STRINGS 



37 



where T = tension, in dynes, 



m = mass per unit length, in grams, 

 / = length of the string, in centimeters. 

 The shape of the vibration of a string is sinusoidal. In addition to the 

 fundamental, other modes of vibration may occur, the frequencies being 

 2, 3, 4, 5, etc., times the fundamental. The first few modes of vibration 

 of a string are shown in Fig. 3.1. The points which are at rest are termed 



SECOND OVERTONE 

 L ^J^ L, 



FOURTH - OVERTONE 



THIRD HARMONIC 

 FOURTH HARMONIC 

 FIFTH HARMONIC 



FIFTH OVERTONE 



SIXTH HARMONIC 



Fig. 3.1. Modes of vibration of a stretched string. The nodes and loops are indicated by 



N and L. 



nodes and are marked A^. The points between the nodes where the ampli- 

 tude is a maximum are termed antinodes or loops and are marked L. 



The above example is the simplest form of vibration of a string. A few 

 of theproblems which have been considered by different investigators ^•^•^•^■^ 

 are as follows: nonuniform strings, loaded strings, stiff strings, nonrigid 



^ Rayleigh, " Theory of Sound," Macmillan and Co., London, 

 '^ Crandall, " Theory of Vibrating Systems and Sound," D. Van Nostrand Co., 

 New York. 



' Wood, " A Text Book of Sound," Bell and Sons, London. 



* Morse, " Vibration and Sound," McGraw Hill Book Co., New York. 



^ Lamb, " Dynamical Theory of Sound," E. Arnold, London, 



