A string stretched between two fixed points sounds its second harmonic at frequency f. Which expression, where n is an integer, gives the frequencies of harmonics that have a node at the centre of the string? A. n 12f B. nf C. 2nf D. (2n 1)f

Answer

Harmonic Frequencies with Center Node on a Stretched String

Harmonic Frequencies with a Node at the Center of a String

When a string is stretched between two fixed points, it vibrates in specific patterns called standing waves. These vibrations produce harmonics — frequencies at which the string naturally oscillates.

Understanding Harmonics

The first harmonic (fundamental frequency) is the lowest possible frequency.
Higher harmonics occur at integer multiples of this fundamental frequency.
A node is a point of zero displacement; an antinode is a point of maximum displacement.

Which Harmonics Have a Node at the Center?

Only even-numbered harmonics (2nd, 4th, 6th, …) produce a node at the center of the string. This is due to the symmetric nature of these vibrational modes.

If the second harmonic has a frequency f, then harmonics with a node at the center occur at:

f_n = n × f, where n = 1, 2, 3, …

This sequence yields:

2nd harmonic: f
4th harmonic: 2f
6th harmonic: 3f
… and so on

Incorrect Options – Explained

A: (n/2)f – Produces fractional values not consistent with harmonic series.
C: 2nf – Skips valid frequencies, does not align with second harmonic multiples alone.
D: (2n + 1)f – Corresponds to odd harmonics with an antinode at the center, not a node.

Final Answer

The frequencies of vibration that have a node at the center are the integer multiples of the second harmonic frequency:

f_n = n × f (n = 1, 2, 3, …)

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