A standing wave propagating with velocity `300ms^(-1)` in an open pipe of length 4 m has four nodes. The frequency of the wave is

Options

A. 75 Hz
B. 100 Hz
C. 150 Hz
D. 300 Hz

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the standing wave in the open pipe
In an open pipe, standing waves are formed with nodes and antinodes. The number of nodes in the pipe is given as 4.

### Step 2: Determine the relationship between nodes and wavelengths
In a standing wave, the distance between two consecutive nodes is equal to half the wavelength (λ/2). Therefore, if there are 4 nodes, we can visualize the arrangement as follows:
- Node 1
- Antinode
- Node 2
- Antinode
- Node 3
- Antinode
- Node 4

This means there are 3 segments of λ/2 in the length of the pipe.

### Step 3: Calculate the total length of the pipe in terms of wavelength
Since the length of the pipe (L) is 4 meters, and there are 3 segments of λ/2, we can write the equation:
\[ L = \frac{3\lambda}{2} \]
Substituting the length of the pipe:
\[ 4 = \frac{3\lambda}{2} \]

### Step 4: Solve for the wavelength (λ)
To find λ, we rearrange the equation:
\[ 3\lambda = 8 \quad \Rightarrow \quad \lambda = \frac{8}{3} \text{ meters} \]

### Step 5: Use the wave velocity to find the frequency
The wave velocity (v) is given as 300 m/s. The relationship between wave velocity (v), frequency (ν), and wavelength (λ) is given by:
\[ v = \nu \lambda \]
Rearranging for frequency:
\[ \nu = \frac{v}{\lambda} \]

### Step 6: Substitute the known values
Substituting the values we have:
\[ \nu = \frac{300 \text{ m/s}}{\frac{8}{3} \text{ m}} = 300 \times \frac{3}{8} = \frac{900}{8} = 112.5 \text{ Hz} \]

### Step 7: Finalize the answer
Thus, the frequency of the wave is:
\[ \nu = 112.5 \text{ Hz} \]

### Summary
The frequency of the standing wave in the open pipe is **112.5 Hz**.

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