THEORY OF VIBRATIONS 23 



and it appears that the value of x is the projection of OR, the 

 diagonal of the parallelogram determined by OQ l} OQ 2 . 



If Tij = 7i 2 , the two component vibrations have the same period, 

 the angle QiOQ 2 is constant, and the resultant vibration is 

 simple-harmonic of the same period. 



But if Wj, 7*2 are unequal, the angle QiOQ 2 will vary between 

 and 180, and OR will oscillate between the values c^ a 2 . 

 In Lord Kelvin's "tidal clock," the "hands" OQ l} OQ 2 revolve 

 in half a lunar and half a solar day, respectively, and the sides 

 QtR, Q 2 R of the parallelogram are formed of rods jointed to 

 these and to one another. The projection of R then indicates 

 the tide-height due to the superposition of the lunar and solar 

 semidiurnal tides. 



If the periods Sir/r^, 27r/n z are very nearly though not 

 exactly equal, the angle QiOQ 2 will vary very little in the course 

 of a single revolution of OQ l or OQ 2 , and the resultant vibration 

 may be described, in general terms, as a simple vibration whose 

 amplitude fluctuates between the limits c^ a z . The period 

 of a fluctuation is the interval in which one arm OQ l gains four 

 right angles on the other, or STT/^ n 2 ). Inverting, we see 

 that the frequency of the fluctuations is the difference of the 

 frequencies of the two constituent vibrations. We have here 

 the reason for the alternation of " spring " and " neap " tides, 

 according as the phases of the lunar and solar semidiurnal 

 tides agree or are opposed. In acoustics we have the important 

 phenomenon of " beats " between two tones of slightly different 

 pitch. The contrast between the maximum and minimum 

 amplitudes is of course greatest when the amplitudes a lf a^ of 



Fig. 10. 



the primary vibrations are equal. We then have 

 x = o^cos (n^t + ej + a? cos (nj -f e 2 ) 



= 2acos {(7*! - n^) t + (e l - e 2 )} cos {J (n^ + w 2 ) t + J (e l -I- e 2 )}. (3) 

 This may be described, in the same general manner as before, 



