Release:
2015, Vol. 1. №2(2)About the authors:
Aleksandr I. Filippov, Dr. Sci. (Tech.), Professor, Department of General and Theoretical Physics, Sterlitamak branch of the Bashkir State University; eLibrary AuthorID, filippovai@rambler.ruAbstract:
In the second part of the article the first coefficient of the asymptotic expansion has been found, which provides the main part of the amendment specifying the geometry of the wave front of the pressure field in a three-layered anisotropic permeable medium. It is stated that the equation for the first coefficient of the asymptotic expansion in the central region, as well as for the zero one, contains the values of the normal derivative of the pressure field in a related field at its adjoining border (trace of derivative). It is shown in order to obtain a unique solution of the problem for the first coefficient of expansion the condition at x = 0 should be weakened and replaced by the condition for the integral of the unknown function (nonlocal integral one). The problems of this kind are not traditional for mathematical physics that is why this problem is non-classical one. To find the nonlocal condition the formulation of the problem for the remainder after the first expansion coefficient has been carried out. The seeking condition is determined by the requirements of the trivial solution of the problem for the remainder integrally averaged in the range of the central layer. In the space of a sine-Fourier transform the exact solution of the original problem has been found. The correctness of the developed method that allows constructing approximate analytical solution of a wide range of physical problems is confirmed by the comparison of the asymptotic solutions with the expansion coefficients of the exact solution of the parameterized problem in the Maclaurin series according to the formal parameter.Keywords:
References:
1. Filippov, A. I., Akhmetova, O. V. Predstavlenie l'tratsionno-volnovyh polej v sloistoj anizotropnoj srede v vide ploskoj volny (chast' I) // Bullettin of the Tyumen State University. Physical and mathematical modeling. Oil, Gas , Energy. № 1(1). Т. 1. 2015. Pp. 65-76.
2. Kulik, V. M. Ploskaja volna deformatsii v izotropnom sloe vjazkouprugogo materiala// Prikladnaja mekhanika i tekhnicheskaja fi zika. 2006. T. 47. № 3 (277). Pp. 104-111.
3. Shostak, A. S., Pershanin, D. A. Zondirovanie neodnorodnykh material'nykh sred ploskimi volnami gorizontal'noj i vertikal'noj poljarizatsii // Izvestija vysshikh uchebnykh zavedenij. Fizika. 2012. T. 55. № 8. Pp. 128-129.
4. Capenko, N. E. Padenie ploskoj elektromagnitnoj volny na neodnorodnyj sloj // Gornyj informatsionno-analiticheskij bjulleten' (nauchno-tekhnicheskij zhurnal). 2010. № 6.
Pp. 153-158.
5. Akhmetova, O. V., Filippov, A. I., Filippov, I. M. Kvazistatsionarnye polja davlenija pri linejnoj fi l'tratsii v neodnorodnom anizotropnom plaste v asimptoticheskom priblizhenii // Izvestija Rossijskoj akademii nauk. Mekhanika zhidkosti i gaza. 2012. № 3. Pp. 89-100.
6. Filippov, A. I., Akhmetova, O. V., Zamanova, G. F. Asimptoticheskie predstavlenija uprugikh volnovykh polej v pronitsaemykh plastakh // Akusticheskij zhurnal. 2013. T. 59. № 5. Pp. 548-558.
7. Filippov, A. I., Akhmetova, O. V., Zamanova, G. F., Kovalskij, A. A. Ustanovivshiesja dvumernye fi l'tratsionno-volnovye polja pri zadannykh garmonicheskikh vozmuschenijakh davlenija na granitse// Elektronnyj nauchnyj zhurnal Neftegazovoe delo. 2014. № 5.
Pp. 181-208.
8. Filippov, A. I., Akhmetova, O. V. Odnomernye monokhromaticheskie ploskie fi l'tratsionnye volny // Inzhenerno-fi zicheskij zhurnal. 2015. T. 88. № 2. Pp. 285-290.
9. Vasil'eva, A. B., Butuzov, V. F. Asimptoticheskie metody v teorii singuljarnykh vozmuschenij // M.: Vysshaja shkola, 1990. 208 p.
10. Bejtmen, G., Jerdeji, A. Tablitsy integral'nykh preobrazovanij. T. 1. M.: Nauka, 1969. 344 p.
