Mathematical simulation of physical conditions at a moving boundary in problems of wave reflection leads to a finite-difference equation with variable delay. If the reflector velocity is smaller than the wave-propagation velocity, the existence and uniqueness of the equation solution is proved using the principle of contraction maps and the method of successive approximations. It is shown that the solution can be expressed as a function of the composite argument that depends on one variable. The required accuracy of numerical calculations is ensured by partial sums when the argument is represented by an infinite converging series.