Laplace's equation is the Prandtl-Glauert equation in the limit as the
freestream Mach number goes to zero. It was actually first derived by Euler.
The derivation is very simple, requiring only the equation of continuity,
and the assumptions of irrotational and constant density flow.
The continuity equation becomes then:
Since the flow is irrotational: 
Substitution into the continuity equation yields:
It is interesting to note that Laplace's equation does not require the assumption
of small perturbations, while the Prandtl-Glauert equation does. In fact,
near the stagnation point of an airfoil where velocities become small, the
full potential equation reduces to Laplace's equation, not the Prandtl-Glauert
equation.
Note also that all of the time dependent terms in the full potential equation
are multiplied by 1/a2 so that this form of the equation holds for unsteady
phenomena as well.