Secondary Field of a Finite Elastic Cylindrical Shell in a Liquid in the Far Zone

Abstract

A novel method for solving the problem of a secondary hydroacoustic field of a finite elastic cylindrical shell in the far zone is proposed. This method involves determining the roots of the dispersion equation for natural oscillations of a shell in a liquid, calculating the forced oscillations of a composite shell structure affected by an incident field, using the exact radiation impedance of the final shell in the liquid, calculating the secondary field including the field scattered on the elastic shell and the field reflected from a perfectly rigid shell.

This paper is devoted to a method for calculating the secondary hydroacoustic field of a finite elastic cylindrical shell immersed in a liquid in the far zone. The primary field represents a sound field radiated by an oscillating shell affected by internal excitation sources, with discrete forces applied to the shell. The secondary field (diffracted field) represents a field that occurs resulting from affecting the shell by an external incident sound field generated by a probing source. As a result, forced shell oscillations are excited and a radiated scattered (diffracted) hydroacoustic field arises. The scattered field consists of the following two components: first, the radiated field generated by the forced shell oscillations; second, the field reflected from the shell, as from a perfectly rigid body. The radiation and diffraction processes can be considered both in the near-zone field and in the far-zone field. Here, a finite shell with free ends is considered.

Many papers [1–20] have been devoted to studies on the oscillations, radiation, and diffraction of cylindrical shells. These papers have appeared regularly since the middle of the last century, which indicates the incompleteness and, therefore, the relevance of the study. The radiation impedance of a cylindrical shell is used in the calculations of forced oscillations and shell radiation in the near and far-zone fields.

Let us perform an analysis of the state of the art for solving problems connected with impedance, oscillations, and the secondary field of finite elastic cylindrical shells based on the best known publications.

The authors of well-known book [1] have presented the solution of the wave equation for a diverging cylindrical wave in the following form:

$$p = {{A}_{0}}H_{n}^{{(1)}}\left( {rk} \right){{e}^{{i{{k}_{z}}z}}}\cos (n\varphi ){{e}^{{ - i\omega t}}},$$

(1)

where $H_{n}^{{(1)}}$(arg) is the Hankel function; k = ω/c is the wave number; ω = 2πf is the oscillation frequency; c is the speed of sound in the medium (liquid); k_z is the axial wave number; n is the circumferential harmonic in angle φ; t is time; and r is the radial coordinate.

Solution (1) is not correct, because c instead of k should amount to k_r, and frequency ω according to the following relationship

$$\omega = kc = c\sqrt {k_{{}}^{2} + k_{z}^{2}} $$

at k_z ≠ 0, is not correctly determined. The authors of [1], have considered only the plane problem.

The authors of [2], with reference to [1], have presented a solution to the problem of radiation from an oscillating infinite cylindrical shell, the strain on the surface of which is given according to the following relationship:

$$w(z) = {{w}_{0}}\cos \left( {\frac{{2\pi m}}{\lambda }z} \right),$$

(2)

where λ is the strain wave length, m is the longitudinal harmonic, and z is the axial coordinate. The near-field sound pressure can be expressed according to the following relationship:

$$p = \frac{{i\rho \omega H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} \lambda }} \right. \kern-0em} \lambda }} \right)}}^{2}}} } \right)w\left( z \right)}}{{\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} \lambda }} \right. \kern-0em} \lambda }} \right)}}^{2}}} H_{n}^{{(2)}}{\kern 1pt} '\left( {a\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} \lambda }} \right. \kern-0em} \lambda }} \right)}}^{2}}} } \right)}},$$

(3)

where ρ is the density of the medium. In relationship (3), it is taken that k_z = 2πm/λ. However, in relationship (3), k_z should be the axial wave number of the sound wave propagating in the liquid, rather than the bending form of the shell oscillations (deformations), 2πm/λ (2). These parameters are heterogeneous in physical nature, and, thus, they cannot be interchanged. Therefore, relationship (3) cannot be considered justified.

The problems of sound diffraction on elastic cylindrical shells in the near-zone field were first considered in [3, 4]. The authors of [3] considered diffraction on a bounded shell with Navier boundary conditions. The authors of [4] considered diffraction on an infinite shell. The equations of shell oscillations in displacements have been taken from [5]. The sound pressure on the shell surfaces caused by the deformation of bounded and infinite shells has been represented by a relationship wherein the axial wave number k_z is an unknown integration variable within (–∞, ∞). The relationship for sound pressure is the same as relationship (5), which is discussed below. The authors of [6, 7] considered the problems of sound diffraction on an infinite perfectly rigid cylinder.

Fundamental monograph [8] on acoustics gives the following relationship for the radiation impedance of an oscillating cylindrical shell (in the opinion of many authors dealing with an infinite shell)

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

(4)

The disadvantage of relationship (4) consists in the fact that it has not been completed, because axial wave number k_z therein is not defined, it is not indicated how to determine this parameter.

From [8], the following is known concerning wave number k_z. First, a reference is given to [2], where k_z = 2πm/λ. Second, resulting from an incorrect determination of the constant-phase traveling wave front velocity, it is obtained that k_z/k_r = tan θ. Based on this, it has been taken that k_z = k sin θ, where θ is the angle of wave propagation that allegedly depends on the frequency (on k). At high frequencies, such that k² $\gg$ $k_{z}^{2}$, the radiated waves propagate perpendicular to the surface of the cylinder; as the frequency (k) decreases, angle θ exhibits an increase, and at k = k_z an acoustic short circuit occurs. Third, in the case of k < k_z and $k_{r}^{2}$ = k² – $k_{z}^{2}$ < 0, when the value of k_r becomes imaginary, it is recommended to replace the Hankel function by the Macdonald function.

The fact that exceeding k_z > k is possible was admitted by many authors [3, 4, 8, 9, 12]. As a result, referring to the values of k_z = 2πm/λ, without criticizing the possibility of changing k_z as an integration variable within –∞ < k_z < ∞ [3, 4] and recognizing the possibility of k_z > k, the author of [8] has not been able to give a relationship for determining axial wave number k_z.

The authors of [9] have presented the oscillation-driven displacement for the case of a finite elastic cylindrical shell, in the form of a Fourier integral, whereas the solution of the wave equation has been taken in the following form:

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

The radiated near-zone field caused by the oscillations of the finite elastic shell can be represented (without harmonic summation) in the following form:

$${{p}_{{sw}}} = \frac{{i\rho \omega }}{{4{{\pi }^{2}}}}\int\limits_{ - \infty }^\infty {\int\limits_{ - \infty }^\infty {\frac{{{{H}_{n}}\left( {a\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)w\left( {z{\kern 1pt} '} \right){{e}^{{i\xi (z - z{\kern 1pt} ')}}}}}{{\sqrt {{{k}^{2}} - k_{z}^{2}} H_{n}^{'}\left( {\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)}}dz{\kern 1pt} '{\kern 1pt} d{{k}_{z}}} } .$$

(5)

The relationships for the sound pressure of the radiated near-zone field caused by shell oscillations presented in [3, 4, 9] (upon integration within the limits of (–∞ < k_z < ∞) coincide with each other. However, because of the fact that these relationships are not justified, they do not inspire confidence.

For example, the authors of [9] performed changing variables k_z = ksin(α₁ + iα₂) followed by the determination of α₁ and α₂, when considering the radiation of a pulsating cylinder under the integration over k_z within the limits of (–∞ < k_z < ∞).

The authors of [4, 9] also solved the problem of sound diffraction on an infinite impedance shell in the near-zone field. With no justification it has been accepted that k_z = ksinψ, where ψ is the angle of incidence of the sound wave.

The authors of [10] have solved for the first time the problem of radiation from an oscillating finite cylindrical shell in the far-zone field using the Kirchhoff formula. The problem of forced shell oscillations was not solved. The displacements were specified in the form of a harmonic function. As the impedance, the authors used the radiation impedance of an infinite cylindrical shell (3), wherein the parameter 2π/λ was replaced without justification by ξ = –k cos θ. The replacement has been explained by the fact that the Kirchhoff formula includes integrals containing exponents exp(ikz cos θ) (12). These integrals are unreasonably called Fourier transforms, and their ratio has been hypothetically taken as the radiation impedance of the finite shell. Such a choice of impedance cannot be considered theoretically justified.

The radiation impedance of a limited cylindrical shell is presented in [11]. The oscillatory velocity of the shell is set in the same form as (2), only λ = L, where L is the length of the shell. The authors have presented the pressure of the radiated field in the following form:

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

(6)

where they have taken as Z₁ the relationship for an infinite cylindrical shell (4). The radiation impedance of a bounded shell in the final form can be represented by the following relationship:

$${{Z}_{3}} = \frac{{\rho {{\omega }^{2}}H_{n}^{{(2)}}\left( {r\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} L}} \right. \kern-0em} L}} \right)}}^{2}}} } \right)}}{{\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} L}} \right. \kern-0em} L}} \right)}}^{2}}} H_{n}^{{(2){\kern 1pt} '}}\left( {a\sqrt {{{k}^{2}} - {{{\left( {{{2\pi m} \mathord{\left/ {\vphantom {{2\pi m} L}} \right. \kern-0em} L}} \right)}}^{2}}} } \right)}}.$$

(7)

Relationship (7) is just the same as (3) taking into account λ = L. It should be noted that there is no reference to [2]. The remarks made concerning (3) can be applied to this case as well. Here, relationship (6) is the reason for obtaining the erroneous result. It was erroneously stated in [11] that, when L → ∞, the impedance of a bounded shell tends to the impedance of an infinite shell.

The authors of [12] considered the problem of sound diffraction on a finite perfectly rigid cylinder in the far-zone field using the Kirchhoff formula:

$${{P}_{{{\text{rad}}}}} = {{P}_{0}}\frac{{{{e}^{{ikR}}}i}}{{R\pi }}\sum\limits_{n = 0}^\infty {{{\varepsilon }_{n}}\cos n\varphi \int\limits_{ - L}^L {\frac{{J_{n}^{'}\left( {ak\sin \psi } \right)}}{{H_{n}^{'}\left( {ak\sin \psi } \right)}}dz} } ,$$

(8)

where ψ is the angle of incidence of the sound wave. As the impedance in the Kirchhoff formula, one takes the impedance for an infinite cylinder. Instead of the total pressure, only the pressure of the scattered field is taken, and the derivative of the total pressure, contrary to the boundary condition, is not equated to zero. Resulting from this, relationship (8) is erroneous.

The problem of sound diffraction on a solid cylinder bounded at the ends by hemispheres has been considered in [13] in the far-zone field. Therein, as was done in [12], instead of the summary of the total field, only a part thereof, i.e., the scattered field was taken erroneously in the Kirchhoff formula.

Numerical calculation methods have been proposed in a number of papers [8, 14, 15], but they significantly exceed analytical methods from the standpoint of complexity and laboriousness. A modern numerical method is represented by the use of the finite element method (FEM) based on the ANSYS software package. The authors of [15] have performed the simulation of the scattered field using the FEM method for the case of a cylindrical shell model with a length of 10 m and a diameter of 0.9 m. In the same paper, it was stated that it is impossible to calculate the far hydroacoustic field of real underwater objects using the FEM method, since the number of elements inherent in the calculation model increases in proportion to the cube of the distance. According to the model adopted, the shell was represented by 20 000 elements, whereas the liquid was represented by 180 000 elements.

This analysis has shown that the problem under consideration has not yet been completely solved, and there are no analytical formulas that make it possible to calculate the total secondary field of a finite cylindrical shell in the far zone at arbitrary angles of incidence and observation.

This is the rationale for the relevance of the problem under consideration.

This paper represents a continuation and development of paper [16], wherein the solution of the problem of sound diffraction on a finite perfectly rigid cylindrical shell in the far-zone field has been presented.

These studies are aimed at developing the theoretical foundations of an analytical method for calculating the total diffracted field of a finite elastic cylindrical shell in the far zone at arbitrary observation angles. The novelty thereof consists in the fact that the proposed method is complete; i.e., it includes the calculation of forced cylindrical shell oscillations excited by an incident field and scattering on the shell of both an elastic and a rigid body. This method is proposed for the first time. The calculations of forced shell oscillations and of the secondary field include the radiation impedance of the cylindrical shell exhibited in a different manner in the case of the dispersion equation and in the case of the Kirchhoff formula. For the first time in the dispersion equation and in the calculation of the secondary field sound pressure, the exact radiation impedance is used, wherein the axial wave number k_z is determined based on the solution of the wave equation.

The usefulness of the method consists in the fact that it can be used in solving various applied problems of hydroacoustics, including the acoustic design of shell structures.

The problem of sound diffraction on a finite elastic cylindrical shell immersed in a liquid in the far-zone field is considered. The scattering of sound waves is considered on the cylindrical surface of the shell without taking into account end stubs (spherical, etc.). In the case of a number of applied problems, there are no stubs and it is not required to take them into account. If necessary, scattering by stubs can be considered separately [8, 12]. This makes sense at small angles of incidence (ψ ≈ 0°–5°) and under taking into account the longitudinal oscillations of a cylindrical shell.

The designations of the shell parameters are the following: a is the radius, L is the length, and h is the thickness. The source of sound radiation represents a monopole with volumetric velocity V located at a large distance H from the shell. The forced shell oscillations are excited by the sound field incident on the surface of the shell.

The sound pressure of the field incident on the shell at angle ψ with respect to the z axis of the shell [16] can be expressed as follows:

$${{p}_{0}} = {{A}_{0}}{{e}^{{ikz\cos \psi }}}\sum\limits_{n = 1}^\infty {{{\varepsilon }_{n}}{{i}^{n}}{{J}_{n}}(kr\sin \psi )\cos n\varphi } ,\quad {{A}_{0}} = \frac{{i\rho \omega V{{e}^{{ - ikH}}}}}{{4\pi H}}.$$

(9)

The radiated field determined by the solution of the wave equation in cylindrical coordinates (without summing the harmonics) can be represented as

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

(10)

where B_n is the desired function determined from the boundary conditions, $H_{n}^{{(2)}}$(arg) is the Hankel function of the second kind (below, index (2) is omitted), and k_z is the desired coefficient. This coefficient can be determined from the solution of the wave equation that can be represented as k² = $k_{r}^{2}$ + $k_{z}^{2}$. The mathematical solution to this equation is k_r = k sin θ, k_z = k cos θ. The physical meaning of angle θ is the angle between the propagation direction of the radiated cylindrical wave having wave number k and the z axis in the cylindrical coordinate system. The novelty of this solution consists in the explicit determination of the axial wave number k_z. The usefulness thereof consists in the unambiguous determination of the radiation impedance of the cylindrical shell (4), regardless of its length, i.e., being either finite or infinite. The impedance relationship (4) with k_z = k cos θ eliminates the existing disagreements and its various interpretations and becomes the basis for solving all the problems of hydroacoustics for any cylindrical shell in the liquid.

By using the boundary conditions on the cylindrical shell surface, $\left. {\frac{{\partial p}}{{\partial r}}} \right|$ = $\rho {{\omega }^{2}}w(z)\cos n\varphi$, let us find function B_n

$${{B}_{n}} = \frac{{\rho {{\omega }^{2}}w(z) - k\sin {{A}_{0}}{{e}^{{ikz\cos \psi }}}{{\varepsilon }_{n}}{{i}^{n}}J_{n}^{'}(ak\sin \psi )}}{{\sqrt {{{k}^{2}} - k_{z}^{2}} H_{n}^{'}\left( {a\sqrt {{{k}^{2}} - k_{z}^{2}} } \right){{e}^{{i\gamma z}}}}}.$$

The pressure of the sound field scattered near the shell can be written as follows:

$${{p}_{s}} = \left[ {\frac{{\rho {{\omega }^{2}}w(z) - k\sin {{A}_{0}}{{e}^{{ikz\cos \psi }}}{{\varepsilon }_{n}}{{i}^{n}}J_{n}^{'}(ak\sin \psi )}}{{\sqrt {{{k}^{2}} - k_{z}^{2}} H_{n}^{'}\left( {a\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)}}} \right]{{H}_{n}}\left( {r\sqrt {{{k}^{2}} - k_{z}^{2}} } \right)\cos n\varphi .$$

(11)

The scattered sound field in the far zone can be determined using the Kirchhoff formula [10]:

$${{p}_{N}}{\text{ }} = \frac{1}{{4\pi }}\int {\int {\left[ {p\frac{{\partial G\left( {N,A} \right)}}{{\partial n}} - \frac{{\partial p}}{{\partial n}}G\left( {N,A} \right)} \right]} } {\text{ }}ds,$$

$$G\left( {N,A} \right) = \frac{{\exp ( - ik{{R}_{1}})}}{{{{R}_{1}}}},\quad {{R}_{1}} = \left| {N,A} \right|.$$

Resulting from the performed transformations of this relationship, which have been omitted for brevity, the formula for the scattered field in the far zone can be presented in the following form:

$${{P}_{N}} = - \left( {\frac{{{{e}^{{ - ikR}}}}}{R}} \right)\frac{{{{e}^{{\frac{{i\pi n}}{2}}}}}}{2}\left[ { - \mu J_{n}^{'}\left( \mu \right)\int\limits_0^L {p{{e}^{{ikz\cos \theta }}}dz + a{{J}_{n}}\left( \mu \right)\int\limits_0^L {\frac{{\partial p}}{{\partial r}}{{e}^{{ikz\cos \theta }}}dz} } } \right],$$

(12)

where μ = aksinθ. It should be noted that formula (12) represents a development of the relationship given in [10]. It should be noted that no complete derivation of this relationship has been given in [10], and in the final relationship and intermediate calculations, there were typos and errors that have been corrected here.

The total field on the shell surface is p = p₀ + p_s. Let us substitute the relationships for p₀ (9) and p_s (11) into (12), let us replace exp(iπn/2) = iⁿ to obtain the pressure of the secondary field scattered by the finite cylindrical shell in the far zone, as follows:

$${{p}_{N}} = - E\frac{{{{i}^{n}}}}{2}\left\{ { - \mu J_{n}^{'}\left( \mu \right)\left[ {{{A}_{0}}{{\varepsilon }_{n}}{{i}^{n}}{{T}_{n}}\int\limits_0^L {{{e}^{{ik\beta z}}}dz + \int\limits_0^L {\frac{{a\rho {{\omega }^{2}}{{H}_{n}}\left( {\mu \text{*}} \right)w(z){{e}^{{ikz\cos \theta }}}}}{{\mu \text{*}{\kern 1pt} H_{n}^{'}\left( {\mu \text{*}} \right)}}dz} } } \right] + {{Y}_{n}}} \right\},$$

$${{T}_{n}} = {{J}_{n}}\left( \eta \right) - \frac{{\eta J_{n}^{'}\left( \eta \right){{H}_{n}}\left( {\mu \text{*}} \right)}}{{\mu \text{*}{\kern 1pt} H_{n}^{'}\left( {\mu \text{*}} \right)}},\quad {{Y}_{n}} = a{{J}_{n}}\left( \mu \right)\int\limits_0^L {\rho {{\omega }^{2}}w(z){{e}^{{ikz\cos \theta }}}dz} ,$$

where $\mu \text{*} = a\sqrt {{{k}^{2}} - k_{z}^{2}}$; $\eta = ak\sin \psi$; $E = \frac{{{{e}^{{ - ikR}}}}}{R}$; $\beta = \cos \psi + \cos \theta$.

The total far-zone field p_N consists of the sum of two parts, one of which, p_t corresponds to the reflected field as in the case of reflection from an perfectly rigid body, and the second of which, p_w, is caused by the oscillations of the elastic shell; p_N = p_t + p_w.

Taking into account the summation of harmonics, the far-zone field reflected by a perfectly rigid cylindrical shell can be presented in the following form:

$${{p}_{t}} = \frac{{{{e}^{{ - ikR}}}}}{R}\frac{{{{A}_{0}}\mu }}{{2\mu \text{*}}}\sum\limits_{n = 0}^\infty {\frac{{{{\varepsilon }_{n}}{{i}^{{2n}}}J_{n}^{'}\left( \mu \right)}}{{H_{n}^{'}\left( {\mu \text{*}} \right)}}} \left[ {{{T}_{\eta }}} \right]\int\limits_0^L {{{e}^{{ikz\beta }}}dz} ,$$

(13)

where ${{T}_{\eta }} = \mu \text{*}{\kern 1pt} H_{n}^{'}\left( {\mu \text{*}} \right){{J}_{n}}\left( \eta \right) - \eta J_{n}^{'}\left( \eta \right){{H}_{n}}\left( {\mu \text{*}} \right)$, $\beta = \cos \psi + \cos \theta$.

The scattered far-zone field caused by the oscillations of an elastic cylindrical shell can be expressed as

$${{p}_{w}} = - \frac{{{{e}^{{ - ikR}}}a\rho {{\omega }^{2}}}}{{2R}}\sum\limits_{n = 0}^\infty {{{i}^{n}}\left[ {{{J}_{n}}\left( \mu \right) - \frac{{\mu J_{n}^{'}\left( \mu \right){{H}_{n}}\left( {\mu \text{*}} \right)}}{{\mu \text{*}{\kern 1pt} H_{n}^{'}\left( {\mu \text{*}} \right)}}} \right]\int\limits_0^L {w(z){{e}^{{ikz\cos \theta }}}dz} } .$$

(14)

In order to calculate the radiation pattern in the far-zone field, in the general case, it is required to use both relationships (13) and (14). These relationships can be simplified if one is confined to determining the pressure only in the direction of specular reflection, θ₁ = π – ψ, and in the direction of the shadow ray, θ₂ = π + ψ.

For the calculation according to formula (14), it is necessary to know shell deformations w(z) that should be determined from the calculation of the forced shell oscillations in the liquid excited by the incident field. Let us assume that the external sound pressure on the shell that excites the oscillations thereof occurs in the radial direction along the w coordinate [17–20].

For each circumferential harmonic n taking into account the omitted time dependence e^iωt, the equation for the forced shell oscillations in the liquid can be represented as

$$\left[ {\begin{array}{*{20}{c}} {{{L}_{{11}}} + \omega _{*}^{2}}&{{{L}_{{12}}}}&{{{L}_{{13}}}} \\ { - {{L}_{{12}}}}&{{{L}_{{22}}} + \omega _{*}^{2}}&{{{L}_{{23}}}} \\ { - {{L}_{{13}}}}&{{{L}_{{23}}}}&{{{L}_{{33}}} + \omega _{*}^{2}} \end{array}} \right] \cdot \left\{ {\begin{array}{*{20}{c}} u \\ {v} \\ w \end{array}} \right\} = \frac{a}{q}\left\{ {\begin{array}{*{20}{c}} 0 \\ 0 \\ {{{p}_{0}} + {{p}_{s}}} \end{array}} \right\}.$$

(15)

The complete solution of Eq. (15) can be represented as the sum of the general solution of the corresponding homogeneous equation containing eight constants and the particular solution of forced oscillations excited by total pressure p = p₀ + p_s

$$\begin{gathered} u{\text{ }} = {\text{ }}U\cos (n\varphi ) + {{u}_{p}},\quad {v} = V\sin (n\varphi ) + {{{v}}_{p}},\quad w = W\cos (n\varphi ) + {{w}_{p}}, \\ U = {\text{ }}\sum\limits_{j = 1}^8 {{{C}_{{jn}}}} \frac{{\Delta _{{jn}}^{{\left( 2 \right)}}}}{{\Delta _{{jn}}^{{\left( 1 \right)}}}}{{e}^{{i{{\alpha }_{{jn}}}\xi }}},\quad V = \sum\limits_{j = 1}^8 {{{C}_{{jn}}}} \frac{{\Delta _{{jn}}^{{\left( 3 \right)}}}}{{\Delta _{{jn}}^{{\left( 1 \right)}}}}{{e}^{{i{{\alpha }_{{jn}}}\xi }}},\quad W = \sum\limits_{j = 1}^8 {{{C}_{{jn}}}} {{e}^{{i{{\alpha }_{{jn}}}\xi }}}, \\ \end{gathered} $$

(16)

where n are the circumferential harmonics of the Fourier series, n = 0, 1, 2, 3, …; α_jn are the roots of the dispersion equation; j = 1–8 are the ordinal numbers of the roots; С_jn are the desired coefficients; $\Delta$_jn are the minors of the matrix in shell oscillation equation (15); ω = 2πf is the angular frequency of oscillations; f is the oscillation frequency; and ξ = z/a. Solution (16) includes dispersion equation roots α_jn and coefficients С_jn to be determined. In order to obtain the dispersion equation, let us present solution (16) in the following form:

$$u = U{{e}^{{i\alpha \xi }}}\cos n\varphi ;\quad {v} = V{{e}^{{i\alpha \xi }}}\sin n\varphi ;\quad w = W{{e}^{{i\alpha \xi }}}\cos n\varphi ;\quad \xi = {z \mathord{\left/ {\vphantom {z a}} \right. \kern-0em} a}.$$

Resulting from the substitution of these solutions into Eq. (15) at p₀ = 0, one can obtain the following equation for the free shell oscillations:

$$\left[ {\begin{array}{*{20}{c}} {{{L}_{{11}}} + \omega _{*}^{2}}&{{{L}_{{12}}}}&{{{L}_{{13}}}} \\ { - {{L}_{{12}}}}&{{{L}_{{22}}} + \omega _{*}^{2}}&{{{L}_{{23}}}} \\ { - {{L}_{{13}}}}&{{{L}_{{23}}}}&{{{L}_{{33}}} + \omega _{*}^{2}} \end{array}} \right] \cdot \left\{ {\begin{array}{*{20}{c}} U \\ V \\ W \end{array}} \right\} = \frac{a}{q}\left\{ {\begin{array}{*{20}{c}} 0 \\ 0 \\ {{{p}_{{s0}}}} \end{array}} \right\},$$

(17)

where p_s0 is the sound pressure in the liquid caused by free shell oscillations taking into account an effect of the added mass of the liquid. Assuming that the radiation occurs in the radial direction along coordinate w, i.e., normal to the surface of the shell, one can take that r = a, θ = 90° and, therefore, in (4) one obtains k_z = 0.

$${{p}_{{s0}}} = \frac{{{{\rho }_{0}}{{\omega }^{2}}wH_{n}^{{(2)}}(ak)}}{{kH_{n}^{{(2)'}}(ak)}}.$$

Exponents α included in the solution are the roots of the dispersion equation for free oscillations. After appropriate transformations, one can obtain the dispersion equation for the oscillations of a cylindrical shell in the liquid in the following form [21]:

$$\frac{{{{\Delta }_{0}}(\alpha )}}{{{{\Delta }^{1}}(\alpha )}} - \frac{{{{\rho }_{0}}{{\omega }^{2}}aH_{n}^{{(2)}}(ka)}}{{qkH_{n}^{{(2)'}}(ka)}} = 0.$$

(18)

However, the difference for the result obtained consists in the fact that earlier the authors of [21] did not justify obtaining this equation, based on the erroneous assumptions for the determination of p_s0 [9]. The roots are determined here in the same way as was done in [21].

Let us define a particular solution of Eq. (15), taking it in the form of (u_p cos nφ, ${{{v}}_{p}}$ sin nφ, w_p cos nφ)e^iγξ , where γ = kα cos ψ is the phase of the incident field.

When substituting solutions into Eq. (17), the elements of the matrix can be presented in the following form:

$${{L}_{{11}}} = {{\gamma }^{2}} - \frac{{1 - \nu }}{2}{{n}^{2}} + {{b}_{1}}{{\gamma }^{2}},\quad {{L}_{{12}}} = \frac{{1 + \nu }}{2}\gamma n = - {{L}_{{21}}},\quad {{L}_{{13}}} = \gamma \nu - \frac{{{{z}_{1}}{{b}_{1}}}}{r}{{\gamma }^{3}},$$

$${{L}_{{22}}} = \frac{{1 - \nu }}{2}\left( {1 + 4{{\delta }^{2}}} \right){{\gamma }^{2}} - {{n}^{2}}\left( {1 + {{b}_{2}} + 2\frac{{z{{b}_{2}}}}{p} + {{\delta }^{2}} + \frac{{{{a}_{2}}}}{{{{r}^{2}}}}} \right),$$

$${{L}_{{23}}} = {{L}_{{32}}} = - n\left[ {1 + {{b}_{2}} + \frac{{{{z}_{2}}{{b}_{2}}}}{r} - \left( {2 - \nu } \right){{\delta }^{2}}{{\gamma }^{2}} + {{n}^{2}}\left( {{{\delta }^{2}} + \frac{{{{z}_{2}}{{b}_{2}}}}{r} + \frac{{{{a}_{2}}}}{{{{r}^{2}}}}} \right)} \right],$$

(19)

$${{L}_{{31}}} = - {{L}_{{13}}},\quad {{L}_{{33}}} = - 1 - {{b}_{2}} - {{n}^{4}}\frac{{{{a}_{2}}}}{{{{r}^{2}}}} - {{\delta }^{2}}{{\left( {{{\gamma }^{2}} - {{n}^{2}}} \right)}^{2}} - 2\frac{{{{z}_{2}}{{b}_{2}}}}{r} - {{\gamma }^{4}}\frac{{{{a}_{1}}}}{{{{r}^{2}}}},$$

$${{\delta }^{2}} = \frac{{{{h}^{2}}}}{{12{{a}^{2}}}},\quad \omega _{*}^{2} = \frac{{{{\omega }^{2}}{{a}^{2}}\rho _{*}^{{}}\left( {1 - {{\nu }^{2}}} \right)}}{E},\quad q = \frac{{{{E}_{1}}h}}{{\left( {1 - {{\nu }^{2}}} \right)a}},\quad i = \sqrt { - 1} ,$$

where a₁, b₁ are the stringer parameters; а₂, b₂, z₂ are the frame parameters; E₁= E₀(1 + iχ) is the complex modulus of elasticity; χ is the value of losses in the shell material; r = a; ν is the Poisson ratio; and ${{\rho }_{*}}$ is the density of the shell material.

Denoting the matrix in (15) as

$${{L}_{{i,j}}} = \left[ {\begin{array}{*{20}{c}} {{{L}_{{11}}} + \omega _{*}^{2}}&{{{L}_{{12}}}}&{{{L}_{{13}}}} \\ { - {{L}_{{12}}}}&{{{L}_{{22}}} + \omega _{*}^{2}}&{{{L}_{{23}}}} \\ { - {{L}_{{13}}}}&{{{L}_{{23}}}}&{{{L}_{{33}}} + \omega _{*}^{2}} \end{array}} \right],$$

let us determine the vector of a particular solution to Eq. (15)

$$\left\{ \begin{gathered} {{u}_{p}} \\ {{{v}}_{p}} \\ {{w}_{p}} \\ \end{gathered} \right\} = \frac{a}{q}{{\left[ {{{L}_{{i,j}}}} \right]}^{{ - 1}}}\left\{ \begin{gathered} 0 \\ 0 \\ {{p}_{0}} + {{p}_{s}} \\ \end{gathered} \right\}.$$

(20)

Based on (20), one can obtain the radial displacement as follows:

$${{w}_{p}} = \frac{{{{\Delta }_{1}}}}{{{{\Delta }_{0}}}}a\left( {{{p}_{0}} + {{p}_{s}}} \right),$$

(21)

where Δ₁ is the minor and Δ₀ is the determinant of matrix L_i, j,

$${{\Delta }_{1}} = \left( {{{L}_{{11}}} + \omega _{*}^{2}} \right)\left( {{{L}_{{22}}} + \omega _{*}^{2}} \right) + L_{{12}}^{2},$$

$$\begin{gathered} {{\Delta }_{0}} = \left( {{{L}_{{11}}} + \omega _{*}^{2}} \right)\left( {{{L}_{{22}}} + \omega _{*}^{2}} \right)\left( {{{L}_{{33}}} + \omega _{*}^{2}} \right) - {{L}_{{12}}}{{L}_{{23}}}{{L}_{{13}}} - {{L}_{{13}}}{{L}_{{12}}}{{L}_{{23}}} \\ + \;{{L}_{{13}}}\left( {{{L}_{{22}}} + \omega _{*}^{2}} \right){{L}_{{13}}} - {\text{ }}\left( {{{L}_{{11}}} + \omega _{*}^{2}} \right)L_{{23}}^{2} + L_{{12}}^{2}{{L}_{{33}}}. \\ \end{gathered} $$

After the substitution of the relationships for incident field pressure p₀ and that for scattered field pressure p_s into relationship (21) and performing the corresponding transformations, one can obtain

$${{w}_{p}} = \frac{{{{A}_{0}}{{\varepsilon }_{n}}{{i}^{n}}\left[ {{{J}_{n}}\left( \eta \right) - \frac{{\eta J_{n}^{'}\left( \eta \right){{H}_{n}}\left( {\mu \text{*}} \right)}}{{\mu \text{*}{\kern 1pt} H_{n}^{'}\left( {\mu \text{*}} \right)}}} \right]{{e}^{{ikz\cos \psi }}}}}{{\frac{q}{a}{{Z}_{{\text{m}}}} - \frac{{a\rho {{\omega }^{2}}{{H}_{n}}\left( {\mu \text{*}} \right)}}{{\mu \text{*}{\kern 1pt} H_{n}^{'}\left( {\mu \text{*}} \right)}}}},$$

(22)

where Z_m = Δ₀/Δ₁ is the mechanical impedance of the shell. Let us substitute w(z) = w_p from (22) into (14) and, taking into account (13) and (14), let us determine the final relationship for p_N..

The technique for compiling the equations of forced oscillations for a composite shell structure, including a set of cylindrical shells interconnected by rings, for the secondary field is similar to the technique described in [21] for the primary field, but differs from it in the relationships for the right-hand sides of the equations of forced oscillations caused by the incident field.

As a result, a system of matrix equations of forced oscillations with respect to ring displacements z_k can be obtained, as follows:

$$\left[ {{{M}_{0}} - H_{0}^{4}G_{1}^{1}\left( 0 \right){\text{ }}C_{1}^{1}H_{0}^{2}} \right]{{z}_{0}} - H_{0}^{4}G_{1}^{1}\left( 0 \right)C_{1}^{2}H_{1}^{1}z_{1}^{{}}$$

$$ = H_{0}^{4}\sum\limits_{s = 1}^{{{s}_{\nu }}} {\left\{ {\Phi _{{s1}}^{{}}F_{{s1}}^{{}} - G_{1}^{1}\left( 0 \right)\left[ {C_{1}^{1}f_{{s1}}^{{}} + C_{1}^{2}f_{{s1}}^{{}}{{e}^{{i{{\theta }_{{sk}}}{{\ell }_{k}}}}}} \right]} \right\}{\text{,}}} $$

$$H_{k}^{3}G_{k}^{1}\left( {{{\ell }_{k}}} \right)C_{k}^{1}H_{{k - 1}}^{2}z_{{k - 1}}^{{}}$$

$$ + \;\left[ {M_{k}^{{}} + H_{k}^{3}G_{k}^{1}\left( {{{\ell }_{k}}} \right)C_{k}^{2}H_{k}^{1} - H_{k}^{4}G_{{k + 1}}^{1}\left( 0 \right)C_{{k + 1}}^{1}H_{k}^{2}} \right]{{z}_{k}} - H_{k}^{4}G_{{k + 1}}^{1}\left( 0 \right)C_{{k + 1}}^{2}H_{{k + 1}}^{1}{{z}_{{k + 1}}}$$

$$ = \sum\limits_{s = 1}^{{{s}_{\nu }}} {\left\{ {H_{k}^{4}\Phi _{{s,k + 1}}^{{}}F_{{s,k + 1}}^{{}} - H_{k}^{3}\Phi _{{sk}}^{{}}F_{{sk}}^{{}}{{e}^{{i{{\theta }_{{sk}}}{{\ell }_{k}}}}} + H_{k}^{3}G_{k}^{1}\left( {{{\ell }_{k}}} \right)\left[ {C_{k}^{1}f_{{sk}}^{{}} + C_{k}^{2}f_{{sk}}^{{}}{{e}^{{i{{\theta }_{{sk}}}{{\ell }_{k}}}}}} \right]} \right.} $$

(23)

$$ - \;\left. {H_{k}^{4}G_{{k + 1}}^{1}\left( 0 \right)\left[ {C_{{k + 1}}^{1}f_{{s,k + 1}}^{{}} + C_{{k + 1}}^{2}f_{{s,k + 1}}^{{}}{{e}^{{i{{\theta }_{{sk}}}{{\ell }_{k}}}}}} \right]} \right\},$$

$$k = 1,2 \ldots n--1;$$

$$H_{n}^{3}G_{n}^{1}\left( {{{\ell }_{n}}} \right)C_{n}^{1}H_{{n - 1}}^{2}{{z}_{{n - 1}}} + \left[ {M_{n}^{{}} + H_{n}^{3}G_{n}^{1}\left( {{{\ell }_{n}}} \right){\text{ }}C_{n}^{2}H_{n}^{1}} \right]{{z}_{n}}$$

$$ = H_{n}^{3}\sum\limits_{s = 1}^{{{s}_{\nu }}} {\left\{ {\Phi _{{sn}}^{{}}F_{{sn}}^{{}}{{e}^{{i{{\theta }_{{sn}}}{{\ell }_{n}}}}} - G_{n}^{1}\left( {{{\ell }_{n}}} \right)\left[ {C_{n}^{1}f_{{sn}}^{{}} + C_{n}^{2}f_{{sn}}^{{}}{{e}^{{i{{\theta }_{{sn}}}{{\ell }_{n}}}}}} \right]} \right\}{\text{.}}} $$

The general matrix equation for a shell structure consisting of a set of compartments and rings exhibits a ribbon diagonal arrangement of block matrices with a size of 4 × 4, and in total can be on the order of several hundred. Resulting from solving this system, the required displacement vectors z_q for the rings can be determined.

After determining the displacement vectors z_q for rings q, based on Eq. (23) one can plot the frequency response curves for the oscillations in preset sections (rings) of the shell structure, as well as the shapes of forced oscillations for each shell and the entire shell structure as a whole.

The shape of oscillations for each shell can be determined according to the relationship [21]

$${\text{ }}\zeta _{q}^{{}}\left( y \right) = {\text{G}}_{{\text{q}}}^{1}\left( y \right)\left[ {C_{q}^{1}\left( {H_{{q - 1}}^{2}{{Z}_{{q - 1}}}} \right) + C_{q}^{2}\left( {H_{q}^{1}{{Z}_{q}}} \right)} \right] + {{W}_{p}},\quad 0 \leqslant y \leqslant \ell _{q}^{{}}.$$

Relationships (13), (14), (22), and (23) and the method as a whole are in fact new and have no analogues. The method is precise, because it is based on an exact theory. No assumptions and conditions limiting the accuracy of the solution have been used. Algorithms and computer programs in Fortran have been developed. According to these programs, the calculations have been conducted.

As an example, Fig. 1 shows the frequency response curve for the bending oscillations of a cylindrical shell at two angles of incidence for n = 1. The shell parameters are as follows: a = 4 m, L = 70 m, h = 0.04 m; the frequency calculation pitch amounts to 0.25 Hz. The coordinate axes represent the values of radial accelerations w in dB and the values of frequency in Hz.

Fig. 1.

Frequency response curves for shell oscillations: (1) ψ = 80°; (2) ψ = 30°.

Comparing the frequency response curve in Fig. 1 with similar frequency response curves obtained by excitation of shell oscillations by discrete forces [21], one can see that the resonant peaks are not as pronounced, and at large angles of incidence ψ they disappear completely; i.e., the shape of frequency response curves depends on the angle of incidence.

Figure 2 shows the shape of the flexural shell oscillations in the region of the first flexural oscillation resonance, f = 3.75 Hz; for the case of n = 1, ψ = 30°, the solid line corresponds to the imaginary component (Im), whereas the dashed line corresponds to the real (Re) component.

Fig. 2.

The shape of shell oscillations.

Figure 3 shows the radiation pattern of the secondary field at incidence angle ψ = 80° for frequency f  = 100 Hz. The coordinate axes represent the value of reduced sound pressure in N/m² (in fractions of 10⁵) and the observation angle ψ in degrees. Figure 4 shows a change in the maximum pressure of the radiation pattern depending on the oscillation frequency at the two angles of incidence according to the sum of eight harmonics, n = 0–7.

Fig. 3.

Radiation pattern.

Fig. 4.

Frequency response curve for the maximum sound pressure: (1) ψ = 30°; (2) ψ = 80°.

CONCLUSIONS

Thus, a novel analytical solution of the diffraction problem is obtained, which makes it possible to determine the total secondary (diffracted) far-zone field of a finite elastic cylindrical shell in the liquid.

The solution obtained is strictly theoretically justified (based on the theory of shell oscillations, the wave equation, and the Kirchhoff formula). The solution was obtained without any assumptions, restrictions, or conditions. This method represents a theoretical basis for practical calculations in the acoustic design of shell structures in a liquid.

The novelty of the solution is determined by the following. When calculating the oscillations and the secondary field, a theoretically justified exact radiation impedance of a finite cylindrical shell in the liquid has been used. The roots of the dispersion equation for the oscillations of a shell in the liquid are determined. The frequency response curves and the forced oscillation shapes of a finite elastic cylindrical shell in the liquid excited by an incident field have been determined. The proposed method has no analogues. The experimental verification of the method is an independent problem.