Neglecting surface tension and viscosity, the equation was first derived by W. H. Besant in his 1859 book with the problem statement stated as An infinite mass of homogeneous incompressible fluid acted upon by no forces is at rest, and a spherical portion of the fluid is suddenly annihilated; it is required to find the instantaneous alteration of pressure at any point of the mass, and the time in which the cavity will be filled up, the pressure at an infinite distance being supposed to remain constant (in fact, Besant attributes the problem to Cambridge Senate-House problems of 1847). Besant predicted the time required to fill an empty cavity of initial radius  to be
Lord Rayleigh found a simpler derivation of the same result, based on conservation of energy. The kinetic energy of the inrushing fluid is  where  is the time-dependent radius of the void, and  the radial velocity of the fluid there. The work done by the fluid pressing in at infinity is , and equating these two energies gives a relation between  and . Then, noting that , separation of variables gives Besants result. Rayleigh went further than Besant, in evaluating the integral (Eulers beta function) in terms of gamma functions.  Rayleigh adapted this approach to the case of a cavity filled with an ideal gas (a bubble) by including a term for the work done compressing the gas.
For the case of the perfectly empty void, Rayleigh determined that the pressure  in the fluid at a radius  is given by:
When the void is at least one quarter of its initial volume, then the pressure decreases monotonically from  at infinity to zero at . As the void shrinks further a pressure maximum, greater than  appears at
very rapidly growing and converging on the void.
The equation was first applied to traveling cavitation bubbles by Milton S. Plesset in 1949 by including effects of surface tension.