82 The N.Z. Journal of Science and Technology. [Mar. 
when t — o, h — — KY = — H, which is the static head corresponding to 
a velocity V. 
This is an equation of an oscillatory movement having a periodic time 
l 
of 27 r/fi, represented by - • Y • e 
Q 
at 
sin fit, upon which is super¬ 
imposed another smaller oscillatory movement represented by KYe af cos fit, 
having the same frequency but differing by 7r/2 in phase from the former, 
together with a third ripple K . Y . e at . ^ • sin fit, which is also an oscil- 
P 
latory movement of the same character as the other two and in phase 
with the first or larger oscillation, all three movements being subject to 
an attenuation or damping controlled by the term e at , the value of a being 
negative. 
Maxima and minima occur when v — o, and the time of the first 
maximum is obtained by placing v = o in equation (2), as a result of which 
it will be seen that the maximum occurs when 
fit = tan -1 - 
a 
The positive maxima will occur at intervals of %cffi thereafter. The first 
negative maximum will occur at fit -\- ir/fi, and will recur at intervals of 
Zv/fi. 
In the foregoing equation the third term is for practical purposes 
negligible compared with the first two, also d 2 /fi is small compared with fi, 
and if these be ignored the equation reduces to the following terms :— 
h = e at . V . fi . sin fit — K . V . cos fitj 
If now we substitute for KY its value H and for fi its approximate 
value \/— . - for the second term, in which the exact value of fi is 
A l 
negligible compared with the first, we get 
h = e at (YV a . I ./A g . sin fit — H cos fit) .(3) 
and by a well-known transformation we obtain 
h = e at ^J% • - • Y 2 + H 2 sin (fit — <f>) 
A g 
H 
where 
<£ = tan- 1 / a l 
V A ' g 
The amplitude of the first or maximum surge is obtained by substituting 
for fit its corresponding value tan -1 fi/a, and for t a value derived from 
the relation fit — tan -1 fi/a. This is by far the most satisfactory way of 
ascertaining the amplitude, and its calculation is easily obtained with the 
aid of tables of the values of e~ x . 
If, however, an approximate equation which is clear of the exponential 
be desired, such an equation may be derived from (4), provided that at 
