Oirves showing tnis vai'iatdon oi zJo" with -, — snd -rfor cic=85 are also 



'-3V. -I 



shown in the overlay in figure 2a, 



When either the wave directiorx. or the direction in which the orthogonal 

 is being draiiro, is from shallow to deeper water, the direction of turriing is 

 opposite and equation (15) becoraes 



Z/<^ sin a' ... 



/ia = -i-- (16) 



CiLrves Bhocring this solution ajre included in the overlay in figure 2b. 



These two overlays may be used either as ordinary graphs, or in ccn- 

 junction TJ-ith a di'afting machine in the same aethcd. as the original 

 Isaacs' overlay. In the past, the turning point for the change in angle 

 {Aoc ) has been assumed to be at the aiid-contour <, Hovvever;, the correct- 

 orthogonal (circular arc) is not tangent to two straight lines that meet 

 at the raid -contour 5 but to two lines that meet some small distance above 

 (toward deeper water from) the irdd-contoui^ (see figure 3)« The ratio 

 of the dist.ance of this point from the deeper contour to the full distance 

 between contours is given by 



z ■^ 



■^3/7 oCf — -Lay) oc^ 

 If, for example J cx,=(oO° and a^^S'S" 3 "this ratio is 0«48 instead of 0,5., 

 and if a> = 6(9t <y^-= SO' s "t-his ratio is 0»UI instead of 0.5. The magnitude 

 of the error in the distance along the contours is given by the difference 

 between the tangent of the average angle (the exact solution) and the 

 aire rage tangent of the two angles, (i.e. tan SlzL^^ _ i£nJ[jL±:k±2' ) 



The accuracy is improved considerably by taking the turning point not at 

 the mid -contour r, but at a point toward deeper -vrater^ such that,, in Figixi^e 

 3. the distance AO = 03 . This point may be estimated by eye with sufficient 

 accuracy, 



A direct construction of Ri^may be made with a drafting iriacliine by 

 changing the direction of the cithogonal by 4^ once at the first contour 

 and again at the second contour, rather than once by the fuH value of 

 4a. at the mid-contour . The angles being correct, this constr^lGtion 

 "Kdll define the position and direction of the orthogonal at the second con- 

 tour. 



Hcr,7ever, it is n-^t. felt that the increase in accuracy whd.ch would re- 

 sult from this procedure warrants a chiinge xsx the standardized method of 

 constructing orthogonals using one angle change. Barely are contours 

 defined accurately, and the modified procedure suggest/ed,, in requiring two 

 angle changes instead of one, complicates the mechanics of orthogonal con- 

 struction. It is thought that sufficient accuracy is obtained by naking 

 the one angle change at a point estimated by eye such that the distances 

 OA and OB are equal, 



29 



