On solving equations of Lagrangian hydrodynamics

The paper shows that coordinates of liquid particles and pressure can be expressed in terms of one arbitrary function $\psi$ so that the incompressibility condition is satisfied for any choice of this function. Introducing this function features as the unknown one significantly simplifies obtaining analytical and numerical solutions of the hydrodynamic equations written in Lagrangian variables. An incompressible flow can be written in Eulerian or Lagrangian variables. Both forms of these equations have been known for a long time but scientists usually prefer to use the Euler variables. This is explained by the unusualness of Lagrange equations. They include nonlinear terms in a form that is inconvenient for numerical and analytical calculations. Until now, hydrodynamicists did not try to exclude the incompressibility condition from the Lagrange equations with the aim of reducing the number of unknown variables. Therefore, in this paper we show that the incompressibility condition can be satisfied automatically if the particle coordinates $x(a,b,t)$ and $y(a,b,t)$ are expressed in terms of the same arbitrary function of coordinates and time. In Lagrangian variables, such a function plays the same role as the function of the current in Euler variables. In this paper, as an example of the exact solution, a solution of the problem of standing waves in a liquid layer is presented. The problem is solved using Lagrange variables. To do this, it is necessary to select an area of the Lagrangian variables in the form of an infinite strip the lower edge of which corresponds to a solid wall. Similarly, the problem is solved for a traveling wave.