We have reduced the conservation equations, under several assumptions to a linearized potential equation which is mathematically much simpler to solve. As in most engineering analyses the physics of the problem are preserved by specifying the proper boundary conditions. If a solution of the equation is to make sense, it must satisfy the boundary conditions that we specify.

a : angle of attack
n : unit vector normal to the (upper) surface at a given point
q : slope of the surface at a given point is tan q
 

The linearized potential eqn. is a 2^nd order partial differential equation: 2 boundary conditions are required.

1. The perturbation must die away as we move far from the source (from the airfoil).


 
2. There can be no flow through the surface of the airfoil. The normal component of the velocity is zero at the surface.

at the surface
Another way of saying this is that

or,

is the equation of the airfoil surface.

Thus, 
Approximately, since is very small.
 

Planar Wing Approximation

For thin wings and airfoils with small camber, at small angles of attack, it does not make much difference whether the boundary surface is taken at the airfoil surface or at the y=0 plane. [When you calculate you will find that the values of  are small and you will wonder why you went through the trouble of calculating it].

Thus, the boundary condition becomes,

for small q (in radians) or,

Note that we have only applied the b.c. at y=0. We have not assumed that the slope of the surface is zero.