Small Perturbation Solution

The small perturbation solution refers to an analytical solution where only a small change (or several small changes) occurs. In this case, it refers to a case where only a ``small shock'' occurs, which is up to M_x=1.3. This approach had a major significance and usefulness at a time when personal computers were not available. Now, during the writing of this version of the book, this technique is used mostly in obtaining analytical expressions for simplified models. This technique also has an academic value and therefore will be described in the next version (0.5.x series).

The strength of the shock wave is defined as

$$\hat{\cal P} = {P_y - P_x \over P_x} = {P_y \over P_x} - 1$$ (5.39)

By using equation (5.23) transforms equation (5.39) into

$$\hat{\cal P} = {2k \over k +1 } \left({M_x}^2 - 1 \right)$$ (5.40)

or by utilizing equation (5.24) the following is obtained:

$$\hat{\cal P} = {{2k \over k -1} \left({ \rho_y \over \rho x} - 1 \right) \over {2 \over k -1} - \left( {\rho_y \over \rho_x} - 1 \right) }$$ (5.41)