Superposition of Elementary Solutions

From the property of linear governing equations, if ф1 and ф2 are each solution of the Laplace equation, ф1 + ф2 or any linear combination will be a solution of the Laplace equation. The same is true for the stream functions. This is the basis of the principle of superposition. Note that the elementary solutions for ф and f are singular at their centers (infinite velocity), hence they are also called singular solutions and the superposition of singular solutions is called the method of singularities. These singular solutions are regular everywhere in the flow field when the centers are excluded, for example by enclosing them within the obstacle.

Mathematically one can prove that it is possible to solve for the flow past an arbitrary obstacle by superposing elementary solutions, but, in general, the resulting pressure field is not the sum of the individual pressure fields, due to the nonlinear­ity of Bernoulli’s equation. For small disturbances, the Bernoulli equation can be linearized, allowing for superposition of pressure fields as well.