86 
The N.Z. Journal of Science and Technology. 
[Mar. 
The accompany table gives the values of the amplitude of the first surge, 
or its height above static level, for a wide range of conditions calculated 
by formula (5) for comparison with the more accurate values calculated 
by means of equations (3) or (4). 
Table showing Amplitude, or Height above Static Level, of First 
Surge, as calculated by Equations (3) and (5). 
Constants : A = 1,050 sq. ft. ; a = 50-745 sq. ft. ; r = 1-946 ft. ; friction- 
head ! + kv 2 ) ; k = 0-052. 
V. 
1 
C. 
1c. 
H. 
a. 
I 
fi • 
t. 
at. 
1-2 
6-0 
1-2 
6-0 
6-0 
1-2 
6-0 
6952 
6952 
695-2 
695-2 
695-2 
6952 
6952 
110 
110 
110 
110 
80 
80 
80 
0-052 
0-052 
0-052 
0-052 
0-052 
0-052 
0-052 
0-5 
12-495 
0-1175 
2- 94 
3- 92 
0-88 
21-92 
0-000963 
0-0048 
0-00226 
0-00131 
0-0151 
0-001695 
0-0084 
0-015 
0-015 
0-0475 
0-0475 
0-0475 
0-015 
0-015 
109-2 
125 
3405 
38- 8 
39- 6 
112 
139 
-0-1052 
-0-6 
-0-077 
-0-44 
-0-598 
-0-1898 
-1-175 
V. 
e a t 
fit 
Radians. 
fit 
Degrees. 
Sin fit. 
Cos fit. 
a / at 
A g 
Amplitude. 
Form 3. Form 5. 
1-2 
6-0 
1-2 
6-0 
6-0 
1-2 
6-0 
0-9903 
0-5488 
0-9259 
0-644 
0-55 
0-8272 
0-3081 
1-64 
1-88 
1-6182 
1-8402 
1-8789 
1- 6834 
2- 0839 
93° 41' 
107° 45' 
92° 43' 
105° 26' 
107° 34' 
96° 27' 
119° 24' 
0-9979 
0-9524 
0-9989 
0-9639 
0-9529 
0-9937 
0-8714 
-00642 
-0-3048 
-0-0474 
-0-2661 
-0-3032 
-01123 
-0-4909 
3-23 
3-23 
1-022 
1-022 
1-022 
3-23 
3-23 
3- 51 
12-22 
1-23 
4- 30 
3-87 
3-26 
8-52 
3- 475 
12-42 
1-13 
4- 30 
. 3-91 
3-21 
10-55 
It will be seen that the correspondence is near enough for practical 
purposes, and justifies the assumptions made in the process of simplification. 
It must be borne in mind, however, that equation (5) fails for long tunnels 
and low values of the coefficient C in Chezy’s formula, under which condi¬ 
tions at is liable to exceed unity. The last line in the table of comparison 
is a case in point, and it will be seen that in such cases equation (5) yields 
too high a value. Throughout the table the friction-head has been taken 
at ~ + 0-052v\ 
c r 
As regards the time of the first maximum, the equation fit = tan -1 fi/a is 
simple in form and easily calculated with the aid of trigonometrical tables, 
and there does not seem to be much object in reducing it to its elements 
and clearing it of the trigonometrical function. In any case the equation 
does not admit of a satisfactory reduction which is generally applicable, 
because fi/a is invariably situated between 7r/4 and 3 tt/4, so that expansion 
by Gregory’s series is not permissible. On the whole, the best way is to 
leave the equation in the form given above. 
My thanks are due to Mr. Lawrence Birks and Mr. McLennan for making 
the observations recorded in fig. 2, and to Mr. A. C. Owen for making the 
calculations employed in this paper. 
