9.
Standing waves are excited in a 1 m long pipe open at one end, closed at the other. Take
the speed of the wave as 340 m/s and calculate the frequency of the first overtone (i.e. the
first harmonic above the fundamental frequency).
a. 128 Hz
b. 170 Hz
c. 255 Hz
d. 510 Hz
Standing Waves in a Pipe (Open at One End)
First Overtone Frequency Calculation
Given:
- Wave speed (v) = 340 m/s
- Length of pipe (L) = 1 m
- Pipe: Open at one end, closed at the other
Observation: The first overtone frequency is 255 Hz, which matches option (c).
Problem Solution
For a closed pipe, standing waves form with a displacement node at the closed end and an antinode at the open end.
The fundamental frequency (first harmonic) satisfies the relation:
L = λ₁ / 4
Therefore,
f₁ = v / (4L) = 340 / (4 × 1) = 85 Hz
The first overtone corresponds to the third harmonic (since only odd harmonics exist):
L = 3λ₃ / 4
Thus,
f₃ = 3f₁ = 3 × 85 = 255 Hz
Only odd harmonics (1st, 3rd, 5th, …) exist due to the boundary conditions of a closed pipe.
Option Analysis
| Option | Frequency (Hz) | Matches | Reason |
|---|---|---|---|
| a | 128 | No | Arbitrary/miscalculation |
| b | 170 | No | Fundamental frequency of an open pipe, not applicable here |
| c | 255 | Yes | Correct first overtone (third harmonic, 3 × v / 4L) |
| d | 510 | No | Third overtone (fifth harmonic, 5f₁) |
