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XXII. On the Effect of Internal Friction on Resonance. 

 By J, Hopkinson, D.Sc, B.A* 



AS a typical case which may be taken as illustrating the 

 nature of the phenomena in more complex cases, let us 

 cousider the motion of a strings of a column of air, or an elastic 

 rod vibrating longitudinally, one extremity being fixed, whilst 

 the other is acted on so that its motion is expressed by a simple 

 harmonic function of the time. 



Let / be the length of the string, a the velocity with which a 

 wave is transmitted along it, f the displacement of a point of the 

 string distant x from the fixed extremity at the time t. In the 

 hypothetical case, in which there is no friction, no resistance of 

 a surrounding medium, and the displacements are indefinitely 

 small, the equation of motion is 



d^_ 9 d*£ 



= a 



dt 2 dx 2 



(1) 



with the conditions that at the extremities f = when x = 0, 

 and f = Asin?^ when x=l, also that at some epoch f shall be 

 a specified function of x. 



It we start with the string straight and at rest, we have the 

 condition £ = for aJl values of x from zero to very near / when 

 t = Q, and we readily find 



.. A . . .«#-«. pirx . pirat /Q s 



t= -smnt.sm \-A\j n smV • sln — r ' • • W 



. nl a p I I 



2nal 



sin — 

 a 



where C = ( — 1) 



2 -V 



nl 

 When — is very nearly a multiple of it (i. e. when the note 



sounded by the forcing vibration at the extremity is almost the 

 same as one of the natural notes of the string), we have two notes 

 sounded with intensity, viz. one the same as the forcing vibra- 

 tion, the other native to the string. That this is the case may 

 be readily seen with a two-stringed monochord, the strings being 

 nearlv in unison : one string being sounded, the motion of the 

 other is seen by the eye to be intermittent, the period of varia- 

 tion being the same as that of the beats of the two strings 



nl 

 sounded to°;ether. But should — be an exact multiple of it, 



D a r 



two terms in the value of f become infinite, and our whole me- 

 thod of solution is invalid. A somewhat similar difficulty, of 



* Communicated by the Author. 



