Final Form of D'Alembert’s Solution (with initial conditions)

Today, we’ll explore D'Alembert's solution to the one-dimensional wave equation. What do you recall about this equation?

It describes how waves, like sound or light, travel in a medium.

Exactly! The equation is $\partial^2 u / \partial t^2 = c^2 \partial^2 u / \partial x^2$. D'Alembert's solution gives us wave behavior in a specific form. Can anyone tell me what that form looks like?

Isn’t it $u(x, t) = f(x + ct) + g(x - ct)$?

Correct! This representation shows how shapes of waves travel without distortion. We’ll now dive deeper into applying initial conditions to this solution.

Great! Now, let's think about what happens when we have initial conditions. How do we define them in the context of wave equations?

We define initial displacement $\phi(x)$ and initial velocity $\psi(x)$.

"Correct! By applying these initial conditions, we can express D'Alembert’s solution more completely. It's given by:

So far, we have seen the mathematical formulation. How do the terms in D'Alembert’s solution relate to physical wave behavior?

The terms show how the wave moves based on initial conditions, right?

Exactly! The displacement terms provide insight into how the wave propagates as per the initial settings. The integral shows how velocity dynamically influences the wave shape.

Can we use this to predict wave behavior?

Definitely! Understanding these aspects enables us to solve diverse wave problems accurately. Let's summarize what we learned.

Let’s apply our knowledge to solve a practical wave equation problem. We have the wave equation with $u(x, 0) = ext{sin}(x)$ and $\partial u / \partial t (x, 0) = 0$. What steps can we take?

First, we identify $c = 2$, then $\phi(x) = ext{sin}(x)$ and $\psi(x) = 0$.

Correct! Now, can you apply D'Alembert’s solution using these conditions?

"Using D'Alembert's formula, we get: