ON THE HYDRODYNAMICAL THEORY OF SEICHES. 637 
The boundary conditions are, that & vanish at the ends, and that E=&,C=C at O. 
These lead readily to the following :— 
we=aA weeny We oe hs in n(t—7); 
(nF a 28 tage 
ea ep 
AERO ent 
And the period equation is 
a2{Y,(n8)Jo(na) - J,(nB)¥ (na) } {V4 (mB")J,(na’) ~ J,(mB')Y (na!) } 
+.a2{¥,(mB")Io(na’) — Jy(mB')¥ 4(na")}{ V,(nB)JI,(na) ~ Jy(nB)Y,(na) = 0 
UNSYMMETRIC LAKE SHELVING AT Boru ENDs. 
§ 47. 
es DS 6 p 
Fie. 9. 
In this case B=0, 8’ =0. 
Therefore, since L J,(w)/Y,(w)=0, the equations of § 46 reduce to 
w=0 
WE = at sin n(t —7), 
v_ 2A Jy(w) 
“h J, (na) 
J,(w’) 
we oe ,(na’) 
C= _ 207A Jy(w) 
~~ sin (t— 7). 
h J, (na’) 
sin n(t — 7) ; 
sin (t —7) ; 
: A) re). 
Period equation 
a?J,(na)J,(na’) + a'23,,(na’)J,(na) =0. 
(79). 
The nodes are given by * 
Jo(w) =0 in the part O A; 
AC Ol nee coe eh els 
where for the »-nodal seiche 
w=na, wW=n,a. 
* Roots of the Bessel Functions—In what follows I shall denote the positive roots of the equation Jo(z)~0 by j,, 
. aa .; and the positive roots of J, (z)=0 (excluding the zero root j,=0) by jo,Jg,Jg) +++ ++ +++ So that 
we have approximately j,=2-405 an 832 , j,=5'°520 , j,=7°016 , j; =8'654 ,j,= 10173, 7,=11° 792 )Jg=13'323 , jy= 
14°931 ,j,9>=16°471, etc. 
For large values of % , Jn=(2n+1)x/4, approximately : ¢.g. this formula gives j,,=18°064 instead of the correct 
Value 18°071 ; so that the error after n=11 is less than ‘1 ae 
