14.3.2. Step 2: Solve the Simplified PDE

Given the equation from previous steps, we have the simplified form:

$\frac{u}{\xi \eta} = 0$

In this part, we start with the result obtained from the earlier manipulations of the wave equation. The key equation we look at now indicates that the second mixed partial derivative equals zero. This means that any wave solution can be expressed as a function that is separable with respect to two new variables: $\xi$ and $\eta$. So, we are going to solve the simplified partial differential equation (PDE) that we derived in the previous step.

Think of this like simplifying a puzzle. If you have many pieces that make a complicated picture, sometimes you can group similar pieces together to see the overall shape easier. Here, we are making the equation simpler to solve, just like grouping the puzzle pieces.

The general solution of this PDE is represented as:

$u(\xi, \eta) = F(\xi) + G(\eta)$

From our statement that $u = 0$ leads us to believe that the functions we are looking at can be expressed individually. This means that the overall behavior of the wave can be completely recovered by understanding two separate functions, $F(\xi)$ and $G(\eta)$. Each of these functions corresponds to waveforms traveling in opposite directions; $F(\xi)$ is for one direction, while $G(\eta)$ corresponds to the other.

Imagine you are at a busy intersection looking at traffic from two different roads. One road has cars moving straight towards you (let's say it's like $F(\xi)$), while cars from the other road are moving away from you. Each set of cars tells a different story of traffic flow, but together they provide a complete picture of what’s happening at the intersection.

Finally, we return to our original variables to express the solution in the context of the wave equation:

$u(x, t) = f(x + ct) + g(x - ct)$

Once we have our general solution in the new variables, we need to transform it back to our original variables, which correspond to the physical quantities we're interested in. By substituting back the definitions of $\xi$ and $\eta$ in terms of the original variables $x$ and $t$, we recover the wave solution that can now be interpreted in a more concrete context—where the functions $f$ and $g$ represent waves propagating through space over time.

This is akin to translating a recipe from one language back to your native language. The recipe (solution) remains the same in terms of the ingredients, but it’s essential that it’s phrased in a way you understand so you can actually cook (apply the solution) using it.

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