In an organ pipe whose one end is at x =0, the pressure is expressed by `P = P_(0) "cos" (3 pi x)/(2) sin 300 pi t` where x is in meter and t in sec. The organ pipe can be :-

Options

A. Closed at one end, open at another with length = 0.5 m
B. Open at both ends, length = 1m
C. Closed at both ends, length = 2 m
D. Closed at one end, open at another with length = 2/3 m

Explanation

To solve the problem, we need to analyze the given pressure equation for the organ pipe and determine its characteristics based on the boundary conditions.

### Step-by-Step Solution:

1. **Identify the Pressure Equation**:
The pressure in the organ pipe is given by:
\[ P = P_0 \cos\left(\frac{3\pi x}{2}\right) \sin(300\pi t) \]
Here, \( P_0 \) is the amplitude of the pressure wave, \( x \) is the position in meters, and \( t \) is the time in seconds.

2. **Determine the Wavelength**:
The term \(\frac{3\pi x}{2}\) indicates the wave number \( k \). The wave number \( k \) can be expressed as:
\[ k = \frac{2\pi}{\lambda} \]
From the equation, we have:
\[ k = \frac{3\pi}{2} \implies \lambda = \frac{2\pi}{\frac{3\pi}{2}} = \frac{4}{3} \text{ meters} \]

3. **Identify the Length of the Pipe**:
The length of the pipe \( L \) can be determined by the relationship between the wavelength and the pipe's boundary conditions. For a pipe closed at one end and open at the other, the length of the pipe is given by:
\[ L = \frac{\lambda}{4} = \frac{4/3}{4} = \frac{1}{3} \text{ meters} \]
However, since we are given that the pressure function indicates a node at the closed end and an antinode at the open end, we need to check the maximum and minimum pressure conditions.

4. **Check Boundary Conditions**:
- At \( x = 0 \):
\[ P = P_0 \cos(0) \sin(0) = P_0 \cdot 1 \cdot 0 = 0 \]
This indicates that there is a node at \( x = 0 \), which means this end is closed.
- At \( x = L \) (where \( L = \frac{2}{3} \)):
\[ P = P_0 \cos\left(\frac{3\pi \cdot \frac{2}{3}}{2}\right) \sin(300\pi t) = P_0 \cos(\pi) \sin(300\pi t) = -P_0 \sin(300\pi t \]
This indicates that there is an antinode at \( x = \frac{2}{3} \), which means this end is open.

5. **Conclusion**:
The organ pipe is closed at one end (at \( x = 0 \)) and open at the other end (at \( x = \frac{2}{3} \)). The length of the pipe is \( \frac{2}{3} \) meters.

### Final Answer:
The organ pipe is closed at one end and open at the other end, with a length of \( \frac{2}{3} \) meters.