230 Mb EARNSHAW, ON FLUID MOTION. 



Fir -at) F'(r-at) r. ^, .- 



= ^ '- + — ^ '- for the diverging wave. 



r- r 



f(r + at) f'{r + at) „ ,, 



and = — '-^— ^ — :; + - — ^ ' for the converging wave. 



r" T 



31. Since in general the motion of each particle (Art. 3) is perpen- 

 dicular to the wave-surface in which it is situated, the motions will be 

 directed to or from focal lines, which may be fixed or moveable: and 

 if the motion of the wave were perpendicular to its front we might 

 always deduce the law of variation of the velocity of a particle; 

 but inasmuch as the direction of a wave's motion entirely depends on 

 its form, and is never in the direction of a normal except at those 

 points where the curvature is a maximum or minimum, or when the 

 waves are spherical, no general law of the velocity can he deduced; but 

 we must first find the value of <p by the method of Art. 29, and 

 then the velocity may be obtained. 



32. When the form of the waves is spherical the law of variation 

 of density may be found. 



For if p be the equilibrium density, and p the density at a 

 point in the wave-surface, 



a'f.-z= — d,<p = - . F' {r — at) for the diverging wave ; 



P 



•••l«S^(j)=fr-^'(^-«^)^ 



f(r-at) 



P = p'.e »- 



For a given part of this wave (as the Jront or the middle,...) 

 r- at is constant, and therefore F' {r - at) is constant = Aa suppose, 



.-. p = p. e^ ; 

 wherefore as the wave travels through the medium, the density of a 

 given part of it varies as 6% which rapidly diminishes. 



