Form of the solution

Remember from your undergraduate fluids that if we have a relationship of the form

then the nondimensional ratio will be a function of the nondimensional combinations (or -groups) that we can make from the four function arguments. Now is already a nondimensional number. From the other three, we can only get rid of the mass dimension by taking the ratio , which we already know is proportional to the speed of sound a₁. We can then get rid of the velocity dimension by taking a ratio with the speed u₁. So our only other nondimensional combination is the Mach number M₁=u₁/a₁, and dimensional analysis then tells us that

Of course, the same holds for any other nondimensional ratio we solve for. So the form of our solution will be:

For the important case of normal air, i.e. , the book provides all these functions in tabular form in Table A.2. The first column in the table is the Mach number M₁, the second column the corresponding p₂/p₁, the third , etcetera.

For other values of than 1.4, you will have to use the actual formulae, which we will derive now.