Worked Example

Let's start with the basics. What is the one-dimensional wave equation?

Is it the equation that describes how waves propagate through a medium?

Exactly! The wave equation models the movement of waves, such as sound or water waves, along one dimension. Can anyone tell me the standard form of this equation?

I think it goes like this: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$!

Great! Here, $u(x,t)$ represents displacement, and $c$ is the wave speed. Remember this equation as it is foundational in understanding wave behavior.

Now, let’s take a look at our specific problem. We have the wave equation: $\frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}$. What boundary conditions do we have?

We have $u(0,t) = 0$ and $u(\pi,t) = 0$.

Right! These fixed boundary conditions mean the ends of our string do not move. And what about the initial conditions?

The initial displacement is $u(x,0) = \sin(x)$, and the initial velocity is $\frac{\partial u}{\partial t}(x,0) = 0$.

Perfect! We will use these to find our solution.

To solve this using separation of variables, we assume a solution of the form $u(x,t) = X(x)T(t)$. Can anyone derive this relationship?

We break the equation into two parts: one depending on $x$ and the other on $t$.

Exactly! By substituting back into our wave equation, we can separate the variables. Now, given that the initial displacement is $\sin(x)$, how does this help us express the solution?

It tells us that the solution will involve the sine function for the spatial part!

Great! So what would our solution be?

I think it would be $u(x,t) = \cos(2t)\sin(x)$!

Exactly right! This shows the wave behavior with respect to time and space.

To summarize, we covered how we set the boundary and initial conditions for our wave equation problem. We used separation of variables to arrive at our solution, which demonstrates how waves propagate in the string.

So, we basically applied the principles from earlier sections and now we see how they connect to real wave scenarios!

Precisely! Understanding this worked example helps solidify our grasp on solving wave equations with real-world applications.