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**Sample text**

0]F[t-] (X,= (155) This is the formula for the potential at coordinates (x, y, z) due to a single point source at coordinates (ý, il, t). The boundary condition for the flow over a thin wing was given in equation (79). w =h + U(156) where h (x, v, t) describes the time dependent deformation of a thin wing in the (x, y) plane. The obvious question remains; how do we use equation (155) to solve for the flow over a wing? The aiswer is not simple and is the subject of the remainder of this text. We still need to formulate the source doublet in Section X and then we formulate the pressute doublet in Section MI.

The elementary solution to the acoustic potential equation is a stationary point source with a spafial decay of (1 /r). We used a modified form of this solution (109) to obtain an elementary solution to the aerodynamic potential equation. Then a complication arose. We discovered that a simple translation of the stationary source in the x cbreztion does not satisfy the acoustic potential equation. This mathematical complication is the result of compressibility (also referred to as the Doppler effect).

We substitute equation (82) into equation (79) to obtain I'I W-= F + -[X](83) "J=Ok=O J/=Ok3 33 Ueaized Houndy Condtions ftom Fim Piciples The complex modulus of w is denoted as iv such that w = iieit. k'-yI) 1 j (84) The main reason for developing equation (84) is to provide an example of the boundary condition formulation which may be used in the doublet lattice method. Input for the example doublet lattice program in Appendix A is in this form. 34 SECTION V Transformation to the Acoustic Potential Equation In Section Zl, we started with the Euler equations and derived the aerodynamic potential equation.