Mr. J. M c Cowan on the Solitary Wave. 47 



To determine the form oif{z + ix) we note that (i.) as the 

 wave is to be solitary it must be non-periodic as regards a?, 

 (ii.) it must be finite and continuous throughout the liquid 

 including the bounding surfaces, (iii.) when x is infinite 

 ( -f or — ) it must vanish or have a finite value independent 

 of z or a, and (iv.) if we take sfr = at the bottom, it must be 

 an odd function of (z + ix). 



Thus we obtain 



f{z + ix)=^a 2i+1 tan 2 ^ l im(z + Lx), ... (2) 



with the restriction mz < ir. 



The condition for a free surface only remains to be satisfied, 

 and for this the coefficients a l9 a s . . . may be determined by 

 the method of successive approximation, but for the present 

 at least we shall content ourselves with examining to what 

 degree of accuracy this condition can be satisfied by taking 

 the first term only. 



2. The First Approximation. 

 Take, then, 



^r + *<£= — \J(z + ix) + Tiaism\m(z + ix:), . . (3) 

 which gives 



r tt ,tt sin. mz ,.v 



>ir= — Uz+Ua- : — ... (4) 



cos mz + cosh, mat 



and 



, TT TT ainhmx ._, N 



<£=— ILi;+Ua — — r . . . . (5) 



cos mz + cosn mx v ' 



Let q be the whole velocity and u, w its components parallel 

 to the axes of x and z respectively, then 



, =u + iw= — U{1 — hnasec 2 hn(z-{-ix)\ : . (6) 



a.z + LX 



which gives 



XT f , 1 + cos mz cosh ?7i# 1 



M =-U4 l-flW 7 — : r r 2 l r . . . (7) 



(cos mz + cosn mx) z j v y 



sin w?2. sinh m,# 

 y= — U \ I— ma— 



and 



TT I . sin mz. sinh m.# I 

 = -U 1-ma- - r r- 2 l. . . (8) 



(COS m£ ■+• COSh m#) J J v ' 



■ _ T _ f. m 2 a 2 — 2ma(l + cos ???£ cosh m#) 1 ■;• 



<7 2 = IM 4 H ; : ; ^ I . (9) 



2 (cos »*c + cosn mxy v y 



