239-240] TRIANGULAR SECTION. 433 



When A/A is small, this reduces to 



c-(ferA)* (v), 



in agreement with Art. 167 (8), since the mean depth is now denoted by \h. 

 When, on the other hand, A/X is moderately large, we have 



(vi). 



The formula (i) indicates now a rapid increase of amplitude towards the 

 sides. We have here, in fact, an instance of * edge- waves,' and the wave- 

 velocity agrees with that obtained by putting /3=45 in Stokes' formula. 



The remaining types of oscillation which are symmetrical with respect to 

 the medial plane y are given by the formula 



< = C (cosh ay cos pz + cos fry cosh az) cos kx . cos (art + c ) ( vii), 



provided o, p, tr are properly determined. This evidently satisfies (ii), and the 

 equation of continuity gives 



a 2 -/3 2 =F (riii). 



The surface-condition, Art. 239 (4), to be satisfied for z=*h, requires 



o- 2 coshaA= ^asinhaA,"! 

 o^cos ph = -gp sin p/i j 



Hence aAtanhaA-fA tanA = (x). 



The values of a, p are determined by (viii) and (x), and the corresponding 

 values of a- are then given by either of the equations (ix). 



If, for a moment, we write 



#=aA, y=ph (xi), 



the roots are given by the intersections of the curve 



#tanh#-fy tany = (xii), 



whose general form can be easily traced, with the hyperbola 



There are an infinite number of real solutions, with values of /3A lying in 

 the second, fourth, sixth, ... quadrants. These give respectively 2, 4, 6, ... 

 longitudinal nodes of the free surface. When h/\ is moderately large, we have 

 tanh ah = 1, nearly, and /3A is (in the simplest mode of this class) a little greater 

 than ITT. The two longitudinal nodes in this case approach very closely to 

 the edges as X is diminished, whilst the wave-velocity becomes practically 

 equal to that of waves of length X on deep water. As a numerical example, 

 assuming /3A = I'l x TT, we find 



aA=10-910, M = 10-772, c=l'0064 x 



The distance of either nodal line from the nearest edge is then '12A. 

 I. 28 



