The equation of a plane progressive wave is `y=Asin(pit-(x)/(3))`. Find the value of A for which the wave speed is equal to the maximum particle speed.

Answer

To solve the problem, we need to find the value of \( A \) for which the wave speed is equal to the maximum particle speed in the given wave equation \( y = A \sin(\pi t - \frac{x}{3}) \).

### Step-by-Step Solution:

1. **Identify the wave equation parameters**:
The wave equation is given as:
\[ y = A \sin(\pi t - \frac{x}{3}) \]
From this equation, we can identify:
- The angular frequency \( \omega \) is the coefficient of \( t \), which is \( \pi \).
- The wave number \( k \) is the coefficient of \( x \), which can be found from the term \( -\frac{x}{3} \). Thus, \( k = \frac{1}{3} \).

2. **Calculate the wave speed \( v \)**:
The wave speed \( v \) is given by the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \( \omega \) and \( k \):
\[ v = \frac{\pi}{\frac{1}{3}} = \pi \times 3 = 3\pi \]

3. **Calculate the maximum particle speed \( v_{\text{max}} \)**:
The maximum particle speed is given by:
\[ v_{\text{max}} = A \cdot \omega \]
Substituting the value of \( \omega \):
\[ v_{\text{max}} = A \cdot \pi \]

4. **Set the wave speed equal to the maximum particle speed**:
According to the problem, we need to find \( A \) such that:
\[ v = v_{\text{max}} \]
This gives us the equation:
\[ 3\pi = A \cdot \pi \]

5. **Solve for \( A \)**:
Dividing both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ 3 = A \]
Therefore, the value of \( A \) is:
\[ A = 3 \]

### Final Answer:
The value of \( A \) for which the wave speed is equal to the maximum particle speed is \( A = 3 \).