65. LAMINA BETWEEN WALLS 



sinh w = g sinh 2, g real and g > 1. [65a] 



sinh cf) cos vjj - g sinh x cos y, cosh ^ sin ip = g cosh x sin y, [65b, c] 



from w = (fi + iip, 2 = x + iy and hyperbolic formulas in Section 32. The functions and ip 

 are many-valued. It is readily seen that continuous values can be chosen so as to satisfy 

 the following description. 



has the sign of x and i/* that of y; if y = 0, t/i = 0. Thus the x- and y-axes represent 

 planes of symmetry. As a; -» <», -> » also; furthermore, coth cp -> coth a; -» 1, so that, since 

 from [65b, c] coth tan \p = coth x tan y, ip -* y. For all values of a;, = y on the lines y = 0, 

 y = 7t/2, y = 7)-; then one of Equations [65b, c] is satisfied automatically and the other fixes 

 in terms of x. Furthermore, on the y-axis, wherever sin y > 1/g, in [65c], g cosh x sin y > 1 

 and this equation can be satisfied only if > 0; then, to make a; = 0, cos 0=0. In particu- 

 lar, cos = and = 7r/2 on the segment defined as follows: 



cos ' — = y =3 _ + cos * — . 

 2 g 2 g 



" -1 ^ < < " 

 cos ' — = V ^ — 



On this segment of the y-axis is discontinuous, since here cosh = ^ sin y > 1 but must 

 change sign with x as the y-axis is crossed. 



For a physically possible case, a plane lamina must be inserted along the segment of 

 the y-axis in question; and walls may also be inserted along the lines y = and y = 77. Then 

 the flow is represented between these walls, with unit velocity at infinity where -» y, past 

 a lamina of width L - 2 cos~^ (1/^) placed perpendicular to the walls and midway between 

 them; see Figure 98a. 



1} 



2 cos 



-1 1 



■1} 



2cos-'l 

 ff 



Figure 98 — (a) Lamina between walls, or (b) a grating of laminas; 

 (c) slot in a partition between walls. 



On the walls, = Vi sinh = ^ sinh x. [65d] 



On the median line, y = Tr/2, ifj = n-/2, cosh <^ = g cosh x. [65e] 



On the lamina, = n/2, cosh = (7 sin y. [65f] 



146 



