91Ó°ÊÓ

The wave equation is a quintessential partial differential equation that describes the distribution of waves, such as sound or light, through different mediums. In its simplest form, the classic wave equation is given by: \[ u_{tt} = c^2 u_{xx} \] where \( u \) represents the wave function, \( t \) is time, \( x \) is spatial dimension, and \( c \) is the speed of the wave. For the exercise at hand, the modified wave equation is:\[ u_{xx} = u_{tt} + 2u_t \] which adds a damping term \( 2u_t \). This term accounts for energy loss in the wave as it propagates, a scenario often encountered in real-world situations. Here, the challenge is to observe how this modifying term alters the behavior of the solution compared to the classical wave equation, particularly in a bounded domain.

Boundary conditions are essential for solving differential equations as they provide additional information that specifies a unique solution. In one-dimensional wave equations, boundary conditions usually specify the behavior of the wave at the spatial edges. For our problem, the boundary conditions are specified as:

• \( u(0, t) = 0 \)
• \( u(\pi, t) = 0 \)

These conditions imply that the wave is fixed at both \( x = 0 \) and \( x = \pi \), meaning the ends are physically restrained or "clamped." This results in nodes at these points, allowing us to use sine functions—\( \sin(nx) \)—as part of our solution since they naturally satisfy these boundary conditions. Boundary conditions are crucial as they influence the selection of valid solutions from the general solution set to the wave equation.

Separation of variables is a powerful mathematical technique used to simplify the process of solving partial differential equations. The idea is to assume that the solution can be expressed as a product of functions, each a function of a single variable. For our wave equation, we propose:\[ u(x, t) = X(x)T(t) \]Substituting this form into the wave equation allows us to separate the functions and form two ordinary differential equations:

• \( X''(x) = -\lambda X(x) \)
• \( T''(t) + 2T'(t) = -\lambda T(t) \)

Here, \( \lambda \) is a separation constant. This separation reduces the complexity significantly as each equation can be solved individually. For \( X(x) \), we get a typical Sturm-Liouville problem, while \( T(t) \) becomes a second-order homogeneous differential equation. This technique is pivotal in simplifying and ultimately solving the wave equation within the boundary limits.

Initial conditions are specific requirements given at the starting point of the time variable, specifically \( t = 0 \), which help determine the unique solution to our differential equation. In this exercise, the initial conditions are:

• \( u(x, 0) = \sin x + \sin 3x \)
• \( u_t(x, 0) = 0 \)

The first condition specifies the initial shape or displacement of the wave. The second condition indicates that there is no initial velocity. These conditions serve as the basis to find particular values for constants \( A_n \) and \( B_n \) in the general solution derived from separation of variables. Solving these initial conditions involves matching the coefficients in the assumed general solution with the given initial profile of the wave, ensuring that the solution accurately reflects the initial state of the system.