Laguerre polynomials in describing the profiles of forward and backwardwaves for thewave equation on a segment under the robin condition or under the joined mass condition

A formula that describes the profiles of forward and backward waves for the solution of the initial-boundary value problem for the wave equation on a segment is obtained. The following combinations of boundary conditions are considered: 1) the first kind condition in the left end point and the third kind condition in the right end point, 2) the second kind condition in the left end point and the third kind condition in the right end point, 3) the first kind condition in the left end point and so-called joined mass condition in the right end point, 4) the second kind condition in the left end point and joined mass condition in the right end point. This formula contains a finite number of arithmetic operations, elementary functions, quadratures, and such transformations of the independent argument of the initial data as multiplication by a number and taking the integer part of the number.