Question

The vibrations of four air columns are shown below. The ratio of frequencies is:

• A. \( 1:2:3:4 \)
• B. \( 1:3:2:4 \)
• C. \( 1:4:3:2 \)
• D. \( 1:4:2:3 \)

Hint:

For standing waves in an air column, the frequency follows \( f_n = n f_1 \), where \( n \) depends on the harmonic mode. Count the number of nodes and antinodes carefully.

Answer 1

The Correct Option is D

Step 1: Understanding the Harmonic Frequencies - The frequency of vibration in an air column depends on the number of nodes and antinodes formed in the standing wave. - The fundamental frequency is given by: \[ f_n = n \times f_1. \]
Step 2: Identifying the Harmonics Observing the diagrams: - The first column shows the fundamental mode: \( f_1 \). - The second column shows the fourth harmonic: \( f_4 = 4f_1 \). - The third column shows the second harmonic: \( f_2 = 2f_1 \). - The fourth column shows the third harmonic: \( f_3 = 3f_1 \). Thus, the ratio is: \[ 1:4:2:3. \] Thus, the correct answer is: \[ \boxed{1:4:2:3}. \]