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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.
V (95) +_ (96) Of course, there is no change for derivatives in the lateral directions. 36 Transfoimutioci to the Acoustc P (aia •yy=Oyo° =:0o0 Equadicc (97) (98) By substituting equations (94) through (98) into equation (85), we obtain equation (86). The interpretation of the two approaches is different. The first approach is trivial. We simply set the velocity to zero. The second approach is a linear transformation. The aerodynamic potential equation (85) is identically the same as the acoustic potential equation in a uniformly moving frame.
Here, we normally assume the thickness effects are not time dependent. h (x, y, t) = hm (x, y, t) ± ht (x, y) (80) STherefore, when analyzing the dynamic response of a wing, we superimpose the dynamic response due to hm on a separate time invariant solution using h,. This has an important influence in our choice of the doublet sheet to model the aerodynamics for the time dependent flow over the wing midplane and our choice of source panels to model the aerodynamics for the time invariant component of flow the wing thickness envelope.
A COMPILL4TIBN OF THE MATHEMATICS LEADING TO THE DOUBLET LKI TICE METHOD