Acoustic conductance of an anisotropic spherical shell submerged in a liquid 1. Wave model of contact interaction between a solid hollow sphere and a liquid. Degenerate solutions

With the example of a thick-walled spherical shell made of an anisotropic material, the decrease in the vibration intensity across the thickness of shells related to a change in the amplitude of wave propagation is considered. The motion rate of the outer surface of the shell, contacting a liquid medium, as a function of shell thickness and material properties in the circumferential and radial directions is investigated by using a model of radial vibrations caused by a harmonic source located on the inner surface of the shell. The model takes into account the interaction of surface of the shell with the surrounding medium. By using the operational method, with the use of the Laplace transformation, a general solution to the wave problem for radial vibrations is obtained. The amplitude solution is found in an expansion form with the use of the theory of residues. The degenerate solution of the transcendental equation for eigenfrequencies, which, in the general case, is a combination of Bessel functions, is investigated in the cases where the radial and circumferential elastic moduli differ considerably.

Appendix

Numbering of the formulas with Latin letters in the Appendix is made with reference to the formulas in the basic text.

The modified Bessel function I _μ in Eq. (8) is obtained from the ordinary Bessel function J _μ by multiplying its argument by $i = \sqrt {{ - 1}}$ and the function itself by an exponential quantity depending on the number μ:

$$ {I_\mu}\left( {\frac{p}{c}r} \right) = {e^{ - \frac{{\mu \pi i}}{2}}}{J_\mu}\left( {ir\frac{p}{c}} \right). $$

(8a)

The McDonald function is derived from the Hankel function of the first kind of an imaginary argument by multiplying it by a constant quantity. For noninteger values of m, we have the following relationship for the functions introduced [14]:

$$ {K_\mu}\left( {\frac{p}{c}r} \right) = \frac{{\pi i}}{2}{e^{{{{\mu \pi i}} \left/ {2} \right.}}}H_\mu^{(1)}\left( {ir\frac{p}{c}} \right) = \frac{\pi }{2} \cdot \frac{{{I_{ - \mu }}\left( {\frac{p}{c}r} \right) - {I_\mu}\left( {\frac{p}{c}r} \right)}}{{\sin \mu \pi }}. $$

(8b)

The formulas of differentiation of the functions of complex argument considered are

$$ {I'_\mu }(z) = {I_{\mu - 1}}(z) - \frac{\mu }{z}{I_\mu }(z),\quad {K'_\mu }(z) = - {K_{\mu - 1}}(z) - \frac{\mu }{z}{K_\mu }(z). $$

(8c)

The calculation of the Wronskian with these functions can be presented in the form

$$ W\left\{ {{K_\mu }(z),{I_\mu }(z)} \right\} = {I_\mu }(z){K_{\mu + 1}}(z) + {I_{\mu + 1}}(z){K_\mu }(z) = \frac{1}{z}. $$

(8d)

For an isotropic material of the shell, μ = 3/2 and

$$ {I_{{{1} \left/ {2} \right.}}}(z) = \sqrt {{\frac{2}{{\pi z}}}} \sinh z,\quad {I_{{{{ - 1}} \left/ {2} \right.}}}(z) = \sqrt {{\frac{2}{{\pi z}}}} \cosh z, $$

$$ {I_{{{3} \left/ {2} \right.}}}(z) = {I_{{{{ - 1}} \left/ {2} \right.}}}(z) - \frac{{{I_{{{1} \left/ {2} \right.}}}(z)}}{z} = \sqrt {{\frac{2}{{\pi z}}}} \left( {\cosh z - \frac{{\sinh z}}{z}} \right), $$

(8e)

$$ {K_{{{1} \left/ {2} \right.}}}(z) = {K_{{{{ - 1}} \left/ {2} \right.}}}(z) = \sqrt {{\frac{\pi }{{2z}}}} {e^{ - z}},\quad {K_{{{3} \left/ {2} \right.}}}(z) = {K_{{{{ - 1}} \left/ {2} \right.}}}(z) + \frac{{{K_{{{1} \left/ {2} \right.}}}(z)}}{z} = \sqrt {{\frac{\pi }{{2z}}}} {e^{ - z}}\left( {1 + \frac{1}{z}} \right). $$

The solution of Eq. (7) at the half-integer parameter μ = 3/2 is expressed in elementary functions:

$$ U(r) = {C_1}\left( {\frac{{\cosh \lambda r}}{{\lambda r}} - \frac{{\sinh \lambda r}}{{{\lambda^2}{r^2}}}} \right) + {C_2}\left( {\frac{{1 + \lambda r}}{{{\lambda^2}{r^2}}}} \right){e^{ - \lambda r}},\left( {{\lambda^2} = {{{{p^2}}} \left/ {{{c^2}}} \right.}} \right). $$

(8f)

At μ → ∞, we have the asymptotic formulas

$$ {I_\mu}\left( {\mu z} \right)\sim \frac{1}{{\sqrt {{2\pi \mu }} }} \cdot \frac{{{e^{\mu \eta }}}}{{{{\left( {1 + {z^2}} \right)}^{{{1} \left/ {4} \right.}}}}},\quad {K_\mu}\left( {\mu z} \right)\sim \sqrt {{\frac{\pi }{{2\mu }}}} \cdot \frac{{{e^{ - \mu \eta }}}}{{{{\left( {1 + {z^2}} \right)}^{{{1} \left/ {4} \right.}}}}}, $$

$$ {I'_\mu}\left( {\mu z} \right)\sim \frac{1}{{\sqrt {{2\pi \mu }} }} \cdot \frac{{{{\left( {1 + {z^2}} \right)}^{{{1} \left/ {4} \right.}}}}}{z}{e^{\mu \eta }},\quad {K'_\mu}\left( {\mu z} \right)\sim - \sqrt {{\frac{\pi }{{2\mu }}}} \cdot \frac{{{{\left( {1 + {z^2}} \right)}^{{{1} \left/ {4} \right.}}}}}{z}{e^{ - \mu \eta }}, $$

(8′)

$$ \eta = \sqrt {{1 + {z^2}}} + \ln \left[ {{{z} \left/ {{\left( {1 + \sqrt {{1 + {z^2}}} } \right)}} \right.}} \right]. $$

The nontrivial solution of the homogeneous equations derived from Eq. (19) by inserting r = R ₁, R ₀ into its left-hand side, with account of formulas (8d) and (8e) from the Appendix, is reduced to solution of the transcendental equation

$$ \tan \left( {{z_0} - {z_1}} \right) = \frac{{\left( {{z_0} - {z_1}} \right)\left( {{\nu^0} + {z_0}{z_1}} \right)}}{{{\nu^0}\left( {1 + {z_0}{z_1}} \right) + {{{z_0^2z_1^2}} \left/ {{{\nu^0} - z_0^2 - z_1^2}} \right.}}}, $$

(19a)

where z ₀ = pR ₀/c, z ₁ = pR ₁/c, and ν⁰ = -χ. The parameter p in the steady-state mode of free vibrations of an isotropic shell is a real number, and the roots p _k obtained from Eq. (19а) represent the spectrum of eigenfrequencies.

By inserting λ = iξ with a real ξ in expression (42), the hyperbolic functions are transformed into trigonometric functions of real variable, and its roots are found from the transcendental equation

$$ \tan \left[ {\xi \left( {1 - \varepsilon } \right)} \right] = \xi . $$

(42a)

Using the asymptotic behavior of the tangent function, we introduce into calculation the distance δ _k on the ξ axis from a root ξ _k at k = 0, 1, 2… to the abscissa of the nearest vertical straight line on the right which determines the asymptote of the tangent (see Fig. 1).

At ε ≠ 0, there exists such δ _k that the equality (1-ε)ξ _k  + δ _k  = (k + 1/2)π at k = 0, 1, 2,… is valid for the roots ξ _k . We should note that, at ε = 0, the tangent and the straight line y = ξ do not intersect in the interval 0 ≤ ξ ≤ π/2 between the origin of coordinates and the first asymptote, except when ξ = 0. Therefore, the discrete numbering of roots at ε = 0 is performed for the index k = 1, 2,…, but this case is analyzed in [8]. In the case considered here, in view of the replacement accepted above, we have the equality tan[((1-ε)ξ _k ] = tan [(k + 1/2)π - δ _k ] ≡ c tan δ _k . The parameter δ _k , after the approximate replacement cotan δ _k  ≅ 1/δ _k - δ _k /3, is found from Eq. (42а), which acquires the form

$$ \delta_k^2 - \frac{{3\left( {k + {{1} \left/ {2} \right.}} \right)\pi }}{{2 + \varepsilon }}{\delta_k} + \frac{{3\left( {1 - \varepsilon } \right)}}{{2 + \varepsilon }} = 0. $$

(42b)

The solution of this equation yields two values of the parameter δ _k , from which, at high values of the number k, we can take the least one. The quantity δ _k  = 2(1-ε)/[π(2 k +1)] decreases at k → ∞, which means that the linear function y = ξ intersects the tangent function on the asymptote of tangent curve infinitely distant from the origin of coordinates. With growing k, the second root increases and has to be excluded from consideration, i.e., the above-mentioned approximation of cotangent function at great values of roots δ _k of Eq. (42b) is incorrect. A refinement of the procedure for deriving the values of δ _k corresponding to initial numbers k from equation (42б) is achieved by the iteration method. In this case, the solution of the linear part of this equation is refined by the additional quadratic term δ _k ² obtained from the previous iteration. Thus, the roots of Eq. (42) derived with account of a second iteration from relation (42b) are described by the formula

$$ {\xi_k} = \frac{{\pi \left( {2k + 1} \right)}}{{2\left( {1 - \varepsilon } \right)}} - \frac{2}{{\pi \left( {2k + 1} \right)}} - \frac{{8\left( {2 + \varepsilon } \right)\left( {1 - \varepsilon } \right)}}{{3{{\left[ {\pi \left( {2k + 1} \right)} \right]}^3}}},\quad k = 0,1,2, \ldots . $$

(42c)

The transform of the speed of points of a hollow anisotropic shell in vacuum upon vibration disturbance of its inner surface is found by calculating the constants from homogeneous equation (19) (with a zero right-hand side) and inhomogeneous equation (21). Inserting the expressions obtained for constants into solution (8) multiplied by p, we come to the transform of speed of the shell on an arbitrary radius r. The ratio between the transform and the greatest speed on the internal radius r = R ₁ can be presented as

$$ \tilde{V}\left( {r,\lambda } \right) = \frac{{V(p)}}{{A\omega }} = \sqrt {{\frac{{{R_1}}}{r}}} \cdot \frac{{{\Phi_\mu}\left( {r,\lambda } \right)}}{{{\Phi_\mu}\left( {{R_1},\lambda } \right)}}, $$

(33a)

where

$$ {\Phi_\mu}\left( {r,\lambda } \right) = \chi \left[ {{I_\mu}\left( \lambda \right){K_\mu}\left( {\lambda \frac{r}{{{R_0}}}} \right) - {I_\mu}\left( {\lambda \frac{r}{{{R_0}}}} \right){K_\mu}\left( \lambda \right)} \right] + \lambda \left[ {{I_{\mu - 1}}\left( \lambda \right){K_\mu}\left( {\lambda \frac{r}{{{R_0}}}} \right) + {I_\mu}\left( {\lambda \frac{r}{{{R_0}}}} \right){K_{\mu - 1}}\left( \lambda \right)} \right]. $$

On the outer surface of the spherical shell, at r = R ₀, from Eq. (33а), with account of Wronskian (8d), we obtain the following transform of the relative speed:

$$ \tilde{V}\left( {{R_0},\lambda } \right) = \frac{{\lambda \sqrt {\varepsilon } \left( {{{{{R_0}}} \left/ {c} \right.}} \right)}}{{\left( {{\lambda^2} + {\varpi^2}} \right){\Phi_\mu}\left( {{R_1},N} \right)}}. $$

(33b)

After the calculation of constants from homogeneous equation (20) (with a zero right-hand side) and inhomogeneous equation (21), we can write the following expression for the transform of the speed of points along the shell radius:

$$ V\left( {r,\lambda } \right) = \frac{{A\omega \lambda {R_0}}}{{c\left( {{\lambda^2} + {\varpi^2}} \right)}}\sqrt {{\frac{{{R_1}}}{r}}} \cdot \frac{{{K_\mu}\left( {\lambda \frac{r}{{{R_0}}}} \right){I_\mu}\left( \lambda \right) - {I_\mu}\left( {\lambda \frac{r}{{{R_0}}}} \right){K_\mu}\left( \lambda \right)}}{{{K_\mu}\left( {\varepsilon \lambda } \right){I_\mu}\left( \lambda \right) - {K_\mu}\left( \lambda \right){I_\mu}\left( {\varepsilon \lambda } \right)}}. $$

(33c)

As seen from Eq. (33c), vibrations are absent at r = R ₀. This solution variant corresponds to rigid fixation of the outer surface of the shell.