22 DYNAMICAL THEOKY OF SOUND 



It will be observed that the amplitudes of the various terms 

 are not proportional to those of the corresponding terms in the 

 value of Q, owing to the difference in the denominators. 



This is an illustration of a remark made in 1 that the 

 simple-harmonic type is the only one which is unaltered in 

 character when it is transmitted, the character of the composite 

 vibration represented by (8) being different from that of the 

 generating force. In particular if one of the imposed speeds 

 p lt p z , ... be nearly coincident with the natural speed n, the 

 corresponding element in the forced vibration may greatly 

 predominate over the rest. This is the theory of selective 

 "resonance," so far as it is possible to develop it without 

 reference to dissipative forces. 



10. Superposition of Simple Vibrations. 



The superposition of simple-harmonic motions in the same 

 straight line has many important applications. For instance, 

 the height of the tide at any station is the algebraic sum of a 

 number of simple-harmonic com- 

 ponents, the most considerable 

 (at many stations) being those 

 whose periods are half a lunar 

 and half a solar day, respectively. 



The composition of two 

 simple vibrations may be illus- 

 trated by the geometrical 

 method of Fig. 2. If two 

 points Q lt Q 2 describe concentric 

 circles with the angular velo- 

 cities TH, n^ their projections 



on a fixed diameter will execute simple-harmonic vibrations 

 of the forms 



#! = aj cos (nj + e^, # 2 = a 2 cos (n + e 2 ), (1) 



where Oj , a 2 are the radii of the two circles, and ej , e 2 are the 

 initial inclinations of the radii OQi, OQ 2 to the axis of x. The 

 result of the superposition is 



(2) 



