# Benjamin's Othogonal Curvilinear Coordinate System to analyze Gas Velocity

I have been trying to understand a problem given in a paper for a couple of months but cannot figure out the rationale behind the change of variables of a function. This problem is outlined below.

In his 1959 paper Benjamin uses an orthogonal curvilinear coordinate system to define the boundary-layer of a gas profile. This system is given below:

$$\xi_1=x-\imath\epsilon e^{-\alpha y}e^{\imath \alpha x} \\ \xi_2=y-\epsilon e^{-\alpha y}e^{\imath \alpha x}$$

This coordinate system is used to determine the streamfunction of a counter-current gas liquid film flow.

The streamfunction is defined as follows: $$\psi = \psi_0(\xi_2) + \epsilon\psi_1(\xi_2)e^{\imath\alpha\xi_1}+O(\epsilon^2)$$

The $$\psi_0(\xi_2)$$ corresponds to a streamfunction of a base state with no waviness(flat boundary). This base state is given by a velocity function which is only a function of $$y$$ and does not depend on $$x$$: $$l^4+L^2U_0^4-(L+1)^2U_0^2+2L(L+1)U_0-L^2=0$$ where $$U_0(y)=\psi_0'(y)$$ and $$l=\kappa y[1-\exp(-y/A)]$$ which is also a function of $$y$$.

My problem arises by a confusion regarding how the base state can be a function exclusively of $$\xi_2$$ when there should also be a component of $$\xi_1$$ that is caused by the change of variable from $$y$$. I am trying to find out how to calculate $$\psi_0(\xi_2)$$ from $$\psi_0(y)$$.

Thanks in advance. Any help in understanding what I am missing is appreciated.

P.S. The Jacobian matrix of the transformation is also provided in the paper but I am not sure if it is helpful. I am providing it below in case it is needed: $$\Gamma=\Bigl(\begin{bmatrix}1 & -\imath\epsilon \alpha e^{-\alpha \xi_2} e^{\imath\alpha\xi_1}\\ \imath\epsilon \alpha e^{-\alpha \xi_2} e^{\imath\alpha\xi_1} & 1\end{bmatrix} +O(\epsilon^2)\Bigl)$$

The paper I am atepmting to understand is the following: doi:10.1017/S002211201000618X