When a sudden increase in the discharge occurs in an open
channel, a surge wave is formed. This body of water appears to
move along the initial surface. Depending on the discharge this
surge can be undular, breaking undular, or steep fronted as the
discharge increases.

A theoretical expression has been derived for the undular
form, but no allowances have been made in the theory for the effect
of forces that cause the waves to break. To simplify matters the
surge was assumed to have been arrested, by superimposing on it
a velocity equal and opposite to that of the mean velocity of the
head of the surge. Then it appears as an undular hydraulic jump,
with moving boundaries. The expression for the profile is
derived for permanent flow,^if solved alone, with no allowance for
friction gives a solitary wave profile. Hence two further expressions have been derived for the changes in energy and momentum.
After simplifying and assuming that the channel bed is horizontal
and the channel cross section is rectangular, the resulting non
dimensional equations are:-

1/6 (dY/dX)² = EY² - Y/2³ +1/2 - $Y (41)

dY/dX is the slope of the water surface. E refers to energy, $ to momentum, and Y to depth (y = ycY. yc - critical depth).

d$/dX = g/E²[1/Y₀ - 1/Y]²[1 + 2y/l] (71) C - coefficient of friction. Suffix ₀ - initial conditions. l - width of channel. y - depth of water.

dE/dX = g/YC²[1/Y₀ - 1/Y]²[1 + 2y/l] (72)

With the small channel used in the experiments, allowances had to be made for wall friction (1 + 2Y/l).

Benjamin and Lighthill show that this undular form of surge
is not possible unless losses in energy and momentum occur.

The waves are termed 'cnoidal' waves because the profile
can be represented, to a very close approximation by, the graph
of the square of the Jacobean elliptic function cn x. The term
'cnoidal' was coined by Korteweg and de Vries.

The Equations 41, 71 and 72 were then obtained in a form
suitable for computation, and a number of numerical examples were
solved. The resulting profiles were checked by experiment, and
the agreement between the results was considered to be good.

It is believed that if the calculations were made for greater
initial depths then those possible in the model channel, that there
would be greater agreement with recorded values. This is because
of the uncertainty of the determination of the value of the
coefficient of friction at low Reynolds numbers. At high values
of R the friction coefficient can be determined more accurately.
It is thought that probably the values derived from the Bazin,
Manning, or Gauckler-Strickler formulae would then be suitable.

A considerable number of experimental determinations of wave
profiles were made, and the results listed in graphical and tabular
form. The curves show that until breaking occurs there is a
definite dependence between wave length and amplitude of the waves.

Probably the most significant result of this study of the
undular surge, is the realisation of the importance of the effect of
friction on the shape of the waves constituting the surge. In a
rough-sided channel for a given Y₀, the crest height is greater
and increases more rapidly from wave to wave, and the wave length
is shorter than in a channel with a smoother surface.