Physical meaning of the velocity potential. The Lavrentiev problem of a directed explosion

We now must clarify the direct physical meaning of the velocity potential: without understanding this it is impossible to formulate the Dirichlet boundary value problem: we have to prescribe the velocity potential at the boundary, but we do not know yet what the potential is.

Consider a body in a continuous medium which at t = t0 is at rest. Assume that at t = t0 each particle experiences a pressure pulse such that the pressure varies according to the law

Here θ(x) is a function of the position of the particle, and δ(z) is the generalized Dirac function. According to the simplest definition of this function, which is all we need for now,

for arbitrarily small positive ∊.

The motion begins from a state of rest before the pressure pulse starts. Therefore uf tge system of mass forces acting on the medium is a potential one, the Lagrange–Cauchy integral holds in the ideal incompressible fluid approximation:

We put (4.1) into (4.3) and integrate from t = t0 − ε to t = t0 + ε.