An Analytical Solution for the Vibration and Far-Field Sound Radiation Analysis of Finite, Semisubmerged Cylindrical Shells

The vibration response and far-ﬁeld sound radiation of a semisubmerged, ﬁnite cylindrical shell with low-frequency excitation are studied. The solution to this problem can be divided into two steps. The ﬁrst step is to apply the wave propagation approach to determine the vibration response of the cylindrical shell. In the cylindrical coordinate system, the Fl¨ugge shell equations and Laplace equation are used to describe the cylindrical shell and surrounding ﬂuid so that the vibration responses of the shell can be addressed analytically. The ﬂuid free surface eﬀect is taken into account by applying the sine series to force the velocity potential on the free surface to be zero. Furthermore, compared with the FEM (the ﬁnite element method), the present method is not only reliable but also eﬀective. In the second step, the far-ﬁeld sound radiation is solved by the Fourier transform technique and the stationary phase method in accordance with the vibration responses of the shell from the previous step. The boundary element method is applied to validate the reliability of the acoustical radiation calculation. The circumferential directivity of far-ﬁeld sound pressure is discussed, and it is found that the maximum value of the sound pressure always appears directly under the structure when the driving frequencies are relatively low. Besides, in consideration of simplicity and less computation eﬀort, the present method can be used for the rapid prediction of the vibration and far-ﬁeld sound pressure of a semisubmerged cylindrical shell with low-frequency excitation.


Introduction
Vibroacoustical characteristics of cylindrical shells are the subject studied extensively due to their widespread use in many industrial applications, such as underwater vehicles, aircrafts, and pipelines. ere are many researchers [1,2] who have studied the shell and plate vibration problems. And with respect to the coupling effect of acoustics and structure, previous works [3][4][5][6][7][8][9][10][11] focused mainly on shells that were in complete contact with a fluid. So, the influence of a free surface on the vibration and sound radiation of the cylindrical shell tended to be neglected. Obviously, as far as the structure located close to the water surface or afloat on the water surface is concerned, it is unreasonable to consider the fluid region around the cylindrical shell unbounded. Cylindrical shells partially coupled with the fluid (the shell centerline is parallel to the free surface) are frequently encountered in ocean engineering, such as ships or submarines floating on the sea. e coupling dynamic behavior, in particular the free vibration behavior, has attracted a lot of attention from researchers. In order to study the vibration of a finite cylindrical shell in a partially liquid-filled state, Amabili [10] applied appropriate boundary conditions on the free surface. Afterward, using the above method, Amabili [11] further studied the vibration of a cylindrical shell in a partially immersed state. For the purpose of solving the problem about the vibration of a partially liquid-filled and submerged finite cylindrical shell, Ergin [12] combined the boundary integral method and the image method to study it for the first time. Lakis et al. [13,14] made the element division in the circumferential direction of the finite cylindrical shell and found that the energy method is very suitable for the analysis of the vibration and sloshing of the finite cylindrical shell in a partially liquid-filled state. Li et al. [15] adopted the wave propagation method and Fourier transform technique to study the vibroacoustic characteristics of a semisubmerged infinite cylindrical shell. Guo et al. [16] investigated the free and forced vibration of a semisubmerged finite cylindrical shell by applying the Rayleigh-Ritz method. e sound radiation from structures partially coupled with fluid has received growing attention. Wu et al. [17,18] studied the vibroacoustic characteristics of a fluid-filled cylindrical shell semisubmerged in a coaxial flow with the Fourier transform technique, and the effects of parameters of the cylindrical shell on the vibration were discussed. Zhou and Joseph [19] employed the finite element method (FEM) with the boundary element method (BEM) to analyze the vibration and sound radiation from an underwater ribbed cylindrical shell. Brunner et al. [20,21] proposed a fast multipole BEM coupled with the FEM to investigate the vibroacoustic behavior of a partly immersed finite cylindrical shell and a ship model. Jouaillec and Jacquart [22] adopted the FEM and boundary integral method to study the sound radiation from a point-excited cylindrical shell partially immersed in the fluid. Salaün [23] applied the Fourier series expansion method to study the far-field sound radiation of a semisubmerged cylindrical shell at relatively high driving frequencies.
Generally speaking, there is a lack of an accurate and efficient method that can be used not only in the vibration response study of semisubmerged finite cylindrical shells but also in the far-field acoustic radiation analysis of ones, to the authors' knowledge. us, this analytical method is used to deal with such problems in this paper. It is necessary to split the problem into two parts before applying the analytical method to calculation. In this way, the accuracy can be guaranteed, and the calculation efficiency can be improved. First of all, the vibrational response of the finite cylindrical shell must be obtained: it is a good idea to use the wave propagation method to deduce the fluid-structure coupling vibration equations, considering that the fluid is mainly distributed in the axial interval of the finite cylindrical shell. Secondly, the sound radiation of the finite cylindrical shell must be obtained. In this step, it would be well if there was a semi-infinite cylindrical acoustical baffle on each side of the finite cylindrical shell, combining the Fourier transform with the stationary phase method， the far-field sound pressure of the finite cylindrical shell could be further gained. In order to make the far-field acoustic radiation easier to get across, the circumferential directivity of far-field sound pressure is discussed.

e Motion Equations of the Cylindrical Shell.
ere is a simply supported finite thin circular cylindrical shell in Figure 1, whose length, thickness, radius, Young's modulus, Poisson's ratio, and density are, respectively, L, h, R, E, μ, and ρ. Meanwhile, this cylindrical shell model submerges in such a fluid, whose density and sound velocity are, respectively, ρ f and c f . e hypothesis of this paper is a semi-infinite fluid domain. at is to say, there is a free surface that divides the whole space into two parts: the upper part is a vacuum and the lower part is the fluid domain. Since the cylindrical coordinate system (r, x, φ) is very convenient to solve and calculate in this paper, this coordinate system is adopted, whose origin is point O. Furthermore, the analytical solution for the vibration of the cylindrical shell can be obtained by using this coordinate system. Besides, at the point (R, x 0 , φ 0 ) of the cylindrical shell, there is a harmonic point force f r , whose amplitude and driving frequency are separately F 0 and f 0 . e motion state of the cylindrical shell can be represented by the Flügge shell equations [24]: e meaning of each character in the formula is as follows: U represents the axial displacement of the shell, V represents the tangential displacement, W represents the radial displacement, and [L] denotes the classical Flügge differential operator for the thin shell theory: e meaning of f p is the fluid pressure on the cylindrical shell, which can be defined as where p is the fluid pressure on the outer surface of the cylindrical shell.
Because the motion of the cylindrical shell is a kind of harmonic vibration, the axial displacement, the tangential displacement, the radial displacement, and the fluid pressure can be separately written as follows: e meaning of each character in the aforementioned formulas is as follows: U mn represents the axial displacement amplitude, V mn represents the tangential displacement amplitude, and W mn represents the radial displacement amplitude. F mn refers to the amplitudes of harmonic point force f r , and f mn refers to the amplitudes of the fluid pressure f p on a cylindrical shell. As for m and n, they separately represent the expansion coefficients in the axial and the circumferential directions. Besides, the circular frequency is expressed with ω, and the axial wavenumber is expressed with k m , and k m � mπ/L [25] and i � �� � − 1 √ . Substituting equation (4) into equation (1), the following equations can be further gained after the orthogonalization: Matrix [T] contains multiple elements, whose expressions are as follows: T 32 � T 23 , From the perspective of dimension, Ω belongs to a nondimensional frequency，which is only related to the ring frequency of the cylindrical shell, and Ω � ω ����������� � ρR 2 (1 − μ 2 )/E. It is not difficult to obtain the vibration equation about radial displacement by solving equation (5).
where I mn � |T|/(T 11 T 22 − T 12 T 21 ). e expression for f r is as follows: e meaning of each character in the aforementioned formula is as follows: F 0 is the maximum value of the harmonic point force f r , δ( ) is the delta function, and (x 0, φ 0 ) is the position of f r .
Substituting equation (8) into equation (4), F mn can be further gained after the orthogonalization:

e Motion Equations of the Fluid.
Combined with the incompressibility of the fluid and velocity potential function ϕ(r, x, φ, t), a function meeting the fluid Laplace equation, the characteristics of the fluid field can be reflected.
Consider that the liquid at infinity is still; that is, At the free surface, both the gravity effect and the surface tension of the fluid could be neglected [26]. erefore, the fluid boundary condition at the free surface can be expressed as e separation of variables method can be employed to solve the Laplace equation [10], and the velocity potential function which meets all the conditions could be expressed as follows: where ϕ ma denotes the amplitude of the velocity potential function and K a ( ) is the ath-order modified Bessel function of the second kind. Besides, sound pressure should be equal to 0 for an arbitrary point on the free surface. us, the velocity potential would be equivalent to zero on the free surface. at is why we choose the sine series.

2.3.
e Velocity Continuity Condition. Since the radial velocity on the outer surface of the cylindrical shell should be equal to that of the fluid, By conducting orthogonal processing and integrating about equation (14), one can get where sin(aφ) is a weight function, and equation (15) is a typical Galerkin integral equation. Furthermore, the amplitude of the velocity potential function can be deduced from equation (15): where e load p which the fluid imposed on the cylindrical shell can be calculated by Bernoulli's equation: By substituting equations (13) and (17) into equation (18), the fluid pressure p is expressed as By applying orthogonal processing about equation (3), one can get us, the amplitude of the load on the shell's surface imposed by the fluid pressure can be obtained after solving equation (20).
Since there exist infinite series in equation (22), a truncation process is required to make the solution possible. e truncated numbers of a, m, and n are K, M, and N, where N represents the truncated numbers of N and B, and n � b. en, equation (22) can be rewritten as where For the free vibration analysis, there is no exciting force, and the above problem changes into the solution of eigenvalues.

Theoretical Analysis of the Far-Field Sound Radiation
In this section, the far-field sound radiation of the semisubmerged finite shell will be derived. In Figure 2, point P is a far-field observation point in the acoustic field. In the figure, the length of line segment OP is defined as R 0 , the angle formed by rotating counterclockwise from the vector OP to the positive half of the x-axis is defined as θ， and the vector OP is written as R → 0 . It is obvious that the coordinate system (R 0 , θ, φ) is a typical spherical coordinate system.
In the x-direction, the fluid is free and unbounded, which would bring about such a problem that the domain of the sound field is incompatible with that of the shell's surface velocity. In order to avoid this problem, we have processed the cylindrical shell properly, that is, setting up a semiinfinite rigid column barrier [26] at each end of the shell, respectively. erefore, the radial displacement w(x, φ) of the shell is rewritten as When dealing with the acoustic radiation problem, the fluid is considered as an acoustic medium, and it is compressible. us, the sound pressure P(r, x, φ) in the fluid field should satisfy the Helmholtz equation: where k f � ω/c f is the compression wavenumber, and c f is the sound speed in the fluid.
Since the free surface exists, the sound pressure release boundary condition on the free surface should be met: erefore, the sound pressure expression, which satisfies the sound pressure release boundary, can be expressed as In order to solve the sound-structural coupling model, the Fourier transform and Fourier inverse transform are introduced as follows: e meaning of each character in the aforementioned formulas is as follows: f(k x ) refers to the function of f(x), k x refers to the wavenumber i � �� � − 1 √ , and they are all in the wavenumber domain.
By substituting equation (27) into equation (25) and conducting the Fourier transform technique, equation (25) is further changed to en, the expression of p a (r, x) can be acquired by applying the separation of variables method: Shock and Vibration where A a (k x ) is the sound pressure amplitude in the wavenumber domain, H (1) a ( ) is the ath-order Hankel function of the first kind, and k r � Afterward, the sound pressure in the wavenumber domain can be expressed as Similarly, combined with the Fourier transform technique, the radial displacement can be converted into the wavenumber domain: Accordingly, the expression of the radial displacement is as follows: where At the interface of a cylindrical shell and external fluid, they can be regarded as continuous, so it is not difficult to gain the following equation: Substituting equations (31) and (33) into equation (34), the amplitude of sound pressure can be further gained after the orthogonalization: where λ na � π 0 sin(aφ)exp(inφ)dφ.
Based on the principle of the inverse Fourier transform, the expression of sound pressure can be obtained as follows: According to the theory of far-field acoustics, the Hankel function can be approximately written as [22] where x � k r r, and when r goes to infinity, x also goes to infinity. Substituting equation (38) into equation (37), the sound pressure can be further written as

Shock and Vibration
Converting equation (39) into the expression under the spherical coordinate system (R 0 , r, x), where the conversion relationship is r � R 0 sinθ and x � R 0 cosθ, the sound pressure could be rewritten as follows: e far-field sound pressure is examined in this study, and in the acoustic field [27] and electromagnetic field [28], it is often necessary to calculate the infinite integral, which could lead to a large amount of calculation and inconvenience. In order to solve this problem well, a common approach is to introduce the stationary phase method, which can transform the infinite integral problem into a linear operation one. erefore, the solution of equation (40) is relatively easier.
Two related functions are given here to explain the method-the stationary phase method: According to the aforementioned formulas, the infinite integral in equation (40) is further changed into en, the stationary phase method [27] is used to process equation (40), and the result is where k s refers to the stationary phase point. Actually, k s is the solution of equation (43): erefore, the stationary phase point k s � k f cosθ, and equation (43) is further changed into us, the expression of the far-field sound pressure in the spherical coordinate system is obtained as It should be mentioned that when the stationary phase point k s � k f cosθ is close to the axial wavenumber k m , the expression of w n (k f cos θ) is also convergent; the reason is as follows: To be clear, the calculation formula of the sound pressure level is SPL � 20log10(|p|/p 0 ), and the value of P 0 is 10 − 6 Pa.

Numerical Examples of the Shell Vibration
e relevant calculation parameters used in this paper are as follows: for the cylindrical shell, its elastic modulus E is 206 GPa, Poisson's ratio μ is 0.3, density ρ is 7850 kg/m 3 , length L is 1.284 m, radius R is 0.18 m, and thickness h is 0.003 m; for the harmonic point force, its amplitude F 0 is 1 N; for the fluid, its density ρ f is 1025 kg/m 3 , and sound velocity c f is 1500 m/s. In this paper, the complex elastic modulus E c is used instead of the elastic modulus E for calculation.
e relationship between them is as follows: where η is the damping coefficient of the structure, and its value is 0.01.

4.1.
e Convergence Analysis of the Radial Velocity. In Section 2, in order to calculate the infinite series in equation (23), we introduced the fixed-phase method, which is an approximate algorithm, so it is necessary to prove its convergence. e main proof process is as follows: if M, K, and N are equal, then the quadratic velocity V m of the cylindrical shell can be further calculated, and its expression is V m � (1/2s) s |v n | 2 ds, where the meaning of s is the surface area of the cylindrical shell. e calculation results of V m are Shock and Vibration 7 shown in Figure 3. e quadratic velocity level is defined as QVL � 10log10(V m /V 0 ) (unit: dB), where V 0 � 10 − 12 m 2 /s 2 . It is not hard to see in the figure that when N is close to 12, V m begins to converge, so it is reasonable to take N � 14 in this paper.

e Verification of the Present Method for Vibration
Analysis. In this section, we will continue to demonstrate the reliability of the present method. In finite element software MSC Patran/Nastran, a cylindrical shell model with the external fluid is established (see Figure 4), and the model is simplified. For example, in MSC Patran/Nastran, the cylindrical shell is regarded as the shell element, and the virtual added mass matrix [29] in the virtual mass method is used to simulate the influence of the fluid on the free vibration and forced vibration of the cylindrical shell. In addition, the model is divided into 40 equal parts in both axial and tangential directions, with a total of 1600 elements, all of which are quadrilateral.

Free Vibration.
In order to illustrate the calculation accuracy of the present method, the natural frequencies and mode shapes obtained by the present method are compared with those obtained by the FEM, respectively (see Table 1 and Figure 5), where f a and f b represent the natural frequencies obtained by the present method and the FEM, respectively. e relative error of the natural frequencies between both methods is defined as It is easy to see that the method is reliable and accurate from the comparison results of Figure 5 and Table 1. For the natural frequencies of the first ten orders, the relative errors are less than 1%, where mode number (i, j) indicates the axial mode number and circumferential mode number in Table 1, respectively.
In order to further demonstrate the accuracy and reliability of the present method, the natural frequencies calculated with the present method are compared with those in [11]. e material parameters in [11] are listed as follows: E � 206 GPa, μ � 0.3, ρ � 7680 kg/m 3 , and the dimensions of the shell are L � 0.664 m, R � 0.175 m, and thickness h � 0.001 m. e density of the fluid is ρ f � 1000 kg/m 3 . e relative error of the natural frequencies obtained by the two methods is defined as Err2 � |f pre -f Ref |/f Ref , where f pre refers to the natural frequencies obtained by the present method and f Ref represents the natural frequencies in [11].
In Table 2, the differences between results of the present method and [11] are relatively small (less than 3%), proving that the present method for the free vibration is accurate and reliable.

Forced Vibration.
By using the same model parameters as those in Table 1, the point excitation is exerted at (R, L/2, − π/2), and then the dynamic responses at (R, L/4, − π/2) and (R, L/4, − π/2) can be obtained. ey are evaluated by the radial displacement level, whose calculation formula is RDL � 20log10(|w|/w0) · (dB), where the value of w0 is 10 − 12 m. e comparison between the finite element simulation results about RDL and the theoretical calculation ones is shown in Figure 6.
As shown in Figure 6, it is obvious that this method is reliable, and its calculation results are in good agreement with the finite element ones. More importantly, the computational efficiency of the present method is much higher than that of the finite element method. e former is about 20 times faster than the latter.
It is noteworthy that this method can solve the problem of low frequency very well. When the frequency of simple harmonic point force is relatively high (more than 500 Hz), it is not so suitable. e reason for this phenomenon lies in the following: in software Nastran, the fluid is regarded as virtual mass and infinite in the axial direction of the cylindrical shell; however, when the problem is solved analytically, the fluid is limited in the axial direction of the cylindrical shell; besides, the calculation accuracy of the analytical method depends on the frequency of the harmonic point force.

Numerical Analysis of the Far-Field
Sound Radiation

e Verification of the Accuracy of the Stationary Phase
Method. It can be seen from the first section that the solution content of this paper is mainly vibration response and acoustic radiation. Since the vibration response has been calculated in the previous sections, the task of this section is to solve the sound radiation and compare this result with that of the half-space boundary element method (BEM) [26]. e excitation point is located at (R, L/2, π/2), and its maximum value F 0 is 1 N. When R 0 � 1000 m, θ � π/4, and φ � 0∼π, the sound pressure level calculated by the present method and that calculated by the half-space boundary element method are calculated, respectively, and the differences between them are compared (see Figure 7).
As shown in the figure, the results gained by the two methods are very consistent, and the calculation process of the present method (the stationary phase method) is actually a linear calculation process, so this method is not only reliable but also efficient. Under the same conditions, the computational speed of the present method is 40 times faster than that of the half-space BEM.
Besides, since the driven frequencies are relatively low, the directivity of far-field sound pressure in the circumferential direction is monotonous. Moreover, it can be seen that the maximum value of the sound pressure in the circumferential direction is at the angle of the excitation position (φ 0 � π/2) in Figure 7.

Analysis of the Circumferential Directivity of the Far-Field
Sound Radiation. In order to learn more about the relationship between the circumferential directivity and the excitation position, we discussed the circumferential directivity at different excitation position angles φ 0 . In particular, φ 0 is set to be π/2, π/4, 0, − π/4, and − π/2; when R 0 � 1000 m, θ � π/3, and φ � 0∼π, the circumferential 8 Shock and Vibration    Shock and Vibration directivity of the sound pressure is studied at a frequency of 350 Hz. It is found in Figure 8 that the maximum value of the sound pressure on the circumferential direction is located at the bottom, and it is independent of the excitation position. e physical explanation for the directivity is given below. e sound pressure mentioned in this paper is that at infinity, and the distance from any point of the cylindrical shell to the sound pressure point studied is the same, so the cylindrical shell can be replaced by a point source.
According to the image method [30,31], another equidistant (the distance between the point source and free surface) and opposite phase source is applied to force the sound pressure on the free surface to be zero. In fact, the above physical model is called a dipole model. Furthermore, it is not difficult to deduce the sound pressure of the dipole model [27]: where the meaning of A is the maximum value of the sound pressure of the point source, r is the distance between the point source and the sound pressure point studied, and D represents the distance between the source and the image source (D/R < 2). In fact, the exact value of D is unknown, but it does not affect our analysis of sound pressure directivity.
is is  because the driving frequency is relatively low, and the product of k f and D is less than π/2 (k f D < π/2); thus, when sinφsinθ takes a maximum value, the sound pressure also does. erefore, this is a good explanation for why the extreme value of sound pressure always appears at φ � π/2 and why the circumferential directivity diagram has good symmetry, although the velocity distribution is asymmetrical.
Besides, from equation (48), the sound pressure will also reach a maximum value obviously when θ � π/2. For the purpose of illustrating the viewpoint, we calculated the sound pressure at different angles θ when the excitation position angle φ 0 is equal to π/4. In particular, θ is set to be π/8, π/4, and π/2; when R 0 � 1000 m and φ � 0∼π, the circumferential directivity of the sound pressure is studied at the same driven frequency (350 Hz). Figure 9 presents the circumferential directivity of the sound pressure with different angles θ. From Figure 9, it is seen that the sound pressures increase with the increase of the angle θ. is further proves that the dipole model is suitable to explain the law of directivity.

Conclusion
An approximate analytical method is proposed to study the vibration and far-field acoustic radiation of a semisubmerged, finite cylindrical shell with low-frequency excitation. In order to improve the efficiency of the solving process, the solution to this problem can be divided into two parts. After comparing the results of the present method with those of the FEM and BEM, this method is very reliable and can be used to calculate far-field sound pressure with high efficiency. Besides, the results show the following sound pressure characteristics. e circumferential directivity diagram has good symmetry, although the velocity distribution is asymmetrical. Furthermore, the circumferential directivity of far-field sound pressure is independent of the excitation position, and the maximum value of the sound pressure always appears at the angle φ equal to π/2 or the angle θ equal to π/2 (directly under the structure). In addition, the sound pressures increase with the increase of the angle θ.

Data Availability
All the data included in this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.