[boyle] COMPRESSIONAL WAVES IN METALS 295 



reflected at the common junction of the two, while the bars are actually 

 in contact. This is quite contrary to the compressional wave theory. 



1 1 . Damping of the Longitudinal Waves 



The impact of the hammer on the clamped bar excites by "shock" 

 longitudinal waves of period corresponding to that of " free " vibration 

 of the bar. The length of the bar being one-half the wave-length of 

 the fundamental note of the bar. 



There is a question as to how sustained this vibration will be. 

 The damping of the vibration is due to the energy dissipation by the 

 "viscous" resistance of the solid material of the bar, and will depend 

 on the value of the coefficient of "solid viscosity." When the emitted 

 note is of audible pitch it is possible to determine qualitatively by 

 the ear that in the materials steel, brass, dur-aluminium, the vibrations 

 persist for an appreciable time. 



The differential equation of the vibrating motion in the bar, 

 considering the wave travel is that of plane waves, and neglecting all 

 energy dissipation by lateral motions, is 



dt^ p dx^ dx^dt 



where y is the displacement, E Young's modulus, p the density, and 

 C a constant depending on the material and including the coefficient 

 of viscosity.^ 



It should be noticed that by using Young's modulus we are not 

 entirely neglecting all account of lateral motion, since in any deter- 

 mination of E lateral motion always is possible and takes place. 

 Further the experiments of Mr. Lang show that lateral motion, even 

 to frequencies of 50,000 vibrations per second, has less effect than 

 might have been expected on the velocity of the longitudinal waves 

 in the rod, and therefore on the dissipation of energy. 



From the solution of the equation above, the expression for the 

 velocity of the waves is 



E Ok- 



P 4 



2t 

 where k = — , X being the wave-length and m the present mstance 

 \ 



twice the length of the bar. The damping constant, i.e., damping 



Ck'' 

 with regard to time, is — . It is possible, and even likely, that Cwill 



^Rayleigh, Theory of Sound, Vol. II, para. 346. 



