Finite Difference Analysis of 2-Dimensional Acoustic Wave with a Signal Function Opiyo Richard Otieno 1 , Alfred Manyonge 1 , Owino Maurice 2 & Ochieng Daniel 1 richardopiyo08@gmail.com 1 ,wmanyonge@gmail.com 1 & mauricearaka@yahoo.com 2 1 Dept of Pure & Applied Mathematics Maseno University(Kenya) 2 Dept of Mathematics & Computer Sciences University of Kabianga (Kenya) December 9, 2015 Abstract This paper describes progress on a two dimensional numerical simulation of acoustic wave propagation that has been developed to visualize the propagation of acoustic wave fronts and to provide time-domain signal. In this exercise, we have simulated propaga- tion of sound in such a medium using both explicit and Crank Nicolson finite difference schemes, we have also tested for stability of the developed schemes us- ing Vonn Newmann and Matrix stability analysis to- gether with its associated code in matlab. The stabil- ity analyses of the developed schemes revealed that Explicit scheme was conditionally stable while the Hybrid one (Crank Nicolson Scheme) was uncondi- tionally stable, for all values of courant number r. The rate of convergence of the algorithms depend on the truncation error introduced when approximating the partial derivatives, the Crank-Nicolson method converged at the rate of (k 2 + h 2 ), which is a faster rate of convergence than either the explicit method, or the implicit method. Keywords: Acoustic wave, Finite difference approximation, Signal function, Crank Nicolson, Vonn Newman, Matrix stability analysis. 1 Introduction When determining the acoustic properties of an en- vironment, we are actually interested in the propa- gation of sound, given the properties and location of a sound source. Propagation of light or sound wave is of long standing interest in several branches of ba- sic and applied physics, from old disciplines such as x-ray diffraction in crystallography, to the modern science of photonic crystals. Many problems in nat- ural environment so involve wave propagation in pe- riodic media. For example, nearly periodic sand bars are frequently found in shallow seas outside the surf zone; their presence changes the wave climate near the coast. The technology of remote-sensing, either by underwater sound or by radio waves from a satel- lite, depends on our understanding of scattering by the wavy sea surface. Finite difference method is a key tool in numerical analysis and the motivation to study and learn this method is the fact that in Fluid dynamics, thermody- namics, solid mechanics etc. a large number of differ- ential equations are found. And to solve all of them analytically is very difficult and at times impossible. As a result Finite Difference Methods provide suf- ficiently satisfactory accurate numerical solutions to such equations. Finite-difference modelling of wave propagation in heterogeneous media is a useful tech- nique in a number of disciplines, including seismology and ocean acoustics. Sound is a longitudinal wave that is, waves of alternating pressure deviations from equilibrium causing local regions of compression and rarefaction as a result of vibrating objects. Sound is a wave which can be described as a disturbance that travels through a medium, transporting energy from one location to another location. Many researchers have developed numerical interpre- tations of the wave equation suited to acoustics and seismic propagation. Hugh and Pat [13], developed second order finite difference scheme for modelling the acoustic wave equation in Matlab but their major limitation was, insufficient consideration of boundary conditions. Alford, Kelly and Boore [2], proposed that acoustic wave equation for homogeneous media can be approximated in rectangular co-ordinate sys- tem by the second and fourth order central difference. Although, one-way wave equation method in inho- mogeneous media has been extensively studied in the literature, few detailed studies have been made on the implementation of source term and free bound- ary conditions. For this reason, Xie and Wu [29] inte- grated free surface boundary condition and the source term for one way elastic waves for decomposition of plane wave. Charara and Tarantola [7], in their publication con- 1