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Additional resources for A COMPILL4TIBN OF THE MATHEMATICS LEADING TO THE DOUBLET LKI TICE METHOD
We make an educated guess and look for a solution to equation (101) which is spherically Symmetric. We temporarily designate this solution with an overbar (not to be confused with the steady state solution in Section II). 4) = 4(r, t) (102) A spherically symmetric flow is the same as a pulsating source (or sink) with radial streamlines. Physically, such a flow injects mass into the field. The spherically symmetric form of the acoustic equation is 0 [ a'248r € 7ar r -a] (103) In order to maintain the undisturbed far field condition, we seek an elementary solution which dies off as r - co.
This elementary solution may then be used in computing complex solutions by 1. 113). 39 The Elementay Solution to the Acoustic Potential Equation the principle of superposition. More importantly, the elementary solution may be adapted to solve the aerodynamic potential equation (42). We make an educated guess and look for a solution to equation (101) which is spherically Symmetric. We temporarily designate this solution with an overbar (not to be confused with the steady state solution in Section II).
This is given in equation (109). We introduce the linear Laplace operator V2 and restate the acoustic potential equation (86) as equation (99). ) a22 (99) In equation (99), we have not specified 4 in any particular frame of reference. We now introduce the use of spherical coordinates where r is the radial measure, 0 is the measure of longitude and I is the measure of latitude. 0 = 4(r, 0, Xt) (100) The acoustic equation (99) takes the following form' in spherical coordinates. n1 r2 -sinX [sinl• + 2(si•) 1 (101) We are looking for an elementary solution to the acoustic potential equation.