Question

What other harmonics of the string, if any, are in resonance with standing waves in the air column?
(A) $ {9^{th}} $ harmonic of the string with $ {6^{th}} $ harmonic air column
(B) $ {9^{th}} $ harmonic of the string with $ {3^{rd}} $ harmonic of air column
(C) $ {9^{th}} $ harmonic of the string with $ {2^{nd}} $ harmonic of air column
(D) None

Answer 1

Hint: If the particles of the medium vibrate in a direction perpendicular to the direction of propagation of the wave it is called a transverse wave. Waves in a stretched string is a transverse wave. When two harmonic waves of the same amplitude and same frequency travel in opposite directions, the resultant wave pattern appears to be stationary.

Complete Step by step solution
Consider a string tied to one of the prongs of an electrically maintained tuning fork. The other end passes over a smooth frictionless pulley carrying a suitable load. Now, a wave travels to the pulley and returns by reflection. The resultant wave pattern will be stationary. The points of the string having zero displacements are called nodes and those having maximum displacement are called antinodes. Distance between two successive nodes or two successive antinodes is equal to half the wavelength. The string can vibrate in different modes. The first mode is called the fundamental. It has nodes at two ends and the antinode at the centre. Its frequency is called the fundamental frequency. It is the lowest frequency. The frequencies of all modes will be integer times the fundamental frequency.
Consider the air column in a pipe at one end. The air column inside the pipe can vibrate in different modes. Always a node is formed at the closed end and an antinode at the open end. The frequency of all modes will be an odd multiple of the fundamental frequency. The $ {n^{th}} $ mode is the $ {\left( {n - 1} \right)^{th}} $ overtone and $ {\left( {2n - 1} \right)^{th}} $ harmonic.
In closed type pipes, only odd harmonics like $ f,3\;f,5\;f....... $ where $ f $ stands for the frequency.
Therefore the $ {3^{rd}} $ harmonic of the air column will be in resonance with $ {9^{th}} $ the harmonics of the string.
The answer is: Option (B): $ {9^{th}} $ harmonic of the string with $ {3^{rd}} $ harmonic of air column.

Note
When a sound wave is reflected from a rigid boundary or a denser medium, the phase of the wave is reversed. When a sound wave is reflected from a free boundary or a rarer medium, there is no phase change. The reflection of the sound wave, by an obstruction, is called echo.