On Water-Waves as Asymmetric Oscillations. 777 



The pressure at the surface is proportional to 



-£(l + a 2 F)f + (1 4- 1 a 2 Ic 2 -3ak cos hx- \a 2 k 2 cos 2kx)-Jj -f ak sin hai S. 



The solution 



£_ (<,(*+% + 7 a W^) CQS (£ + ^ + * ^p^Ay CQS (£ _ ^ 



+ a k e (2k+s)y G0S (2k + s)x-\-ake 2 ^ cos sx — ake {s ^ cos sx 

 gives surface pressure 



(s—a 2 k z ) cos (A + 5) x— a*k 3 cos (/c — s)a 



t x 



and 



H-a&^'+^sin (2/c + s> — ake^ 2k+s ^ s'msx — ake^ s ^ sin sx 



£ = ( e v+,)y + I a 2p^A sin (/. + s y r> _ 9 a W* sill (fc - .?) 



the pressure 



( 5 _ cftf) sin (A + .9) x + a 2 & 3 sin (A - *>- 



Hence the disturbance 

 ^+%{AcosX^ + 5> + Csin(^ + s)4 + ^~ s) ^Bcos(^--5> + Dsin(A-s>} 

 is maintained by the pressure 



{ (s-a*&)A-a*t*B\ cos {Jc + s)x+\ (s-a 2 k*)C+a 2 k*lL)}sm(k + s)x 



+ {(-s-a 2 k*)B-a 2 lc 6 A} cos (A- *>+ { (s-a 2 P)T) +a 2 PC\ sin (*-^*)a. 



The pressure acting alone would tend simply to change the 

 intensity of the disturbance if the two components are pro- 

 portional to the amplitudes of the trains of corresponding- 

 wave-lengths, and if the phases differ by a quarter period 

 from those of the trains ; that is, if 



(s-q 2 F) A-a 2 FB _ _ (s-a 2 P)C + a 2 PD 

 A 



( _ S _ a 2 /c 3) B __ a '2p A ( _ >s __ a 2 P) J) + a 2 PC 



— > i — — . — — — n t'iv 



D B -fc sa 7- 



Hence r? 2 = — s 2 ; 



<7 is therefore always complex, and the free train has no 

 tendency to develop a periodicity of amplitude. 



Phil Mag. S. 6. Vol. 21. No. 126. June 1911. 3 E 



