Final answer:
The process requires applying boundary and initial conditions to find the Fourier series solution of the wave equation, then using computational tools to plot W(x, t). Actual calculations and plots cannot be provided here.
Step-by-step explanation:
The student's problem involves finding a series solution for a partial differential equation (PDE) describing a wave system, and then plotting this solution for given ranges of x and t. It is essential to note that the sin function in the initial condition appears to have a typo, assuming the problem's domain is 0 ≤ x ≤ 5 based on the boundary conditions given. The governing equation, boundary conditions, and initial conditions are part of a standard wave equation problem in physics. The series solution would typically involve using the method of separation of variables, and the Fourier series expansion of the initial wave form given the boundary conditions. Subsequently, plots of W(x, t) versus x and t would require computational software.
Since actual calculations and software utilization for such a plot are beyond the scope of this platform and because the provided information is not sufficient to complete the task, no specific solution can be provided. However, as a guide, the student can start by enforcing the boundary conditions to find the permissible values of k, the wave number in the sine term, and then use the initial conditions to determine the coefficients of the Fourier series. Once these coefficients are determined, they will form the series solution for W(x, t), from which W(x, t) can be plotted for the given ranges of x and t using appropriate computational tools.