Question

Pick out the correct statement in the following with reference to stationary wave pattern

• A. In a tube closed at one end, all the harmonics are present
• B. In a tube open at one end, only even harmonics are present
• C. The distance between successive nodes is equal to the wavelength
• D. In a stretched string, the first overtone is the same as the second harmonic
• E. Reflection of a wave from a rigid wall changes the phase by \( 45^\circ \)

Hint:

"Overtone" refers to any frequency higher than the fundamental. In strings and open pipes, the 1st Overtone = 2nd Harmonic. In pipes closed at one end, the 1st Overtone = 3rd Harmonic.

Answer 1

The Correct Option is D

Concept: This question tests fundamental properties of standing waves in different mediums.
• Tubes closed at one end: Only odd harmonics (\(1f, 3f, 5f...\)) are present.
• Stretched strings/Open tubes: All harmonics (both even and odd) are present.
• Phase change: Reflection from a rigid boundary causes a phase shift of \(\pi\) radians or \(180^\circ\).

Step 1: Evaluate statements A, B, and C.
- (A) and (B): For a tube closed at one end (same as open at one end), only odd harmonics exist. Even harmonics are absent. Thus, both are incorrect. - (C): As established in Question 14, the distance between successive nodes is \(\lambda/2\), not \(\lambda\). This is incorrect.

Step 2: Evaluate statements D and E.
- (E): Reflection from a rigid wall (denser medium) results in a phase change of \(180^\circ\) (\(\pi\) rad), not \(45^\circ\). This is incorrect. - (D): In a stretched string, the first harmonic is the fundamental. The next mode is the second harmonic, which is by definition the first overtone. This is the correct statement.