340 REPORT — 1889. 



continuous slope, gradually diminishing from liigli- water to a depth about 

 equal to the tide below low- water ; 



3. That the second action was to groove this beach into banks and 

 low-water channels, which attained certain general proportions (plana 

 5 and 7 A and 2 B, and cross sections) ; 



4. That the slope arrived at after 16,000 tides was the same at the 

 higb speed in both models working at corresponding periods, \/2 to 1 

 (section 3 A and 3 B) ; 



5. That in both models the steepness of the actual slope increased as 

 the tidal period diminished (section 5 and 7 A and 2 and 3 B). 



Owing to the grooving of the surface the exact slopes at the various 

 speeds cannot be exactly compared. One way of effecting a comparison 

 has been to take the highest points on each cross section down the slope, 

 and plot them as a longitudinal section, and in the same way to take the 

 lowest points and plot them as another. These are shown in the two 

 longitudinal sections which accompany each plan. 



The increase of the slope with the diminution of the tidal period, both 

 as regards the banks and channels, is thus rendered apparent ; but these 

 sections do not admit of an accurate comparison. 



Some definite and accurate method of comparing these slopes was 

 essential before any definite conclusions could be arrived at respecting the 

 laws of similarity. To meet this the areas above the successive contours 

 have been taken out. These areas respectively divided by the breadth of 

 the plan give the mean distance of the respective contours from the head 

 of the estuary, and the heights of these contours plotted to this mean dis- 

 tance give a definite mean slope of the sand. There are certain minor 

 objections to this method, but it is eminently definite and practical, and 

 admits of great accuracy, the areas being readily taken out with a plani- 

 meter with very great accuracy even for the most complicated contours. 

 The slopes thus taken out are more readily compared if plotted to scales 

 such that the vertical distances between high and low water are all 

 equal, the horizontal scales being determined so that the vertical exagge- 

 ration is the same in all cases. 



The slopes thus taken out from 5 A (Plans I. and IV. ; Plates III. 

 and VI.), 7 A (Plate VIII.), and from 3 B (Plate IX.) are shown in 

 fig. 1, Plate II. They present a great degree of regularity; and it is 

 seen at once that the results of corresponding periods (33 sees, tank A, 

 and 23 sees, tank B) agree very closely. 



In order to compare the slopes with the conditions of kinetic similarity, 

 all that is necessary is to reduce the horizontal distances in the inverse 

 ratio of the periods, when the slopes should become identical. In doing 

 this the horizontal distances have all been reduced to represent (accord- 

 ing to the kinetic law) a 30-feet tide with the natural period 44,400 seconds, 

 namely, the ratio of the lengths of the estuaries made equal to the ratio 

 of the periods multiplied by the square root of the ratio of the heights. 

 The actual rise and fall of the tide in the models being taken : — 



The horizontal and vertical scales for the five experiments as thus 

 reduced to a 30-feet tide are given in Table I. 



Table II. shows the measured height from low water for each of the 

 contours, together with its mean distance from the contour at the height 

 which reduced is 30 feet above low water. Also the corresponding 

 heights of the contours of the 30-feet natural tide, and the corresponding 



