29.2 Prof. Challis on the Hydro dynamical Theory of Vibrations. 



fi being put in the place of — . Also since to the approxi- 



mation of the first order the condensation (a) on the axis is 



1 d<b 



5 • -rrt we have 



a* at 



Hence 



a— -.2, . /i sin - - \z— Kat + c). 

 a a 



V= -. <T. 

 K 



This is the general relation between the velocity and density 

 at any points of the axis for waves propagated in the positive 

 direction of z. The relation is the same between the velocity 

 parallel to the axis and the condensation at points not on the 

 axis, the velocity at such points being V/j and the condensation 

 of. These velocities coexist with transverse velocities expressed 



b y*i and *f- 



34. Hitherto the reasoning has had reference to the motion 

 relative to a single axis of free rriotion. An important remark is 

 now to be made previous to taking the next step towards the 

 solution of the problem of rectilinear motion perpendicular to a 

 plane. Since the law of longitudinal and transverse vibration 

 relative to an axis was demonstrated generally, and prior to any 

 supposed case of disturbance, it must receive application, what- 

 ever be the disturbance. Hence, as a necessary consequence, 

 any motion arbitrarily impressed on the fluid must be regarded 

 as composed of motions subject to that law. This principle being 

 admitted, and the motion in the instance before us being wholly 

 parallel to a given direction, it follows that the component mo- 

 tions must be so arranged that the transverse motions destroy 

 each other. This will be the case if the axes be all parallel to 

 the given direction and be unlimited in number, and if the phases 

 of vibration be the same for all. For, the disturbing plane being 

 assumed to be of unlimited extent, there will thus be no reason 

 for transverse motion in one direction rather than another, and 

 there will consequently be no transverse motion. Now, let the 

 disturbance of the fluid be conceived to be such that the velocity 

 at the given distance z l from the origin is the arbitrary quantity 

 ¥(/cat) at the time t. Since it may be assumed, in accordance 

 with a recognized analytical principle, that the successive values 

 of this function at the given position can be expressed as nearly 



as we please by S .//-sin — (z l — tcat-{-c), the number of the 



terms under the sign 2, and the values of fi } \, and c for each 



