Standing waves in strings:In musical instruments like sitar, violin, etc. sound is produced due to the vibrations of the stretched strings. Here, we shall discuss the different modes of vibrations of a string which is rigidly fixed at both ends.
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When a string under tension is set into vibration, a transverse progressive wave moves towards the end of the wire and gets reflected. Thus stationary waves are formed.The sonometer consists of a hollow sounding box about a metre long. One end of a thin metallic wire of uniform cross-section is passed over a pulley and attached to a weight hanger .
The wire is stretched over two knife edges P and Q by adding sufficient weights on the hanger. The distance between the two knife edgees can be adjusted to change the vibrating length of the wire.A transversary stationary wave is set up in the wire. Since the ends are fixed, nodes are formed at P and Q, and antinode is formed in the middle.
The length of the vibrating segment is l = lambda/2 . lambda =21. If n is the frequency of vibrating segment, thenn = v/lambda = v/(2l) .We know that v="sqrt(T/m)" where T is the tension and m is the mass per unit length of the wire.Therefore, n = 1/(2l) *sqrt(T/m)Modes of vibration of stretched stringi. Fundamental frequency:If a wire is stretched between two points, a transverse wave travels along the wire and is reflected at the fixed end.
A transverse stationary wave is thus formed .When a wire of AB of length l is made to vibrate in one segment then l lambda 1/2. Thus we get the lowest frequency called fundamental frequency n1 = v/(lambda1) .Thus we get fundamental speed as n*lambda.overtones in stretched stringIf the wire AB is made to vibrate in two segments, then l =(lambda1+lambda2)/2The laws of transverse vibration in a stretched strings are i. the law of length ii. law of tension and iii. the law of massi. For a given wire where m is a constant, when T is also a constant, the fundamental frequency of vibration is inverly proportional to the vibrating length or nl = constant.ii.
For constant l and m, the funcdamental frequency is directgly proportional to the square root of the tension i.e n is proportional to tension of the wire.iii. For constant l and T,the fundamental frequency varies inversely as the square root of the mass per unit length of the wire."&amp;amp;amp;quot;