https://file.aiquickdraw.com/mathbot/file/1729509995652j07ezc4b.jpg

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.

D'Alembert's solution of the one-dimensional wave equation \( u_{tt} = c^2 u_{xx} \) is obtained by introducing two independent variables. Which of the following are the correct variables?

Learn about D'Alembert's solution for the one-dimensional wave equation \( u_{tt} = c^2 u_{xx} \), including the introduction of the characteristic variables \( v = x + ct \) and \( z = x - ct \), which represent traveling wave solutions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation