Geodesic
Exact solution of the fundamental equation of acoustics for a pressure wave developing in two directions
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 79 (2022), pp. 5-13
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The authors proceed from the hyperbolic equation for acoustic pressure. Using the integral Fourier transform along the axial coordinate, an equation in partial derivatives for the kernel of this transformation is found. This equation contains only one spatial coordinate and time. Applying the integral Laplace transform in time to the last equation, we obtain an ordinary differential equation with respect to the radial coordinate for the corresponding image. It turns out that the solution of the last equation is the well-known Macdonald function. For this function, it was possible to find the original image according to Laplace. All this made it possible to write an integral formula for the pressure in a sound wave. If the function of the initial pressure distribution along the pipe axis is taken in the form of a Gaussian impulse, then the integrals included in the representation of the desired solution are taken explicitly. As a result, we obtain an explicit compact formula for the acoustic pressure distribution in the axisymmetric case. It is convenient to use this formula to analyze the distribution of sound disturbances both along the pipe axis and in the radial direction. Therefore, the results are presented as isobars in the $(z, r)$ plane corresponding to different times.
Keywords: wave equation for pressure, non-periodic sound wave, operational calculus
Mots-clés : exact solution. 
@article{VTGU_2022_79_a0,
     author = {V. I. Borodin and A. V. Lun-Fu and M. A. Bubenchikov and A. M. Bubenchikov and D. V. Mamontov},
     title = {Exact solution of the fundamental equation of acoustics for a pressure wave developing in two directions},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {5--13},
     year = {2022},
     number = {79},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2022_79_a0/}
}
TY  - JOUR
AU  - V. I. Borodin
AU  - A. V. Lun-Fu
AU  - M. A. Bubenchikov
AU  - A. M. Bubenchikov
AU  - D. V. Mamontov
TI  - Exact solution of the fundamental equation of acoustics for a pressure wave developing in two directions
JO  - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY  - 2022
SP  - 5
EP  - 13
IS  - 79
UR  - http://geodesic.mathdoc.fr/item/VTGU_2022_79_a0/
LA  - ru
ID  - VTGU_2022_79_a0
ER  - 
%0 Journal Article
%A V. I. Borodin
%A A. V. Lun-Fu
%A M. A. Bubenchikov
%A A. M. Bubenchikov
%A D. V. Mamontov
%T Exact solution of the fundamental equation of acoustics for a pressure wave developing in two directions
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2022
%P 5-13
%N 79
%U http://geodesic.mathdoc.fr/item/VTGU_2022_79_a0/
%G ru
%F VTGU_2022_79_a0
V. I. Borodin; A. V. Lun-Fu; M. A. Bubenchikov; A. M. Bubenchikov; D. V. Mamontov. Exact solution of the fundamental equation of acoustics for a pressure wave developing in two directions. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 79 (2022), pp. 5-13. http://geodesic.mathdoc.fr/item/VTGU_2022_79_a0/

[1] Demir A., Qinar Yanaz O., “Propagation of sound in an infinite two-part duct carrying mean flow inserted axially into a larger infinite duct with wall impedance discontinuity”, Journal of Applied Mathematics and Mechanics, 89 (2009), 454–465 | DOI

[2] Peake N., Abrahams I.D., “Sound radiation from a semi-infinite lined duct”, Wave Motion, 92 (2019), 102407 | DOI

[3] Tiryakioglu B., “Mode Matching Analysis of Sound Waves in an Infinite Pipe with Perforated Screen”, Acoustical Physics, 66 (2021), 580–586 | DOI

[4] Gabard G., Astley R.J., “Theoretical model for sound radiation from annular jet pipes: Far- and near-field solutions”, Journal of Fluid Mechanics, 549 (2006), 315–341

[5] Veitch B., Peake N., “Acoustic propagation and scattering in the exhaust flow from coaxial cylinders”, Journal of Fluid Mechanics, 613 (2008), 275–307

[6] Lun-Fu A.V., Bubenchikov M.A., Bubenchikov A.M., Mamontov D.V., “Passage of Monochro matic Sound Through a Gas Pipeline Wall”, Acoustics Australia, 50 (2021), 119–126 | DOI

[7] Gradshtein I.S., Ryzhik N.M., Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1963, 1100 pp.

[8] Ditkin V.A., Prudnikov A.P., Spravochnik po operatsionnomu ischisleniyu, Vysshaya shkola, M., 1965, 466 pp.