question_answer 58)

                  Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops and 4 loops, die frequencies are in the ratio 1 : 2 : 3 : 4.

Answer:

                  A string under a given tension, can be made to vibrate in more than one segment giving rise to various modes of vibration. But due to boundary condition, the string can oscillate only in special patterns which are called normal modes*. The frequencies of vibration of a string for all these modes of vibration are different as is clear from the following discussion. In fact, a string is said to vibrate in a normal mode if it vibrates according to the equation                 \[y=2A\sin kx\cos \omega t\]               ? (i)                 From eqn. (i), for a point to be a node,                 \[x=n\lambda /2,\] where \[n=0,\,1,\,2,\,3\]    ? (ii)                 As we have already said, the two ends to which the string (of length L) is attached are nodes. If we choose one of the ends as \[x=0\], then the other is \[x=L\]. For this end \[(x=L)\] to be a node, from eqn. (ii)                 \[L=n\lambda /2\] or \[\lambda =\frac{2L}{n},\] where \[n=1,\,2,\,3,\] ?(iii)                 Clearly, \[{{\lambda }_{1}}=2L,\,{{\lambda }_{2}}=L,\,{{\lambda }_{3}}=\frac{2L}{3}.....\]             The frequencies corresponding to these wavelength are given by                 \[v=\frac{\upsilon }{\lambda }=\frac{\upsilon }{2L/n}=n\left( \frac{\upsilon }{2L} \right),\] where \[n=1,\,2\,.....\] ? (iv)                 where \[\upsilon \] is the speed of travelling waves on the string.                 (a) If \[n=1,\,v=\frac{\upsilon }{2L}\] and it is called the fundamental frequency or first harmonic**. The corresponding mode is called fundamental mode of vibration.                 In this case, the string vibrates in one segment.                 (b) If \[n=2,\,{{v}_{1}}=2\left( \frac{\upsilon }{2L} \right)=\frac{\upsilon }{L}\] and it is called second harmonic or first overtone. It is obvious that \[{{v}_{1}}=2v\]. In this case, the string vibrates in two segments.                 (c) If \[n=3,\,{{v}_{3}}=3\left( \frac{\upsilon }{2L} \right)=\frac{3\upsilon }{2L}\] and it is called third harmonic or second overtone. Clearly, \[{{v}_{3}}=3{{v}_{1}}\]. In this case, the string vibrates in three segments.                 (d) If \[n=4,\,{{v}_{4}}=4{{v}_{1}}\]             All these modes of vibrations with their corresponding wavelengths are shown in (a), (b) and (c), respectively.                                 Obviously,  \[{{v}_{1}}:{{v}_{2}}:{{v}_{3}}:{{v}_{4}}=1 &  & :2 & :3:4\]