Laguerre Polynomials in the Forward and Backward Wave Profile Description for the Wave Equation on an Interval with the Robin Condition or the Attached Mass Condition

We obtain a formula describing the forward and backward wave profile for the solution of an initial–boundary value problem for the wave equation on an interval. The following combinations of boundary conditions are considered:

(i) The first-kind condition at the left endpoint of the interval and the third-kind condition at the right endpoint.

(ii) The second-kind condition at the left endpoint and the third-kind condition at the right endpoint.

(iii) The first-kind condition at the left endpoint and the attached mass condition at the right endpoint.

(iv) The second-kind condition at the left endpoint and the attached mass condition at the right endpoint.

The formula contains finitely many arithmetic operations, elementary functions, quadratures, and transformations of the independent argument of the initial data such as the multiplication by a number and taking the integer part of a number.