239-240] TRIANGULAR SECTION. 433 



When k/\ is small, this reduces to 



&amp;lt;? = (fe^)* (v), 



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



When, on the other hand, h/\ 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--=Q are given by the formula 



&amp;lt; = C (cosh ay cos fjz + cos /3y cosh az) cos kx . cos (trt+c) ( vii), 



provided a, /3, a- are properly determined. This evidently satisfies (ii), and the 

 equation of continuity gives 



a 2 -/3 2 = P (viii). 



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



o- 2 coshaA= &amp;lt;7asinhaA,| 

 a 2 cos j3h = gft sin /3A j &quot; 



Hence aA tanh aA+A tan A = (x). 



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

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



If, for a moment, we write 



x=ah, y = fih (xi), 



the roots are given by the intersections of the curve 



x tanh x -f y tan y (xii), 



whose general form can be easily traced, with the hyperbola 



(xiii). 



There are an infinite number of real solutions, with values of fth 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 TT. 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 A= I l x TT, we find 



aA = 10-910, M = 1O772, c= 1-0064 x ( 

 The distance of either nodal line from the nearest edge is then -12^. 



L, 28 



