Question

A string fixed at both ends is vibrating in the lowest mode of vibration for which a point at quarter of its length from one end is a point of maximum displacement. The frequency of vibration in this mode is 100 Hz. What will be the frequency emitted when it vibrates in the next mode such that this point is again a point of maximum displacement?

Answer 1

in the first case point P is an antinode. e.g., string is vibrating in its second harmonic. Let the fundamental frequency be $f_0$.

then, frequency in second harmonic = $2f_0$

100Hz = $2f_0$

$\implies f_0=50Hz$

see 1st figure, here it is clear that if P is antinode then, centre should be node. now string is vibrates again from centre of it( e.g., node) then, wave propagation is shown in 2nd  figure.

so, the point P is  again a point of maximum displacement then frequency will be sixth harmonic or $6f_0$

= 6 × 50 = 300Hz

hence, answer is 300 Hz