Euler continuity equation with high-order terms in time

In this paper, we examine the appearance of high-order terms in the continuity equation for an incompressible fluid obtained by L. Euler in 1752 from the linear Cauchy–Helmholtz equations. According to Lighthill's acoustic analogy, these additional terms in the inhomogeneity of the wave equation lead to the generation of self-oscillations and sound waves. In Lighthill's method, the second-order wave equation is obtained by taking the time derivative of the continuity equation. In this case, second-order terms that are usually neglected, increase their order and become comparable with other terms of the wave equation. Solution of the inhomogeneous wave equation allows one calculate or estimate the intensity of vibrations and self-oscillations, which are sometimes considered spontaneous.