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Extra info for A COMPILL4TIBN OF THE MATHEMATICS LEADING TO THE DOUBLET LKI TICE METHOD

Example text

In time, the mathematical representation of the wing slices out a plane region as it moves forward through the air at velocity U. We will treat this planar region as the trailing wake. (We could have a non-planar wake. ) It is assummed that there is no jump in the non-zero velocity component, w = •z across the wake. Therefore, 4, is symmetric with respect to z in its first approximation (eg. Fourier Series) with respect to z . With 4, symmetric, it follows that ý, •x and 0, are all antisymmetric with respect to z in the first approximation.

Id 1 (143) The definition for R was given in equation (136). Finally, we use equation (143) in equation (132) to obtain 4(X0 ,y0 ,z 0 ,t) =Li()eo(144) From equation (135), we have ail expression for -cwhen E) = 0. We make the substitution in equation (144). UY" R. A PkL3K2 a (145) where R is defined in equation (136). This is the fundamental solution for the potential which arises due to a source moving along the x axis with constant velocity of -UV. In the next section, we transform the coordinates to a moving frame.

6 (x0 YOyZ 0 t) = [][(123) In equation (123), we see the effect of the retarded time. Even though all six sources are pulsed simultaneously, the effect at the point (xo, y0, zo) is picked up (ri/a) later. In other words, six simultaneous pulses are transmitted to the point (x,, yo, z,) with a different delay. Now, instead of pulsing simultaneously, we pulse the sources in a sequence, starting with the sonrce at 4, and ending with the source at 46' Each source is pulsed with a time delay of t = r, relative to t = 0.