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 find the frequency of the standing wave in the open pipe, we can follow these steps:

### Step 1: Understand the relationship between nodes and antinodes in a standing wave.
In an open pipe, the standing wave pattern consists of nodes and antinodes. The number of nodes (N) is related to the number of antinodes (A) by the formula:
\[ A = N - 1 \]
For 4 nodes, the number of antinodes will be:
\[ A = 4 - 1 = 3 \]

### Step 2: Determine the wavelength (λ).
The length of the pipe (L) is related to the wavelength (λ) by the formula:
\[ L = \frac{n}{2} \lambda \]
where n is the number of segments (which is equal to the number of antinodes). Since we have 3 antinodes:
\[ L = \frac{3}{2} \lambda \]
Given that the length of the pipe (L) is 4 m, we can rearrange the equation to find λ:
\[ 4 = \frac{3}{2} \lambda \]
Multiplying both sides by \( \frac{2}{3} \):
\[ \lambda = \frac{4 \times 2}{3} = \frac{8}{3} \text{ m} \]

### Step 3: Use the wave velocity to find the frequency (f).
The relationship between wave velocity (v), frequency (f), and wavelength (λ) is given by:
\[ v = f \lambda \]
We can rearrange this to solve for frequency:
\[ f = \frac{v}{\lambda} \]
Substituting the given velocity (v = 300 m/s) and the calculated wavelength (λ = \( \frac{8}{3} \) m):
\[ f = \frac{300}{\frac{8}{3}} = 300 \times \frac{3}{8} = \frac{900}{8} = 112.5 \text{ Hz} \]

### Step 4: Final answer.
Thus, the frequency of the wave is:
\[ f = 112.5 \text{ Hz} \]