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cases.

In this way, then, we obtain for the problem of the surge wave
dealt with by Mr. Johnson, =j j

D2_cZ2 Del



2 D — d

(which checks the author's Equation (1)), and



X — .(6)



.3 = ^,[d + _;. + _x-]



(7)



In these equations, and also in all subsequent formidas, the author's
notation is used, with the addition oi h = D — d. We may also
set. down for this case the approximate relations (see Equations (4a)
and (4a)0 :



^'3 = \| fi' [<



d + — h'j — v (8)

and t'3 = [l + X ^ If] '^^^ "" '' ^^^

As regards solving separately for the height of surge, h, or D — d,
we may write v^ + v for V, and v for Z7, in Flamant's Equation (2),
combine with Equation (9), and then solve the resulting quadratic ;



if V ^l — b(
^l 9

h = ^TV 16cZ2-f 8 kd-i- dk"^ 4- 3 fc— 4f?] (10)



whence, if v ^j — be denoted by k, there results



as a fair approximation when h is small compared with d.



DISCUSSION ON SDEGES IN AN OPEN CANAL 133

We have, also, from Equation (3&) Mr.

Church.

vd = v^h (10a)

In his report* to the French Academy of Sciences on an experi-
mental investigation of the propagation of waves in open channels,
Bazin gives the data and results of Bidone (1824) in the same field
as well as Darcy's (1856). Bidone's experiments were performed
on a very small scale, his channel being only about 2 ft. wide and
about 40 ft. long in which to create and observe the wave phenomena;
whereas Darcy and Bazin made use, not only of an experimental
channel about 6.5 ft. wide and more than 1 000 ft. long, with depths
of water of 2 ft. and less, but also of a straight reach of a navigation
canal, some 3 000 ft. long and 30 ft. wide, with depths as great as 3 ft.,
as well as of a smaller basin 20 ft. wide.

Such being the fairly large scale on which the Darcy experiments
were made, it has always seemed remarkable to the writer that Bidone's
results should be quoted so frequently in American books on hydraulics,
with little or no mention of Darcy's (in this field of wave motion,
standing waves, etc.).

According to this report of Bazin's, Bidone derived the following
relations from his experiments with surges going up stream and
caused by the abrupt closing of a transverse gate in moving water,
channel rectangular, viz. :

-^^>4-Kv^> ^nH-


^








■ ' ^^Br;


T-riiO'VlTVCM-' i ^'V - ^^ ^


:'




X


- 'S






:^^




V-


-


^'-i-L>-VU>-< ^T Pf^ . .' '^


\i /


- J ■


M-mM*




h


KHVhJ


:it^4^iuH-^


[f • , •■ n




;;-;




tmt




V


■>H^


^Jw^


ft; 'J


Ij-.


•T "




Jr-f^w'X. fr




^■


~" Vv"


'g:Hpipt-i;:_' J


■Lr


3:^_-_^^^M




i"


i >««" -


n -d )'


■f -


''tTlX-w JZ/^ .-^.Lj^^^^^H




*C


ifcr* «




^ ^KIVV^^^^^^^^H


^^ ' " ^oL. ^£)L



Fig. 7. — Pinus strobus (White Pine), About 100 Diameters.

Cross-Section at Annual, Ring, b (Old Cells, Upper

Part with Thick Waui^s, New Cells Below with

Thin Walls), (a) Vessels or Ducts, (6)

Longitudinal Cells (Old), (c) LiOnqi-

tudinal Cells (New), (d) Medul-

LAKY Rays.




Fig. 8. — Pinus strobus (White Pine), About 100 Diameters,

Radial Section at Anntjal Ring Showing (d)

Medullary Rays, and (c) Pitted Tra-

CHEiDS OR Longitudinal Cells.





'I

■li




Pig. 9. — Pinus strobiis (White Pine), About 100 Diameters,
Longitudinal Tangential Section Showing (d) Com-
pound Medullary Rays, and (c) Pitted Tracheids
OR Longitudinal Pointed Cells. The
Longitudinal Cells Extend Beyond
the Limits of the Figure.




Fig. 10. — Quercus suber (Cork Oak) Bark, About 100 Dlvm-
ETERs. Cross-Section at Annual Ring Showing
(f) Annual Ring, (



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