Analysis of Secondary Sound Emission in an Acoustic Analogy with a Propagation Operator Containing Vortex Modes

Abstract

In this paper, we analyze the acoustic analogy method in relation to the sound radiation of a turbulent subsonic jet. This method of describing aerodynamic sound generation by turbulent flows is based on the use of a linear propagation operator with a random source on the right side. The main problem here is the choice of an effective way to separate the left side of the equation, which is responsible for the propagation of sound waves, and the right part, which is responsible for sound generation, so that the noise calculation result corresponds to experimental data and physical ideas about noise generation by turbulence. One of the unsolved problems of the approach, which is common in most acoustic analogies, is the problem of the so-called “shear noise” associated with the excitation of shear flow disturbances by sources and the additional contribution of these disturbances to sound radiation. It is still unclear whether the shear component of the noise is a reflection of real physical processes or is associated with the transformation of equations and inaccurate modeling of sources. Here, within the framework of the problem formulated above, we consider an acoustic analogy, in which the linearized Euler equations are used as the propagation operator. In this description, the propagation operator contains vortex modes, which leads to the appearance of a shear noise component that arises due to the pumping of vortex disturbances by the sources. When modeling sound sources, hypotheses about the quadrupole nature isotropy of sound sources, as well as the spatial uncorrelation of sound source production, are used. To validate the model, the measurement data of the sound emission of the jet using the azimuthal decomposition method are used. The comparison of the model and experiment indicates the absence of a shear component in the jet noise. This makes it possible to conclude that the idea of pumping linear vortex perturbations of the mean flow by nonlinear turbulent pulsations that is used in the considered acoustic analogy does not correspond to the real mechanism of noise generation by a turbulent jet. Possible causes of the discrepancy between the model and the data of acoustic measurements are analyzed. Possible ways of solving this problem, which make it possible to effectively separate the left side of the equation that is responsible for the propagation of sound waves and the right nonlinear part that is responsible for sound generation are considered.

Appendices

Appendix A

We show the isotropy of the tensor:

$${{Q}^{{ij}}} = \sum\limits_{n = 1}^5 {{{a}_{n}}D_{n}^{{ij}}} ,$$

(A1)

where amplitudes ${{a}_{n}}$ are mutually uncorrelated random variables with the same variances and the Cartesian components of tensor $D_{n}^{{ij}}$ have the form

$$\begin{gathered} D_{1}^{{ij}} = \frac{1}{2}\left( {\begin{array}{*{20}{c}} { - 1}&0&0 \\ 0&{ - 1}&0 \\ 0&0&2 \end{array}} \right),\,\,\,\,D_{2}^{{ij}} = \frac{{\sqrt 3 }}{2}\left( {\begin{array}{*{20}{c}} 0&0&1 \\ 0&0&0 \\ 1&0&0 \end{array}} \right), \\ D_{3}^{{ij}} = \frac{{\sqrt 3 }}{2}\left( {\begin{array}{*{20}{c}} 0&0&0 \\ 0&0&1 \\ 0&1&0 \end{array}} \right),\,\,\,\,D_{4}^{{ij}} = \frac{{\sqrt 3 }}{2}\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&{ - 1}&0 \\ 0&0&0 \end{array}} \right), \\ D_{5}^{{ij}} = \frac{{\sqrt 3 }}{2}\left( {\begin{array}{*{20}{c}} 0&1&0 \\ 1&0&0 \\ 0&0&0 \end{array}} \right). \\ \end{gathered} $$

(A2)

We consider a convolution of $q = {{n}^{i}}{{n}^{j}}{{Q}^{{ij}}}$, where ${{n}^{i}}$ is the unit normal vector with Cartesian components of ${{n}^{i}} = \left( {\sin \hat {\theta }\cos \hat {\varphi },\sin \hat {\theta }\sin \hat {\varphi },\cos \hat {\theta }} \right)$, where $\hat {\varphi },\hat {\theta }$ are the spherical angles. The convolution variance is

$$\left\langle {{{q}^{2}}} \right\rangle = \sum\limits_{n = 1}^5 {\left\langle {a_{n}^{2}} \right\rangle \,D_{n}^{{ij}}} D_{n}^{{rs}}{{n}^{i}}{{n}^{j}}{{n}^{k}}{{n}^{s}}.$$

(A3)

For each of the basis quadrupoles, we calculate the ${{n}^{i}}{{n}^{j}}{{D}^{{ij}}}$ convolution:

$$\begin{gathered} {{n}^{i}}{{n}^{j}}D_{1}^{{ij}} = \frac{1}{2}\left( {3{{{\cos }}^{2}}\hat {\theta } - 1} \right), \\ {{n}^{i}}{{n}^{j}}D_{2}^{{ij}} = \frac{{\sqrt 3 }}{2}\sin 2\hat {\theta }\cos \hat {\varphi }, \\ {{n}^{i}}{{n}^{j}}D_{3}^{{ij}} = \frac{{\sqrt 3 }}{2}\sin 2\hat {\theta }\sin \hat {\varphi }, \\ {{n}^{i}}{{n}^{j}}D_{4}^{{ij}} = \frac{{\sqrt 3 }}{2}{{\sin }^{2}}\hat {\theta }\cos 2\hat {\varphi }, \\ {{n}^{i}}{{n}^{j}}D_{5}^{{ij}} = \frac{{\sqrt 3 }}{2}{{\sin }^{2}}\hat {\theta }\sin 2\hat {\varphi }. \\ \end{gathered} $$

(A4)

Substituting (A4) into (A3), we obtain

$$\begin{gathered} \left\langle {{{q}^{2}}} \right\rangle = \left\langle {a_{n}^{2}} \right\rangle \left( {\frac{1}{4}{{{\left( {3{{{\cos }}^{2}}\hat {\theta } - 1} \right)}}^{2}}} \right. \\ \left. { + \,\,\frac{3}{4}{{{\sin }}^{2}}2\hat {\theta } + \frac{3}{4}{{{\sin }}^{4}}\hat {\theta }} \right) = \left\langle {a_{n}^{2}} \right\rangle . \\ \end{gathered} $$

(A5)

Thus, the convolution of tensor ${{Q}^{{ij}}}$ with a normal vector has a variance that does not depend on the direction of the normal, which means that this tensor is isotropic.

Appendix B

We find a solution to the equation

$$\begin{gathered} \frac{{{{{\bar {\omega }}}^{2}}}}{\rho }\frac{d}{{d\rho }}\left( {\frac{\rho }{{{{{\bar {\omega }}}^{2}}}}\frac{d}{{d\rho }}p} \right) \\ + \,\,\left( {\frac{{{{{\bar {\omega }}}^{2}}}}{{c_{0}^{2}}} - {{\alpha }^{2}} - \frac{{{{m}^{2}}}}{{{{\rho }^{2}}}}} \right)p = q\left( {\rho ,m,\alpha ,\omega } \right), \\ \end{gathered} $$

(B1)

where $q\left( {{\mathbf{r}},\omega } \right) = {{\bar {\omega }}^{2}}\nabla \left( {\frac{1}{{{{{\bar {\omega }}}^{2}}}}{\mathbf{f}}} \right)$, the Cartesian components are ${{f}^{i}} = \frac{{\partial {{T}^{{ij}}}}}{{\partial {{r}^{j}}}}$, and the quadrupole source has the form ${{T}^{{ij}}} = D_{n}^{{ij}}\delta \left( {{\mathbf{r}} - {{{\mathbf{r}}}_{s}}} \right)$.

In regions with a constant velocity, this equation reduces to the homogeneous Bessel equation, the solutions of which, with allowance for the boundary conditions at zero and infinity, have the form

$$\begin{gathered} {{p}_{{{\text{in}}}}}\left( {\rho ,m,\alpha ,\omega } \right) = A\,{{J}_{m}}\left( {i\beta \rho } \right),\,\,\,\,\rho < {{\rho }_{1}}, \\ {{p}_{{{\text{out}}}}}\left( {\rho ,m,\alpha ,\omega } \right) = B\,H_{m}^{{\left( 1 \right)}}\left( {i\gamma \rho } \right),\,\,\,\,\rho > {{\rho }_{2}}, \\ \end{gathered} $$

(B2)

where ${{J}_{m}}$ and $H_{m}^{{\left( 1 \right)}}$ are the Bessel and Hankel functions of order m, $\gamma = \sqrt {{{\alpha }^{2}} - {{k}^{2}}}$, $\beta = \sqrt {{{\alpha }^{2}} - {{{\left( {k - {{M}_{{{\text{jet}}}}}\alpha } \right)}}^{2}}}$, $k = {{{\omega \mathord{\left/ {\vphantom {\omega c}} \right. \kern-0em} c}}_{0}}$, and ${{M}_{{{\text{jet}}}}} = {{{{{{V}_{{{\text{jet}}}}}} \mathord{\left/ {\vphantom {{{{V}_{{{\text{jet}}}}}} c}} \right. \kern-0em} c}}_{0}}$.

In the mixing layer, the solution has discontinuities in pressure ${{\left[ p \right]}_{{\,{\text{s}}}}}$ and its derivative ${{\left[ {\partial p} \right]}_{{\,{\text{s}}}}}$ at $\rho = {{\rho }_{{\text{s}}}}$, which are determined by the type of a quadrupole source. We represent the solutions at ${{\rho }_{1}} < \rho < {{\rho }_{2}}$ in the form of $p = A{{p}_{1}} + {{p}_{2}}$, where $A$ is the amplitude of internal solution (B2). ${{p}_{1}}$ is found as a solution of Eq. (B1) with zero right side and boundary conditions at $\rho = {{\rho }_{1}}$:

$${{p}_{1}} = {{J}_{m}}\left( {i\beta {{\rho }_{1}}} \right),\,\,\,\,\frac{{d{{p}_{1}}}}{{d\rho }} = \frac{d}{{d\rho }}{{J}_{m}}\left( {i\beta {{\rho }_{1}}} \right).$$

(B3)

${{p}_{2}}$ in the region of ${{\rho }_{{\text{s}}}} < \rho < {{\rho }_{2}}$ is a solution of Eq. (B1) with a zero right side and boundary conditions at $\rho = {{\rho }_{{\text{s}}}}$:

$${{p}_{2}} = {{\left[ p \right]}_{{\text{s}}}},\,\,\,\,\,\frac{{d{{p}_{2}}}}{{d\rho }} = {{\left[ {\partial p} \right]}_{{\text{s}}}},$$

(B4)

and ${{p}_{2}} = 0$ is in the region of ${{\rho }_{1}} < \rho < {{\rho }_{{\text{s}}}}$.

It is easy to verify that the function of $p = A{{p}_{1}} + {{p}_{2}}$ matches smoothly with solution (B2) on the inner boundary of the layer at $\rho = {{\rho }_{1}}$ and has specified discontinuities at $\rho = {{\rho }_{{\text{s}}}}$. Unknown constants $A$ and $B$ are found from the continuity condition of the solution and its derivative at the outer boundary of the layer at $\rho = {{\rho }_{2}}$. From here, we find constant B, which determines the amplitude of the sound wave outside the jet:

$$B = {{\left. {\frac{{{{p}_{2}}\frac{{d{{p}_{1}}}}{{d\rho }} - {{p}_{1}}\frac{{d{{p}_{2}}}}{{d\rho }}}}{{H_{m}^{{\left( 1 \right)}}\left( {i\gamma \rho } \right)\frac{{d{{p}_{1}}}}{{d\rho }} - {{p}_{1}}\frac{d}{{d\rho }}H_{m}^{{\left( 1 \right)}}\left( {i\gamma \rho } \right)}}} \right|}_{{\rho = {{\rho }_{2}}}}}.$$

(B5)

In the limiting case of a thin mixing layer, we will use the constancy of pressure $p$ and radial component of the shear field ${{\eta }^{\rho }} = \frac{1}{{{{{\bar {\omega }}}^{2}}}}\frac{{dp}}{{d\rho }}$ in the mixing layer, except for the source location of $\rho = {{\rho }_{{\text{s}}}}$, where these variables have discontinuities. In this case, at the outer boundary of the mixing layer at $\rho = {{\rho }_{2}}$, we obtain

$$\begin{gathered} {{p}_{1}} = {{J}_{m}}\left( {i\beta {{\rho }_{1}}} \right),\,\,\,\,\frac{{d{{p}_{1}}}}{{d\rho }} = \frac{{\omega _{{}}^{2}}}{{\bar {\omega }_{{{\text{in}}}}^{2}}}i\beta J_{m}^{'}\left( {i\beta {{\rho }_{1}}} \right), \\ {{p}_{2}} = {{\left[ p \right]}_{{\,{\text{s}}}}},\,\,\,\,\frac{{d{{p}_{2}}}}{{d\rho }} = \frac{{\omega _{{}}^{2}}}{{\bar {\omega }_{{\text{s}}}^{2}}}{{\left[ {\partial p} \right]}_{{\,{\text{s}}}}}, \\ \end{gathered} $$

(B6)

where ${{\bar {\omega }}_{{{\text{in}}}}} = \omega - \alpha \,{{V}_{{{\text{jet}}}}}$, ${{\bar {\omega }}_{{\text{s}}}} = \omega - \alpha \,{{V}_{0}}\left( {{{\rho }_{{\text{s}}}}} \right)$, and prime means differentiation of functions with respect to the argument. Substituting these relations into (B5), we obtain:

$$\begin{gathered} B = i\frac{1}{{\beta {{\rho }_{{\text{s}}}}{{A}_{m}}}} \\ \times \,\,\left( { - \frac{{\bar {\omega }_{{{\text{in}}}}^{2}}}{{\bar {\omega }_{{\text{s}}}^{2}}}{{\rho }_{{\text{s}}}}{{{\left[ {\partial p} \right]}}_{{\,{\text{s}}}}}{{J}_{m}}\left( {i\beta {{\rho }_{{\text{s}}}}} \right) + {{{\left[ p \right]}}_{{\,{\text{s}}}}}i\beta {{\rho }_{\operatorname{s} }}J_{m}^{'}\left( {i\beta {{\rho }_{{\text{s}}}}} \right)} \right), \\ \end{gathered} $$

(B7)

where ${{A}_{m}} = \frac{{\bar {\omega }_{{{\text{in}}}}^{2}}}{{\omega _{{}}^{2}}}\frac{\gamma }{\beta }H{{_{m}^{{\left( 1 \right)}}}^{\prime }}\left( {i\gamma {{\rho }_{{\text{s}}}}} \right){{J}_{m}}\left( {i\beta {{\rho }_{{\text{s}}}}} \right) - H_{m}^{{\left( 1 \right)}}\left( {i\gamma {{\rho }_{{\text{s}}}}} \right)J_{m}^{'}\left( {i\beta {{\rho }_{{\text{s}}}}} \right)$.

In accordance with (B2), we write the sound radiation outside the jet for a point quadrupole source localized at the point: $\rho = {{\rho }_{{\text{s}}}}$, $\varphi = 0$, $z = 0$:

$$p\left( {\rho ,m,\alpha ,\omega } \right) = {{k}^{2}}{{F}_{{\left( n \right)\;m}}}\left( {\alpha ,\omega } \right)\frac{i}{2}H_{m}^{{\left( 1 \right)}}\left( {i\gamma \rho } \right),$$

(B8)

where the dimensionless directivity of the radiation of the nth basis quadrupole are $F_{{\left( n \right)\;m}}^{{}} = - i\frac{2}{{{{k}^{2}}}}B$ and the $B$ value is determined from (B5) in the general case and from (B7) in the limiting case of a thin mixing layer. For an arbitrary point $\rho = {{\rho }_{{\text{s}}}}$, $\varphi = {{\varphi }_{{\text{s}}}}$, $z = {{z}_{{\text{s}}}}$, with allowance for the homogeneity of the process in coordinates φ and z, the factor of $\exp \left( { - i\alpha {{z}_{{\text{s}}}} - im{{\varphi }_{{\text{s}}}}} \right)$ is added to expression (B6); i.e.,

$$\begin{gathered} p\left( {\rho ,m,\alpha ,\omega } \right) = {{k}^{2}}{{F}_{{\left( n \right)\;m}}}\left( {\alpha ,\omega } \right) \\ \times \,\,\frac{i}{2}H_{m}^{{\left( 1 \right)}}\left( {i\gamma \rho } \right)\exp \left( { - i\alpha {{z}_{{\text{s}}}} - im{{\varphi }_{{\text{s}}}}} \right). \\ \end{gathered} $$

(B9)

Using the expressions for ${{\left[ p \right]}_{{\,{\text{s}}}}}$ and ${{\left[ {\partial p} \right]}_{{\,{\text{s}}}}}$ obtained in Appendix B, for a thin mixing layer, we obtain

$$\begin{gathered} {{F}_{{\left( 1 \right)\;m}}} = \frac{1}{{4{{\pi }^{2}}{{k}^{2}}\beta {{\rho }_{{\text{s}}}}{{A}_{m}}}}\frac{{\bar {\omega }_{{{\text{in}}}}^{2}}}{{\bar {\omega }_{{\text{s}}}^{2}}}\left( {3{{\alpha }^{2}} - \frac{{\bar {\omega }_{s}^{2}}}{{{{c}^{2}}}}} \right){{J}_{m}}\left( {i\beta {{\rho }_{{\text{s}}}}} \right), \\ {{F}_{{\left( 2 \right)\;m}}} = \frac{{\sqrt 3 }}{2}\frac{1}{{{{\pi }^{2}}{{k}^{2}}\beta \rho _{{\text{s}}}^{3}{{A}_{m}}}}i\alpha {{\rho }_{{\text{s}}}}\,\left( { - \frac{{\bar {\omega }_{{{\text{in}}}}^{2}}}{{\bar {\omega }_{{\text{s}}}^{2}}}\alpha {{\rho }_{{\text{s}}}}\frac{{{{\Omega }_{0}}}}{{\bar {\omega }_{{\text{s}}}^{{}}}}{{J}_{m}}\left( {i\beta {{\rho }_{{\text{s}}}}} \right) + i\beta {{\rho }_{{\text{s}}}}J_{m}^{'}\left( {i\beta {{\rho }_{{\text{s}}}}} \right)} \right), \\ {{F}_{{\left( 3 \right)\;m}}} = \frac{{\sqrt 3 }}{2}\frac{1}{{{{\pi }^{2}}{{k}^{2}}\beta \rho _{{\text{s}}}^{3}{{A}_{m}}}}m\alpha {{\rho }_{{\text{s}}}}\frac{{\bar {\omega }_{{{\text{in}}}}^{2}}}{{\bar {\omega }_{{\text{s}}}^{2}}}{{J}_{m}}\left( {i\beta {{\rho }_{{\text{s}}}}} \right), \\ {{F}_{{\left( 4 \right)\;m}}} = \frac{{\sqrt 3 }}{2}\frac{1}{{{{\pi }^{2}}{{k}^{2}}\beta \rho _{{\text{s}}}^{3}{{A}_{m}}}}\,\left( {\frac{{\bar {\omega }_{{{\text{in}}}}^{2}}}{{\bar {\omega }_{{\text{s}}}^{2}}}{\kern 1pt} \left( {\frac{{\rho _{{\text{s}}}^{2}}}{2}\left( {\frac{{\bar {\omega }_{{\text{s}}}^{2}}}{{{{c}^{2}}}}\, - \,{{\alpha }^{2}}} \right)\, - \,{{m}^{2}}} \right){\kern 1pt} {{J}_{m}}{\kern 1pt} \left( {i\beta {{\rho }_{{\text{s}}}}} \right)\, + \,i\beta {{\rho }_{{\text{s}}}}J_{m}^{'}{\kern 1pt} \left( {i\beta {\kern 1pt} {{\rho }_{{\text{s}}}}} \right)} \right), \\ {{F}_{{\left( 5 \right)m}}} = \frac{{\sqrt 3 }}{2}\frac{1}{{{{\pi }^{2}}{{k}^{2}}\beta \rho _{{\text{s}}}^{3}{{A}_{m}}}}\,im{\kern 1pt} \left( { - \frac{{\bar {\omega }_{{{\text{in}}}}^{2}}}{{\bar {\omega }_{{\text{s}}}^{2}}}\left( {1\, + \,\alpha \rho _{{\text{s}}}^{{}}\frac{{{{\Omega }_{0}}}}{{{{{\bar {\omega }}}_{{\text{s}}}}}}} \right){\kern 1pt} {{J}_{m}}{\kern 1pt} \left( {i\beta {{\rho }_{{\text{s}}}}} \right)\, + \,i\beta {{\rho }_{{\text{s}}}}J_{m}^{'}{\kern 1pt} \left( {i\beta {\kern 1pt} {{\rho }_{\operatorname{s} }}} \right)} \right), \\ \end{gathered} $$

(B10)

where ${{\Omega }_{0}}$ is the vorticity of the average flow at the point of $\rho = {{\rho }_{{\text{s}}}}$ and the index in brackets corresponds to the number of the basic quadrupole.

Appendix C

We find discontinuities in pressure and pressure derivative at the source location for solutions of the equation

$$\begin{gathered} {{{\bar {\omega }}}^{2}}\frac{1}{\rho }\frac{d}{{d\rho }}\left( {\frac{\rho }{{{{{\bar {\omega }}}^{2}}}}\frac{d}{{d\rho }}p} \right) \\ + \,\,\left( {\frac{{{{{\bar {\omega }}}^{2}}}}{{c_{0}^{2}}} - {{\alpha }^{2}} - \frac{{{{m}^{2}}}}{{{{\rho }^{2}}}}} \right)p = q\left( {\rho ,m,\alpha ,\omega } \right), \\ \end{gathered} $$

(C1)

where $q\left( {{\mathbf{r}},\omega } \right) = {{\bar {\omega }}^{2}}\nabla \left( {\frac{1}{{{{{\bar {\omega }}}^{2}}}}{\mathbf{f}}} \right)$, ${{f}^{i}} = \frac{{\partial {{T}^{{ij}}}}}{{\partial {{r}^{j}}}}$ are the Cartesian components, and the quadrupole source has the form of ${{T}^{{ij}}} = D_{{}}^{{ij}}\delta \left( {{\mathbf{r}} - {{{\mathbf{r}}}_{s}}} \right)$, where $\rho = {{\rho }_{s}}$, $\varphi = 0$, $z = 0$.

We write the field in cylindrical coordinates:

$$\begin{gathered} q\left( {{\mathbf{r}},\omega } \right) = \frac{{{{\partial }^{2}}{{T}^{{\rho \rho }}}}}{{\partial {{\rho }^{2}}}} + \frac{1}{\rho }\frac{\partial }{{\partial \rho }}{{T}^{{\rho \rho }}} + \frac{1}{{{{\rho }^{2}}}}\frac{{{{\partial }^{2}}}}{{\partial {{\varphi }^{2}}}}{{\rho }^{2}}{{T}^{{\varphi \varphi }}} \\ + \,\,\frac{{{{\partial }^{2}}{{T}^{{zz}}}}}{{\partial {{z}^{2}}}} + \frac{1}{\rho }\frac{\partial }{{\partial \rho }}\left( {{{T}^{{\rho \rho }}} - {{\rho }^{2}}{{T}^{{\varphi \varphi }}}} \right) \\ + \,\,2\frac{1}{\rho }\frac{\partial }{{\partial \varphi }}\left( {\frac{\partial }{{\partial \rho }} + \frac{1}{\rho }} \right)\rho {{T}^{{\rho \varphi }}} + 2\frac{\partial }{{\partial z}}\left( {\frac{\partial }{{\partial \rho }} + \frac{1}{\rho }} \right){{T}^{{z\rho }}} \\ + \,\,2\frac{1}{\rho }\frac{\partial }{{\partial \varphi }}\frac{\partial }{{\partial z}}\rho {{T}^{{\varphi z}}} + {{{\bar {\omega }}}^{2}}\frac{d}{{d\rho }}\left( {\frac{1}{{{{{\bar {\omega }}}^{2}}}}} \right) \\ \times \,\,\left( {\frac{{\partial {{T}^{{\rho \rho }}}}}{{\partial \rho }} + \frac{{\partial {{T}^{{\rho \varphi }}}}}{{\partial \varphi }} + \frac{{\partial {{T}^{{\rho z}}}}}{{\partial z}} - \rho {{T}^{{\varphi \varphi }}} + \frac{1}{\rho }{{T}^{{\rho \rho }}}} \right). \\ \end{gathered} $$

(C2)

We use the Fourier transform in the φ and z coordinates. We transform (B2) to the form in which all functions of ρ are under the sign of the derivative with respect to ρ:

$$\begin{matrix} q\left( \rho ,m,\alpha ,\omega \right)=\frac{1}{4{{\pi }^{2}}}\,\,\left( \frac{{{d}^{2}}}{d{{\rho }^{2}}}\left( {{D}^{\rho \rho }}\Phi \right)+\frac{d}{d\rho }\left( \frac{1}{\rho }{{D}^{\rho \rho }}\Phi \right)+\frac{1}{{{\rho }^{2}}}{{D}^{\rho \rho }}\Phi \right) \\ +\,\,\frac{1}{4{{\pi }^{2}}}\left( \frac{d}{d\rho }+\frac{1}{\rho } \right)\left( \frac{1}{\rho }\left( {{D}^{\rho \rho }}-{{\rho }^{2}}{{D}^{\varphi \varphi }} \right)\Phi \right)-\,\,\frac{1}{4{{\pi }^{2}}}\left( {{m}^{2}}{{D}^{\varphi \varphi }}+{{\alpha }^{2}}{{D}^{zz}} \right)\Phi +\frac{1}{4{{\pi }^{2}}}2im\left( \frac{d}{d\rho }+\frac{2}{\rho } \right) \\ \times \,\,\left( {{D}^{\rho \varphi }}\Phi \right)+\frac{1}{4{{\pi }^{2}}}2i\alpha \left( \frac{d}{d\rho }+\frac{1}{\rho } \right)\left( {{D}^{\rho z}}\Phi \right)-\,\,\frac{1}{4{{\pi }^{2}}}2m\alpha {{D}^{\varphi z}}\Phi +\frac{1}{4{{\pi }^{2}}}\frac{d}{d\rho }\left( {{{\bar{\omega }}}^{2}}\frac{d}{d\rho }\left( \frac{1}{{{{\bar{\omega }}}^{2}}} \right){{D}^{\rho \rho }}\Phi \right) \\ -\,\,\frac{1}{4{{\pi }^{2}}}\frac{d}{d\rho }\left( {{{\bar{\omega }}}^{2}}\frac{d}{d\rho }\left( \frac{1}{{{{\bar{\omega }}}^{2}}} \right) \right){{D}^{\rho \rho }}\Phi +\frac{1}{4{{\pi }^{2}}}{{{\bar{\omega }}}^{2}}\frac{d}{d\rho }\times \,\,\left( \frac{1}{{{{\bar{\omega }}}^{2}}} \right)\left( im{{D}^{\rho \varphi }}\Phi +i\alpha {{D}^{\rho z}}\Phi +\frac{1}{\rho }\left( {{D}^{\rho \rho }}-{{\rho }^{2}}{{D}^{\varphi \varphi }} \right)\Phi \right), \\ \end{matrix}$$

(C3)

where $\Phi \left( \rho \right) = \frac{1}{\rho }\delta \left( {\rho - {{\rho }_{s}}} \right)$. We pass to the Cartesian components of tensor ${{D}^{{ij}}}$, which do not depend on ρ and, therefore$,$ can be taken out from under the derivative. After transformations, we obtain

$$\begin{gathered} q\left( {\rho ,m,\alpha ,\omega } \right) = \frac{1}{{4{{\pi }^{2}}}}D_{{}}^{{xx}}\,\left( {{{{\bar {\omega }}}^{2}}\frac{1}{\rho }\frac{d}{{d\rho }}\left( {\frac{1}{{{{{\bar {\omega }}}^{2}}}}\rho \frac{{d\Phi }}{{d\rho }}} \right) - \frac{{{{m}^{2}}}}{{{{\rho }^{2}}}}\Phi - {{\alpha }^{2}}\Phi } \right) \\ + \,\,\frac{1}{{4{{\pi }^{2}}}}\left( {\left( {D_{{}}^{{xx}} - D_{{}}^{{yy}}} \right)\frac{{{{m}^{2}}}}{{{{\rho }^{2}}}}\Phi + \left( {D_{{}}^{{xx}} - D_{{}}^{{zz}}} \right)} \right.\left. {{{\alpha }^{2}}\Phi + 2im{{D}^{{xy}}}\frac{1}{{{{\rho }^{2}}}}\Phi - 2\alpha mD_{{}}^{{yz}}\frac{1}{\rho }\Phi } \right) + \frac{1}{{4{{\pi }^{2}}}}\frac{{{{{\bar {\omega }}}^{2}}}}{\rho } \\ \times \,\,\frac{d}{{d\rho }}\left( {\frac{\rho }{{{{{\bar {\omega }}}^{2}}}}} \right.\left( {2im{{D}^{{xy}}}\frac{1}{\rho }\Phi + 2i\alpha {{D}^{{xz}}}\Phi } \right.\left. {\left. { + \,\,\frac{1}{\rho }\left( {{{D}^{{xx}}} - {{D}^{{yy}}}} \right)\Phi } \right)} \right) - \frac{1}{{4{{\pi }^{2}}}}{{{\bar {\omega }}}^{2}}\frac{d}{{d\rho }}\left( {\frac{1}{{{{{\bar {\omega }}}^{2}}}}} \right)\left( {imD_{{}}^{{xy}}\frac{1}{\rho }\Phi + i\alpha D_{{}}^{{xz}}\Phi } \right). \\ \end{gathered} $$

(C4)

To remove the second derivative with respect to the radial coordinate at the source, we make a change to $\hat {p} = p - {{D}^{{xx}}}\Phi$. In this case, the new and old variables differ only in the layer of $\rho = {{\rho }_{s}}$. Equation (B1) is still valid for new variable. However, the right side of (B4) is changed, namely, the first term is replaced by term $\frac{1}{\pi }\frac{{{{{\bar {\omega }}}^{2}}}}{{{{c}^{2}}}}{{D}^{{xx}}}\Phi$.Thus, for the new variable, the source on the right side of Eq. (B1) contains terms proportional only to the delta function and its first derivative:

$$\begin{gathered} \hat {q}\left( {\rho ,m,\alpha ,\omega } \right) = - \frac{1}{{4{{\pi }^{2}}}}\frac{{{{{\bar {\omega }}}^{2}}}}{{{{c}^{2}}}}{{D}^{{xx}}}\Phi + \frac{1}{{4{{\pi }^{2}}}} \\ \times \,\,\left( {\left( {D_{{}}^{{xx}} - D_{{}}^{{yy}}} \right)\frac{{{{m}^{2}}}}{{{{\rho }^{2}}}}\Phi + \left( {D_{{}}^{{xx}} - D_{{}}^{{zz}}} \right){{\alpha }^{2}}\Phi } \right. \\ \left. { + \,\,2im{{D}^{{xy}}}\frac{1}{{{{\rho }^{2}}}}\Phi - 2\alpha mD_{{}}^{{yz}}\frac{1}{\rho }\Phi } \right) \\ + \frac{1}{{4{{\pi }^{2}}}}\frac{{{{{\bar {\omega }}}^{2}}}}{\rho }\frac{d}{{d\rho }}\left( {\frac{\rho }{{{{{\bar {\omega }}}^{2}}}}} \right.\left( {2im{{D}^{{xy}}}\frac{1}{\rho }\Phi + 2i\alpha {{D}^{{xz}}}\Phi } \right. \\ \left. {\left. { + \,\,\frac{1}{\rho }\left( {{{D}^{{xx}}} - {{D}^{{yy}}}} \right)\Phi } \right)} \right) - \frac{1}{{4{{\pi }^{2}}}}{{{\bar {\omega }}}^{2}}\frac{d}{{d\rho }} \\ \times \,\,\left( {\frac{1}{{{{{\bar {\omega }}}^{2}}}}} \right)\left( {imD_{{}}^{{xy}}\frac{1}{\rho }\Phi + i\alpha D_{{}}^{{xz}}\Phi } \right). \\ \end{gathered} $$

(C5)

We determine the relation between discontinuities in the solution of Eq. (B1) and the functional form of the sources. If the pressure has discontinuity ${{\left[ p \right]}_{{\,{\text{s}}}}}$ at the point of $\rho = {{\rho }_{{\text{s}}}}$, then the first and second derivatives of the pressure have terms containing the delta function and its derivative, $\frac{{dp}}{{d\rho }} = {{\left[ p \right]}_{{\,{\text{s}}}}}\delta \left( {\rho - {{\rho }_{{\text{s}}}}} \right)$ and $\frac{{{{d}^{2}}p}}{{d{{\rho }^{2}}}} = {{\left[ p \right]}_{{\,{\text{s}}}}}\frac{d}{{d\rho }}\delta \left( {\rho - {{\rho }_{{\text{s}}}}} \right)$. If the derivative of pressure has discontinuity ${{\left[ {\partial p} \right]}_{{\,{\text{s}}}}}$, then the second derivative of pressure has a term containing the derivative of the delta function, $\frac{{{{d}^{2}}p}}{{d{{\rho }^{2}}}} = {{\left[ {\partial p} \right]}_{{\,{\text{s}}}}}\delta \left( {\rho - {{\rho }_{{\text{s}}}}} \right)$. Substituting these expressions into the left side of Eq. (B1), we obtain a relation between the terms of the form of the delta function and its derivative at the source and discontinuities of pressure and its derivative in the solution:

$$\begin{gathered} q\left( {\rho ,m,\alpha ,\omega } \right) = {{\left[ p \right]}_{{\,{\text{s}}}}}\delta \left( {\rho - {{\rho }_{{\text{s}}}}} \right) \\ + \,\,{{\left[ {\partial p} \right]}_{{\,{\text{s}}}}}\frac{{{{{\bar {\omega }}}^{2}}}}{\rho }\frac{d}{{d\rho }}\left( {\frac{\rho }{{{{{\bar {\omega }}}^{2}}}}\delta \left( {\rho - {{\rho }_{{\text{s}}}}} \right)} \right). \\ \end{gathered} $$

(C6)

Comparing (B5) and (B6), we find discontinuities of pressure and derivative in the solution of Eq. (B1) for a point quadrupole:

$${{\left[ p \right]}_{s}} = \frac{1}{{4{{\pi }^{2}}\rho _{s}^{2}}}\left( {{{D}^{{xx}}} - {{D}^{{yy}}} + {{D}^{{xy}}}2im + {{D}^{{xz}}}2i\alpha {{\rho }_{s}}} \right),$$

(C7)

$$\begin{gathered} {{\left[ {\partial p} \right]}_{{\text{s}}}} = \frac{1}{{4{{\pi }^{2}}\rho _{{\text{s}}}^{3}}}\left( { - {{D}^{{xx}}}\frac{{\bar {\omega }_{{\text{s}}}^{2}}}{{{{c}^{2}}}}\rho _{{\text{s}}}^{2} + \left( {D_{{}}^{{xx}} - D_{{}}^{{yy}}} \right){{m}^{2}}} \right. \\ \left. { + \,\,\left( {D_{{}}^{{xx}} - D_{{}}^{{zz}}} \right){{\alpha }^{{{{2}^{{^{{^{{}}}}}}}}}}\rho _{{\text{s}}}^{2} + {{D}^{{xy}}}2im - D_{{}}^{{yz}}2m\alpha {{\rho }_{{\text{s}}}}} \right) \\ - \,\,\frac{1}{{4{{\pi }^{2}}\rho _{{\text{s}}}^{2}}}\bar {\omega }_{{\text{s}}}^{2}\frac{d}{{d\rho }}\left( {\frac{1}{{\bar {\omega }_{{\text{s}}}^{2}}}} \right)\left( {D_{{}}^{{xy}}im + D_{{}}^{{xz}}i\alpha {{\rho }_{{\text{s}}}}} \right). \\ \end{gathered} $$

(C8)