A NSV,' :,ETKOD FOR THE GRAPHICAL COHSi'RUCriON 

 OF VrAVS REFRACTION DIAGMMS 



by 



T, Saville, Jr. and K, Kaplan 

 Staff, Baach Erosion Board 



In connection vath the preparation of a design manual for shore 

 otructu::^5 presently bsing praparad by the Beach Erosion Bo^j'd, a re- 

 axarni nation of the nisthcds and validity of the various means of obtainin;' 

 refraction ch-aracteristics becaine necessary. This reexamination has shovm 

 that the ei-ror involved in the use of the crestless method for draTrinr; 

 orthogonals devised by Isaacs (1) (2) and in general use throughout the Uiiited 

 States, may become quite significant when large angles of inoidencs or 

 large contour intervals are used — in extreraa cases errors in axcess of 

 5 to 6 degrees in direction and 10 to 15;t in refraction coefficient being 

 possible. This error is introduced into the scales devised by Isaacs by 

 the assumption that the change in angle, A<x > is small in comparison to 

 the angle of incidence, q- , so that sin ( a - A<x ) — sin ex . 



Ho-»raver, this assumption is not necessary to the derivation of the 

 equation expressing the angle change, and more accurate equations may be 

 derived in a manner sirail??r to that of Isaacs, The basic assumptionfs for 

 this extension of t're original theory remain the same: namely that the 

 velocity varies linearly between contours, that the radius of curvatuire 

 of the orthogonal betv/ean contours is constant (a circular arc), and that /Soc 

 is small so ttet tan Aa ~ sin 4a =. Ao:. 



Referring to figure la, di and d2, Cl and 02^ are the depths and vel- 

 ocities respectively at contours 1 and 2.'- If d^ is greater than d2, then 

 Q2> snd in moving from A to B an orthogonal will follow a circular arc 

 tangent at A and B to OA and OB respectively. Similarlj'* another orthogonal 

 a differential distance d away from A3 is tangent at A' and B' to CA' and 03' 

 respectively. Since d is a differential distance, AA' — 00'^^^ BB's^d- 

 If R is tha distance of wave advance during a tiina interval t, then Ri-^-ZiB-AB , 

 and Rg =■ A'B* . By construction the angle B" B B'- = Aa , and if Aa is 

 considered less than 13 degrees, (for two place accuracy), tan ^or=^5//7^<:rtr^cc, 

 tlien . 



A re - ian Aa = ^ '^"- = E2-S3- /-, ^ 



33' cl ^^> 



Since the velocity is assumed to vary lin'33.rly between contours, the average 

 velocity batv;een A and B is 



C'= ^-^ (2) 



and if t is the time interval required for the wave front to move fVom A A' 



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