A string of length `'l'` is fixed at both ends. It is vibrating in its `3^(rd)` overtone with maximum ampltiude `'a'`. The amplitude at a distance `(l)/(3)` from one end is `= sqrt(p)(a)/(2)`. Find `p`.

Explanation

To solve the problem step by step, we will analyze the vibration of the string and find the required value of \( p \).

### Step 1: Understand the Overtone
The string is vibrating in its 3rd overtone. For a string fixed at both ends, the relationship between the overtone number \( n \) and the wavelength \( \lambda \) is given by:
\[ L = n \cdot \frac{\lambda}{2} \]
For the 3rd overtone, \( n = 4 \) (since the fundamental mode is the 1st overtone, the 2nd overtone is the 3rd mode, and so on). Thus:
\[ L = 4 \cdot \frac{\lambda}{2} \implies \lambda = \frac{L}{2} \]

### Step 2: Write the Expression for Amplitude
The amplitude of the wave at a distance \( x \) from one end is given by:
\[ A(x) = A \sin(kx) \]
where \( k \) is the wave number defined as:
\[ k = \frac{2\pi}{\lambda} \]
Substituting the value of \( \lambda \):
\[ k = \frac{2\pi}{L/2} = \frac{4\pi}{L} \]

### Step 3: Calculate Amplitude at \( x = \frac{L}{3} \)
Now, we need to find the amplitude at a distance \( \frac{L}{3} \):
\[ A\left(\frac{L}{3}\right) = A \sin\left(k \cdot \frac{L}{3}\right) = A \sin\left(\frac{4\pi}{L} \cdot \frac{L}{3}\right) = A \sin\left(\frac{4\pi}{3}\right) \]

### Step 4: Evaluate \( \sin\left(\frac{4\pi}{3}\right) \)
The value of \( \sin\left(\frac{4\pi}{3}\right) \) is:
\[ \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]
Thus, the amplitude becomes:
\[ A\left(\frac{L}{3}\right) = A \left(-\frac{\sqrt{3}}{2}\right) \]

### Step 5: Set Up the Equation
According to the problem, the amplitude at this point is also given by:
\[ A\left(\frac{L}{3}\right) = \frac{\sqrt{p} \cdot A}{2} \]
Setting the two expressions for amplitude equal to each other:
\[ -\frac{\sqrt{3}}{2} A = \frac{\sqrt{p} \cdot A}{2} \]

### Step 6: Simplify and Solve for \( p \)
Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ -\frac{\sqrt{3}}{2} = \frac{\sqrt{p}}{2} \]
Multiplying both sides by 2:
\[ -\sqrt{3} = \sqrt{p} \]
Squaring both sides:
\[ p = 3 \]

### Final Answer
Thus, the value of \( p \) is:
\[ \boxed{3} \]