11.1.7. Worked Example

Problem: Solve the wave equation ∂²𝑢/∂𝑡² = 4, for 0 < 𝑥 < 𝜋, with:

• 𝑢(0,𝑡) = 0, 𝑢(𝜋,𝑡) = 0
• ∂𝑢/∂𝑡 (𝑥,0) = 0
• 𝑢(𝑥,0) = sin(𝑥)

In this example, we are tasked with solving a specific wave equation subject to fixed boundary conditions and an initial displacement. The wave equation given is a second-order partial differential equation that describes how waves propagate over time. The boundary conditions state that at position 0 and π, the wave displacement is fixed at zero. Additionally, the initial conditions specify that the wave’s initial velocity at any point in time is zero and its initial displacement is represented by the function sin(x).

Imagine plucking a guitar string. The ends of the string (0 and C0 in this case) cannot move, just as the fixed boundary conditions ensure the displacement there is zero. The initial shape of the string (the initial displacement) reflects a sine wave, similar to how the vibrating string creates sound waves.

Solution: We use separation of variables:

Since initial displacement is sin(𝑥), we find: 𝑢(𝑥,𝑡) = cos(2𝑡)sin(𝑥)

To solve the wave equation, we apply the method of separation of variables. This technique involves rearranging the wave equation into a form where the variables can be separated, allowing us to handle the spatial and temporal components independently. In this case, we recognize that the initial displacement being sin(x) informs our solution's spatial component. The time component, which represents oscillations, is captured by cos(2t) because it corresponds to the wave’s frequency. Combining these leads to the solution u(x,t) = cos(2t)sin(x), which shows how the wave changes over time in a fixed-length medium.

Think of a swing on a playground. When you push it, it oscillates back and forth (that’s the cos part), while the specific height it reaches at any point in time is influenced by how hard you initially pushed it (that’s the sin part). Together, they explain the motion of the swing just like they describe the wave in our example.

Summary:

• The one-dimensional wave equation models vibrations and wave propagation in strings and similar media.
• The equation is derived from Newton’s laws and string tension considerations.
• D’Alembert’s solution handles infinite domains, while separation of variables is used for bounded domains.
• Boundary and initial conditions are crucial for determining specific solutions.
• The general solution for fixed ends involves Fourier sine series expansions.

In summary, we've solved a wave equation by applying the separation of variables technique, which is particularly effective for problems with fixed boundaries. The results illustrate the fundamental principles of wave motion and how specific initial and boundary conditions dictate the wave's behavior. Recognizing the important role of these conditions helps validate the method we used and illustrates the broader utility of the wave equation in modeling real-world scenarios.

Consider a drum skin. When struck, it vibrates to create sound waves. The wave propagation can be thought of in terms of our equation where the way the drum is initially struck (initial conditions) and the edges of the drum (boundary conditions) significantly affect the sound produced. This connection shows the importance of understanding the underlying equations and conditions to predict and analyze various physical phenomena.

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