The question asks for the independent variables introduced in D'Alembert's solution of the one-dimensional wave equation:

$u_{tt} = c^2 u_{xx}.$

Key Idea of D'Alembert's Solution:

The general solution for this wave equation can be expressed in terms of two independent variables, which represent left- and right-moving waves. These variables are usually introduced as:

$v = x + ct \quad \text{and} \quad z = x - ct.$

These are known as characteristic variables, allowing the equation to separate into two traveling wave solutions.

Correct Answer:

The correct pair of variables is: (a) $v = x + ct$, $z = x - ct$.

This choice corresponds to the forward and backward traveling waves along the $x$-axis at speed $c$.

Do you want a more detailed breakdown of the solution technique?

Here are 5 related questions to deepen your understanding:

1. How does D'Alembert’s solution handle initial conditions?
2. What are the physical interpretations of the variables $x + ct$ and $x - ct$?
3. Can the D'Alembert solution be extended to 2D or 3D wave equations?
4. What is the significance of the wave speed $c$ in the equation?
5. How do different boundary conditions affect the wave solution?

Tip: When working with wave equations, tracking how the wavefronts evolve over time is essential. This is where characteristic variables like $x \pm ct$ become useful for visualizing the wave propagation.