40 



MECHANICAL VIBRATING SYSTEMS 



C. Bar Clamped at Both Ends. — Consider a bar rigidly clamped at both 

 ends. The same tones are obtained as in the case of the perfectly free 

 bar. 



D. Bar Supported at Both Ends. — ■ Consider a bar supported on knife 

 edges at the two edges at the two ends (Fig. 3.2). The fundamental fre- 

 quency is given by 



^' ~ 2/W P 



3.4 



where / = length of the bar, in centimeters. 



All the other quantities are the same as in equation 3.2. 

 The overtones are 



/2 = 4/x 



/3 = 9/, 



Ji = 16/1 etc. 



The nodes are equidistant as in case of the string. 



3.4. Circular Membrane ^'^'8' ^-i". — The ideal membrane is assumed to 

 be flexible, uniform and very thin in cross section, and stretched in all direc- 

 tions by a force which is not affected by motion of the membrane. Com- 

 plete theoretical analyses have been made of square, rectangular and 



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

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

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

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

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

 New York. 



