16.12. Worked Example – Wave Equation

Solve the one-dimensional wave equation: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \quad 0<x<L, \; t>0$ Subject to boundary conditions:

• $u(0,t)=0, \; u(L,t)=0$
• Initial conditions: $u(x,0)=f(x), \; \frac{\partial u(x,0)}{\partial t}=g(x)$

In this chunk, we are given a problem to solve the wave equation, which describes how waves propagate through a medium. The wave equation here is a second-order partial differential equation (PDE) that relates the second time derivative and the second spatial derivative of a function $u$, which usually represents a physical quantity like displacement. The description specifies the behavior of the wave on a segment of a line from 0 to L over time. The boundary conditions indicate that the displacement at both ends (0 and L) must be zero. The initial conditions tell us how the system starts at time t=0, with $f(x)$ defining the starting shape of the wave and $g(x)$ defining the initial velocity of the wave.

Think of a plucked guitar string. When you pluck the string, it vibrates to create sound, which is an example of a wave. The wave travels along the string between two fixed points (the ends of the string). The equation describes how the string vibrates over time and how its shape and velocity evolve as the wave travels.

Assume: $u(x,t)=X(x)T(t)$ Substitute into PDE: $T''(t) X(x) = c^2 X''(x) T(t)$ ⇒ = =−λ $c^2 \frac{T(t)}{T(t)} = -\lambda \frac{X(x)}{X(x)}$

In this step, we use a method called 'separation of variables,' which assumes that the solution to the wave equation can be expressed as a product of a function in space, $X(x)$, and a function in time, $T(t)$. When we substitute this assumption into the wave equation, we can separate the variables, leading to two ordinary differential equations. This is done by rearranging the equation into a form where one side contains only terms involving T, and the other side only terms involving X. The resulting equations can often be solved independently.

Imagine you're organizing a dance performance where different dancers perform various parts simultaneously. If you separate the dance into two groups—one group focuses on specific movements (spatial pattern), while the other group focuses on the timing of the music (temporal pattern)—you can analyze and refine each group's performance separately before combining them for the final show.

Solving gives:

• $X(x)=sin\left(\frac{n\pi x}{L}\right), \; \lambda=(\frac{n\pi}{L})^2$
• $T(t)=A \cos(c \lambda t)+B \sin(c \lambda t)$

The solution for the spatial part $X(x)$ results in sinusoidal functions, indicating that the displacement along the string takes the shape of a sine wave. The constant $n$ corresponds to different modes of vibration, and the parameter $\\lambda$ relates to the frequency of those vibrations. The temporal part $T(t)$ consists of a combination of sine and cosine functions, which describe how the amplitude changes over time. The parameters $A$ and $B$ will be determined based on the initial conditions provided.

Consider how different musical notes (represented by different values of n) create distinct sounds. The vibration patterns of the guitar produce complex sounds over time. The sine and cosine solutions for those vibrations show how each note continues to resonate and fade, similar to how sound from a guitar string evolves as it is plucked.

So the complete solution: $u(x,t) = \sum_{n=1}^{\infty} \left[A_n \cos\left(\frac{n\pi ct}{L}\right) + B_n \sin\left(\frac{n\pi ct}{L}\right)\right] \sin\left(\frac{n\pi x}{L}\right)$ Use Fourier series to determine $A_n, B_n$ from initial conditions.

The complete solution is expressed as an infinite series, known as a Fourier series, which represents the sum of all possible waveforms at each point in time and space. The Fourier series allows us to reconstruct the overall wave behavior from the individual modes captured by the sine functions. The coefficients $A_n$ and $B_n$ will be determined using the initial conditions of the problem (the shape of the wave and its velocity at t=0). This enables us to tailor the general solution to fit the specific physical scenario being modeled.

Pumpkin pie has layers of flavor—from spices to pumpkin sweetness—much like how the Fourier series combines different 'flavors' (sine and cosine components) to create the complete sound of the wave. Each note contributes to the overall taste, just as each sine wave contributes to the overall wave shape over time.

--