Manchester Memoirs, Vol. xlii. (1898), No. 9. 7 



may be important. It is easily seen that the quadratic (27) 

 has in all cases two positive roots. If (as we shall generally 

 suppose) c^ is less than £'/^, one root will lie between o 

 and c^, and the other between EJq and Bj^. The former 

 determines the velocity of waves in the liquid, as diminished 

 by the elastic yielding of the tube ; the latter determines 

 the velocity of longitudinal waves in the tube-wall, as 

 increased by the reaction of the enclosed fluid. If (on the 

 other hand) c^ were greater than £"/o, one root of (27) 

 would lie between o and ^/(>, and the second root between 

 c^ and Bl^. An extreme instance of this is afforded by 

 water in an india-rubber tube ; the fraction 2aK.lhB is then 

 large, and the two values of c are found to be 



{hEliaQof and {BIq)\ 

 approximately. The former of these agrees with the 

 velocity of " pulse-waves " as determined theoretically by 

 Resal.* 



If o- were zero for the substance of the tube, the 

 constants j5 and^ would be identical, and one solution 

 of (27) would be c^ = Bl^, it being otherwise evident that 

 the longitudinal vibrations of the tube would be entirely 

 unaffected by the fluid. Moreover, the difference between 

 B and E is for most solids a small fraction of either. 

 Hence the velocity of longitudinal waves in the tube, 

 which lies (as we have seen) between the limits (^/p)^ and 

 (^/p)*, will not as a rule be more than slightly affected by 

 the liquid. On the other hand, since the product of the 

 roots of (27) is equal to 



__cIEIp_ 



I + 2aKlhB'* 



it appears that the lower root, which determines the wave- 

 velocity in the fluid, must lie between the limits 



vh£fhB-''' ^"d -j-^-^i-^..^^ . (,81. 



* Liouvilky 1876 (quoted by Korteweg). 



