# A STUDY OF WAVE EQUATION BY SEPARATION OF VARIABLES

• V P Sangale Head, Dept. of Mathematics, R. B. Attal College, Georai, Dist. Beed( MS) India.

### Abstract

Objective: In this paper we focus our study on wave equations. Westudying the solution of wave function by using separation of variables technique. The functions of several variablesand having worked through the concept of a partial derivative.

Materials and Methods:We first formulate the wave function u(x,t) where x is length of string. Solving the equation???? ????, ???? = ???? ???? ????(????) in twovariables byusing the methods of Partial Differential Equation. We get the following equation where k is constant.

Results: We are going to check the possible for the constant k in the above equation. First we consider k = 0 then we get

u x, t = (px+r) (at+b) where a,b, p and r, were constants.

Secondly we consider k> 0 then we get

x x F x Ae Be w -w ( ) = + whereA and B are constants and w = k .

was used isidentified by the subscript in u (x, t) n and n l , and arbitrary constants are C and D.

Conclusion:The solutions given in the first two cases are dull solutions.The solution given in the last case really does satisfy the wave equation. We

can find a particular solution function for varying values of time, t,

Keywords: Wave equation, Partial derivatives, Exponential functions, Frequency, Poisson equation

### References

1. L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, AMS (1998).
2. R. K. Gupta, Partial Differential Equations
3. J.F. Annet, Superconductivity, Superfluids and Condensates, Oxford Mas- ter Series in Condensed Matter Physics, Oxford University Press, Reprint (2018)
4. J.M. Ball and R.D. James, Fine mixtures as minimizers of energy , Archive for Rational Mechanics and Analysis, 100, 15-52 (2016).
5. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Volume I, Reprint Elsevier (Singapore, 2013).
6. Mathew J Hancock, The 1-D Wave Equation 18.303 Linear Partial Differential Equation (2006)
7. D. Naik, C. Peterson, A. White, A. Berglund, and P. Kwiat, Phys. Rev. Lett. 84, 4733 (2000).
8. N. Gisin et al, “Quantum cryptography,” Rev. Mod. Phys., Vol. 74, No. 1, January 2002.
9. T. Jennewin, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 84, 4729 (2000).
10. D.R. S.Talbot and J.R.Willis, Bounds for the effective constitutive relationof a nonlinear composite, Proc. R. Soc. Lond. (2004), 460, 2705-2723.
11. N.B. Firoozye and R.V. Khon, Geometric Parameters and the Relaxation for Multiwell Energies, Microstructure and Phase
Transition, the IMA volumes in mathematics and applications, 54, 85-110 (1993).
12. D.Y.Gao and G.Strang, Geometric Nonlinearity: Potential Energy, Com- plementary Energy and the Gap Function, Quarterly Journal of Applied Mathematics, 47, 487-504 (1989a).
Statistics