A COMPILL4TIBN OF THE MATHEMATICS LEADING TO THE DOUBLET LKI

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0@ Again, this formula is used if we know p (xy,z, t) and we need the corresponding function 0 (x, y, z, t). Now X appears twice in the argument list, once in the x and once in the t place holders of the pressure function p (x, y, z, t). This has the influence that as we integrate downstream from x = -*o, we evaluate the pressure at time prior to t. Later in this report, we will refer to this earlier time in the integrand as retarded time. Therefore, if we know the pressure for all time prior to time t, we can evaluate the above integral expression for the potential j (x, y, z, t).

P (X,) dX (, (67) Now we work to cast this expression in the original coordinates. From equations (63) and (65) we have the inverse relations X=4 (68) t =4-'- (69) U We make the change from P to , in equation (67). , =X, S(. dX (70) Now *(x,t) = I(4,t1) -- p(x,x-Ut) (71) Therefore, by substituting equation (71) into equation (70), we obtain (x, t);k[ = By replacing result = + Utd A (72) (Po -p) /po and rearranging terms, we obtain the desired 27 The inearized Ptm Iquadion X P-x tx,t) 0 - ---U (73) In this formula, the reader is reminded that X is the dummy variable of integration representing integration in the x direction.

It is possible to develop a pressure difference if two source sheets are placed in parallel. This can cause numerical problems if the two sheets are brought close together. This is not to say this has not been done. On the contrary, there are many examples where this is exactly what is done. However, for our linear analysis, this results in a waste of computational resources. Instead, we can formulate the limiting condition as two source sheets with opposing 61 Tbe Some ]Doublet strengths are brought close together.

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