Wave Equation with a Free End

Today, we focus on the wave equation, $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$. Can anyone tell me what this equation represents?

It describes wave motion, right?

Exactly! It's fundamental in understanding how waves propagate. Now, what happens when we have boundary conditions, especially at one end?

Is it like having one end fixed and the other free?

Yes! When one end is free, we impose conditions like $u(0,t) = 0$. Let's move into how we solve this using Fourier Sine Transform.

The boundary conditions we need are $u(0,t) = 0$, \lim_{x \to \infty} u(x,t) = 0 ), and \frac{\partial u}{\partial t}(x,0) = 0. Why are these conditions important?

They specify how the wave behaves at certain points and times.

Exactly! Understanding these conditions helps us predict wave behavior. Let's consider the Fourier Sine Transform now. How does it help us with that?

It converts our problem into a different domain, making it easier to solve differential equations?

Absolutely right! We'll see how it simplifies the process of finding solutions.

When we apply the Fourier Sine Transform to our wave equation, we get $\frac{\partial^2 U}{\partial t^2} = -c^2 s^2 U$. What kind of equation is this?

It looks like a second-order ordinary differential equation.

Correct! The solution to this ODE is of the form $U(s,t) = A(s) \cos(cst) + B(s) \sin(cst)$. What can you tell me about coefficients $A(s)$ and $B(s)$?

A(s) is from the initial displacement and B(s) relates to initial velocity, right?

Exactly! Since $\frac{\partial U}{\partial t}(t,0) = 0$, it tells us that $B(s) = 0$. So our solution simplifies. What are we left with?

It's just $U(s,t) = F \{f(x)\} \cos(cst)$.

Correct! Our final solution represents how the displacement evolves over time.