[Malohm]

Hi,

I have a problem understanding a paper from Fung. Maybe someone is ambitious to deal with this old school of aerodynamics

The problem is to describe the unsteady aerodynamic loads on an oscillating wall due to a boundary layer. The approach is to drastically reduce the problem to two layers of potential flow. Above the wall there is a subsonic layer with a supersonic flow field on top. I will add an extract of the respective paper: Fung1963.pdf
(Link to my public drop box. The full paper can be downloaded from AIAA.

As you can see the respective equations of motion are (1) and (2), i.e. the linearized potential flow equations for subsonic and supersonic flow. Equation (3) to (7) are the boundary conditions. Not pretty sure when eq. (8) to (11) come into play, but the describe that to motion of the wall is like a standing wave, while it is shown later on that the pressure might act like a traveling wave.

As the author states it is "easy to see" that the solution of (1) and (2) is Eq. (12) and (13).

This is were I got stuck. Here is my first approach (by the way, is it possible to use latex in this forum?):

The oscillation must be harmonic. So I chose phi to be phi_max times exp(i omega t). Inserting this in (1) and (2) I get a complex EOM. By applying a Fourier transformation to the resulting equations I obtain equations (14) and (15). I append the solution approach as pdf file.

The next questions in my mind are: Can I directly state the the Fourier constant alpha has the physical meaning of the wave number? How do I use the Fourier constants to get to the general solution (12) and (13)?

Any help and discussion is very appreciated!