The uses of these two component operations are 

 summarized in the following: 



Table I 



*Where the angle made by the orthogonal and the contours is greater than 

 80°. the graph is still used by crossing the contour interval in a series of steps 

 and progres'^ing a distance R which has some definite relation to J such as 

 1, 0.5, etc. Equivalent a for such cases comprise the t.ble. Here a« (equiva- 

 lent a) = tan-i -i sin Q, but, as a is between 80° and 90° for these cases, sin a 

 may be given the value of 1 and the equation becomes 



aa=tan-' y 



C3) 



Figure 13 is a reproduction of the type I pro- 

 tractor showing the component scales and their 

 functions. The protractor is 14 inches in diame- 

 ter. 



Figure 22 shows the type II protractor. This 

 protractor involves the use of a moveable arm. 

 When this arm is aligned along the direction of 

 level bottom, Aa is read directly along the pointer 



at the appropriate values of 



a and Aa are 



entered on the graphs in their actual dimensions, 

 and it is thus unnecessary to determine their 



numerical values. The factors -= are entered on 



a circular scale. In this way a^ is indicated 

 directly on the graph for a value of R/J. The 

 type II protractor facilitates the operation but is 

 more difficult to construct. 



III. APPLICATION OF THE METHOD 



A. General Preparation 



Contours are drawn upon the chart at such in- 

 tervals that adequately will represent the details 

 of the bottom topography and which are consistent 

 with assumption 1, above. Normally (as a rule 

 of thumb) there wUl be about as many contours 

 required as the period in seconds of the longest 

 period wave to be studied. These contours must 

 extend to a condition of deep water for the longest 

 period wave. That is, to d=2.56T^, where d is 

 the depth of the deepest contour required and T 



is the period of the longest period wave to be 

 studied. A table is then prepared for each wave 

 period to be studied as shown in table II following: 

 Table II 

 Computation for use of protractor in example 

 Period: 10 seconds io=S12feet 



EXPLANATION OF TABLE 



The first column, d, fathom, is a list of the contours of 

 the chart. Should the chart be. in feet, this column natu- 

 rally is eliminated. In practical problems it would be 

 advisable to have contours at 3, 4, and 7 fathom as the 



values of 7 — should not exceed about 0.20. For brevity 



in the following figures, however, fewer contour intervals 

 have been used. 



Column 2, d, feet, is the result of multiplying column 1 

 by the factor 6. If a stage of tide is used other than that 

 of the datum of the chart, a certain constant must be 

 added or subtracted from this column. 



Column 3, 7- ' is the ratio of the depth to the deep water 



wave length for the wave period. In this case (i. e., 10- 

 second period) the wave length is 512 feet and is computed 

 from the equation Lo = b.\2T'^, where T is the period in 

 seconds, and L„ is the deep water wave length in feet. 



Column 4, 7-' is taken from a graph of this function 



such as that in H. O. Report No. 234, "Breakers and Surf, 

 Principles in Forecasting," plate 1, or HE-1 16-265, "Tables 

 of the Functions of djL and d/LJ', or they can be obtained 

 by interpolation in Table 4 in the appendix of this publi- 

 cation. 



Column 5, -j^ ' is the change in the 7- ratio and is written 



between the lines of the depths between which the change 



occurred. 



Column 6, -j- ave, is the average of the 7- ratio between 



any two depths. 



This is also written between the lines. This figure can 



be obtained simply by adding one-half of the A y~ figure 

 to the previous value for 7 — 



Lio 



Column 7, 7 — > is obtained by dividing column 5 by 

 column 6. 



B. General Procedure 



The general method of drawing a refraction dia- 



20 



