fluid γ. Therefore, it follows that Eq. (5) is erroneous and it is not recommended for use. Equation (5) and similar erroneus equations were used by numerous authors [6, 8, 11–13].

As for Eq. (11), it should be mentioned that it is impossible to analyze it because of the absence of derivation of the Green function in [9]. Equations (5), (8), and (11) have a common feature: the Hankel functions in them are included into the integration function of the improper integral with the integration limits (–∞, ∞).

Junger and Muzychenko–Rybak Error

Let us demonstrate that Eqs. (4) and (10) can be obtained from Eq. (5). Let us set the displacement as follows:

$$w(z) = {{w}_{0}}{{e}^{{iqz}}}.$$
(31)

Let us substitute the displacement (31) into Eq. (5):

$$p(z) = \frac{{\rho {{\omega }^{2}}}}{{2\pi }}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {{{w}_{0}}{{e}^{{iqz}}}{{e}^{{ - i\gamma z}}}\frac{{H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right){{e}^{{i\gamma z}}}}}{{\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}d\gamma dz} } .$$

According to the definition of function δ, let us write the following:

$$\int\limits_{ - \infty }^\infty {{{e}^{{i(q - \gamma )z}}}dz = 2\pi \delta (q - \gamma )} .$$

Let us substitute function δ into the integral:

$${{p}_{s}} = \frac{{\rho {{\omega }^{2}}}}{{2\pi }}{{w}_{0}}\int\limits_{ - \infty }^\infty {\frac{{H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right){{e}^{{i\gamma z}}}}}{{\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}2\pi \delta (q - \gamma )d\gamma } .$$

According to the main property of the δ function, we obtain the impedance

$$\frac{{p(z)}}{{w(z)}} = \rho {{\omega }^{2}}\frac{{H_{n}^{{(2)}}\left( {a\sqrt {{{k}^{2}} - {{q}^{2}}} } \right)}}{{\left( {\sqrt {{{k}^{2}} - {{q}^{2}}} } \right)H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{q}^{2}}} } \right)}}.$$
(32)

The known solutions are obtained from Eq. (32): Eq. (3) for an infinite shell at q = 2πm/λ and Eq. (10) for constrained shell at q = πm/L. As it turns out, Eq. (32) can be applied for infinite and for constrained shells. Since Eq. (5) is erroneous, then, the resultant Eqs. (3), (10), and (12) are also erroneous. The falsity of Eq. (10) is predetermined by its derivation from Eq. (8). It is impossible to reveal the reason for the error in the derivation of Eq. (3), since its derivation is absent.

Correction of the Shenderov Equation

Equation (5) can be corrected even on the basis of the integral of pressure p* (21) without using the Fourier integral. With this aim, instead of Eq. (22), the correct boundary condition should be used:

$$\frac{{\partial p{\kern 1pt} \text{*}}}{{\partial n}} = \rho {{\omega }^{2}}\int\limits_{ - \infty }^\infty {w(z)d\gamma } .$$
(33)

Let us determine the coefficient An from Eq. (21) and boundary condition (33):

$$\int\limits_{ - \infty }^\infty {{{A}_{n}}\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)}}} \left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right){{e}^{{i\gamma z}}}d\gamma = \rho {{\omega }^{2}}\int\limits_{ - \infty }^\infty {w(z)d\gamma } ,$$
$${{A}_{n}} = \frac{{\rho {{\omega }^{2}}w(z)}}{{{{e}^{{i\gamma z}}}\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}.$$

Let us substitute this coefficient An into Eq. (21):

$$p\text{*} = \int\limits_{ - \infty }^\infty {\frac{{\rho {{\omega }^{2}}H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}{{\sqrt {{{k}^{2}} - {{\gamma }^{2}}} H_{n}^{{(2)'}}\left( {a\sqrt {{{k}^{2}} - {{\gamma }^{2}}} } \right)}}} w(z)d\gamma .$$
(34)

If in Eq. (34) we return from the integral of pressure p* to the pressure p, that is, to remove integration on the left- and right-hand sides of the equality, then, finally, we obtain Eq. (20).

CONCLUSIONS

An exact solution to the problem of radiation by an oscillating cylindrical shell submersed into a fluid was obtained. The equation of the sound pressure of the irradiated field was obtained on the basis of the solution to the wave equation. The following main results should be mentioned: (1) The axial wave number of the sound field irradiated by the cylindrical shell was determined equal to kz = k cos θ. (2) The impedance of a sound field irradiated by a cylindrical shell is invariant with respect to the shell type (finite or infinite); that is, it can be applied to any cylindrical shell. (3) No physically stipulated relation exists between the axial sound wave number and the wave numbers of deformation q = 2πm/L and q = 2π/λ. They are not substitutable functionally. (4) The Shenderov equation (5) and equations similar to it, where the Hankel functions are under the sign of the improper integral of the differential f(γ)dγ, and γ = q are erroneous. (5) The obtained equation of sound pressure of the field irradiated by cylindrical shells in a fluid (20) is universal and can be applied for solving all problems of the hydroacoustics of cylindrical shells. A solution to the formulated problem can be considered completed.