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The full potential equation is non-linear and requires numerical solution. We can however, obtain some nice, simple engineering results by restricting our consideration to small changes in flow variables. In this case, we can neglect terms involving products of small fractions, and simplify the potential equation.
 

Applications

Analyze performance of high-speed airfoils and slender, pointed body shapes at small angles of attack in the subsonic and supersonic regimes.
 

Perturbation Potentials

Let us define the potential  such that,

, where

Now, 

Equation (8) for 2-D flows becomes:

Note:

This is still an exact equation, no different from equation (8). Using (14), (11) becomes:

Linearization

Suppose that 

if in addition, 

This true if  is not too large, i.e. excluding hypersonic flow.

Eq (15) with 'a' replaced by ' ':

Let us look at each of the terms. The second derivatives of f are of the same order of magnitude (no reason to expect otherwise).

1. The coefficient of _xx can be written as

 

 
 
 

if


This is true if we exclude the transonic regime, where .
 

2. The coefficient of _yy can be written as

 

 
 
 

if  is not too large.

(exclude hypersonic flow. But then, we already did that)
 

3. The coefficient of _xy is

we have to throw this out, or we won't get a linear equation. So, let us assume that  the other coefficients. This means that the airfoil must be very thin if  is high, or else  must be very low if the airfoil is not so thin.
 

_{(for  > 1)}

_{(for  < 1)}

Similarly, for 3-D,

_{(for  < 1)}

These equations can be used to describe small perturbations in subsonic (but not transonic) and supersonic (but not transonic or hypersonic0 flow.

Note the difference between equations (18) and (19). The sign of the first term of each depends on whether < or > 1. If we choose the equations so that the first term is positive, then there is a vast difference between equations (18) and (19). The character of the solutions to these equations is vastly changed when the sign of the second term changes from + to -.

Equation (19) describes subsonic flow, where the solution is "elliptic". i.e. changes at any point affect the solution at every other point.

Equation (18) describes supersonic flow, where the solution is "hyperbolic". It is also called the "wave equation", since it can be used to describe the propagation of waves (e.g. sound). One feature of the solution is that changes cannot be felt upstream.

Click here for Prof. Sankar's notes on this derivation