97. GRATINGS 



A linear grating consists of a set of parallel cylinders or laminas of any shape, usually 

 infinite in number, all alike and similarly oriented, with their centers equally spaced along a 

 straight line. Examples occur in Sections 46 and 47. 



Another simple case that can be treated with elementary functions is the following: 



a nw 



— In cos — , 



77 aV 



[97a] 



X - — In 



277 



1 / 2770 



— cos 



2 \ aY 



+ cosh 



2770 



V = - — tan tan — tanh — , 



^77 \ aV aY j 



[97b] 



[97c 



2G 



aY 



Y 2770 



, V = — sinh ■ 



1G aY 



G = sin^ 



aY 



sinh' 



770 1 / 2770 277<^ \ 



— = — I cosh - cos 



aF 2 \ aV aY I 



[97d,e] 



[97n 



The field represented by these formulas is clearly periodic in the direction of y, 

 which changes by a when <^ is increased by aY ^ while ^, w, and v return to their initial values. 



Assuming such a definition of tan~ that 9?) = and = at a; = y = 0, let for the 

 moment be kept equal to zero. Then, as increases from zero, y remains zero but x decreases, 

 down to - 00 at = aY /2\ there, let the tan~^ in y decrease discontinuously by 77; then, as 

 increases further, z returns along the line y = a to the point (0, a) where = aF, then 

 recedes again along the same line, returns after a further decrease of tan~^ by 77 along y=2c, 

 and so on. That this interpretation of tan~ is in harmony with continuity is seen by assuming 

 to be slightly less than zero; then z recedes a long way toward ;e = - <xj, tan~^ decreases 

 rapidly almost to -w as passes through the value aF/2, then z returns slightly below y^a, 

 swings around (0, a) as increases past the value aY , recedes again slightly above y = a, 

 and so on. Negative corresponds similarly to negative y. 



The real axis of u, is thus transformed into an infinite set of lines parallel to x, spaced 

 a apart and each extending from a; = -~toa;=0. A set of rigid planes may be inserted along 

 these lines, forming a plane grating having plates of infinite width. 



As decreases to a large negative value, 



a; -♦ + 00, M -» 0, -y -* - F, since sinh/cosh -» - 1. 



Thus, if is assumed to be negative, the formulas represent flow past the grating with a 

 velocity F toward negative y at distant points. A few of the streamlines are shown in Figure 

 164. 



242 



