LENG, X., and CHANSON, H. (2017). "Upstream Propagation of Surges and Bores: Free-Surface Observations."
Coastal Engineering Journal, Vol. 59, No. 1, paper 1750003, 32 pages & 4 videos (DOI:
10.1142/S0578563417500036) (ISSN 0578-5634).
1
Upstream Propagation of Surges and Bores: Free-Surface Observations
by
Xinqian LENG (
1
) and Hubert CHANSON (
1
)
(
1
) The University of Queensland, School of Civil Engineering, Brisbane QLD 4072, Australia
Corresponding author, Email: h.chanson@uq.edu.au, Ph.: (61 7) 3365 3619, Fax: (61 7) 3365 4599, Url:
http://www.uq.edu.au/~e2hchans/
Abstract
In a free-surface flow, a sudden increase in water depth induces a positive surge, also called compression wave or bore.
Herein a physical study was conducted in a relatively large-size rectangular channel with a smooth bed. The upstream
propagation of breaking and undular bores were investigated with a broad range of Froude numbers Fr
1
ranging from
1.1 to 2.3. Both instantaneous and ensemble-average free-surface measurements were performed non-intrusively. The
observations showed the occurrence of undular bores for 1 < Fr
1
< 1.2 to 1.3, breaking bores for Fr
1
> 1.4 to 1.5, and
breaking bores with secondary waves for 1.2-1.3 < Fr
1
< 1.4-1.5. The propagation of a breaking bore was associated
with an upward free-surface curvature immediately before the roller toe for Fr
1
< 2, and an abrupt increase in free-
surface elevation with the passage of the breaking roller. The propagation of undular bores was characterised by a
smooth upward free-surface curvature, followed by a smooth first wave crest and a train of secondary quasi-periodic
undulations. For all tidal bores, the passage of the bore front was always associated with large free-surface fluctuations,
occurring slightly after the arrival of the front. During the generation process, the positive surge formed very rapidly
and the surge celerity increased very rapidly, reaching maximum values excess of the fully-developed bore celerity.
With increasing time, the surge decelerated and the bore propagated at a early constant celerity for (x
gate
-x)/x
gate
> 10.
Keywords: Positive surges, Tidal bores, Tsunami bores, Rivers, Estuaries, Physical modelling, Free-surface
measurements, Celerity.
1. INTRODUCTION
In an open channel, canal, river or estuary, a sudden increase in flow depth induces a positive surge, also called
compression wave or bore (HENDERSON 1966, BRYSON 1969, LIGGETT 1994). In an estuary, the flood tidal wave
may become a tidal bore during the early flood tide in a narrow funnelled channel under large tidal ranges (TRICKER
1965, CHANSON 2011a) (Fig. 1). A related geophysical application is the up-river propagation of tsunami (SHUTO
1985, TANAKA et al. 2011). During the 26 December 2004 and 10 March 2011 tsunami disasters, the rapid advances
of the tsunami waters caused massive inland damage, when tsunami bores followed rivers and canals (TANAKA et al.
2012, TOLKOVA et al. 2015). In each situation, after its onset, the bore may be analysed as a hydraulic jump in
translation (RAYLEIGH 1908, LIGHTHILL 1978).
Following the milestone investigation of BAZIN (1865), physical studies of positive surges and bores included the
works of FAVRE (1935), BENET and CUNGE (1971), YEH and MOK (1990), TRESKE (1994), HORNUNG et al.
(1995), CHANSON (2005,2010a,2010b,2011b), KOCH and CHANSON (2008,2009), DOCHERTY and CHANSON
(2012), GUALTIERI and CHANSON (2011,2012), KHEZRI and CHANSON (2012), and LENG and CHANSON
(2015a,b). Table 1 presents a comparative summary of detailed laboratory studies. Mathematical and numerical studies
of tidal bores encompassed depth-averaged models (BARRÉ DE SAINT VENANT 1871, BOUSSINESQ 1871,1877,
PEREGRINE 1966, MADSEN et al. 2005, PAN et al. 2007, TOLKOVA et al. 2015), and more recently computational
fluid dynamics (CFD) models (FURUYAMA and CHANSON 2010, LUBIN et al. 2010).
The shape of the surge is a function of its Froude number Fr
1
(MONTES 1998, CHANSON 2012):
1
1
1
1
B
A
g
UV
Fr
(1)
where V
1
is the initial velocity positive downstream, U is the surge celerity positive upstream, g is the gravity
acceleration, A
1
is the initial flow cross-section area and B
1
is the initial free-surface width. An undular surge is
typically observed for Fr
1
< 1.3 to 1.5 (Fig. 1A & 1B) (FAVRE 1935, PEREGRINE 1966, TRESKE 1994, KOCH and
CHANSON 2008, CHANSON 2010a). For Fr
1
> 1.4 to 1.6, the leading edge of the bore is characterised by a breaking
roller (Fig. 1C) (HORNUNG et al. 1995, KOCH and CHANSON 2009, KHEZRI and CHANSON 2012). The integral
form of the equations of conservation of mass and momentum gives a series of relationships between the flow
properties in front of and behind the bore front (LIGHTHILL 1978, CHANSON 2012). For a sloping rectangular
frictionless channel, the application of the continuity and momentum principle yields to a modified Bélanger equation:
LENG, X., and CHANSON, H. (2017). "Upstream Propagation of Surges and Bores: Free-Surface Observations."
Coastal Engineering Journal, Vol. 59, No. 1, paper 1750003, 32 pages & 4 videos (DOI:
10.1142/S0578563417500036) (ISSN 0578-5634).
2
)1(
1
Fr
8)1(
2
1
d
d
2
1
2
1
2
(2)
where d is the flow depth, the subscripts 1 and 2 refer to the initial flow conditions and new conjugate flow conditions
respectively, and ε is a dimensionless coefficient defined in terms of the bed slope S
o
= sin as:
)1Fr(dWg
SWeight
2
1
2
1
o
(3)
with Weight being the weight force, W the channel width, ρ the water density, g the gravitational acceleration and the
angle between the invert and horizontal (LENG and CHANSON 2015b). For a horizontal frictionless rectangular
channel, Equation (2) yields the classical Bélanger equation:
1Fr81
2
1
d
d
2
1
1
2
(4)
and, for a rectangular channel, the Froude number becomes: Fr
1
= V
1
/(g×d
1
)
1/2
.
Herein a physical investigation was conducted with a focus on the generation and upstream propagation of bores. New
experiments were conducted in a large facility. The observations included detailed free-surface measurements at a
number of longitudinal locations for a broad range of flow conditions. For some flow conditions, experiments were
repeated 25 times and the results were ensemble-averaged. A comparative analysis between single measurements and
ensemble-averaged data is developed, together with a re-analysis of both field and laboratory data. It is the aim of this
work to characterise some seminal features of undular and breaking surges.
2. EXPERIMENTAL SETUP AND SURGE GENERATION
2.1 Presentation
New experiments were conducted in a 19 m long 0.7 m wide rectangular flume with smooth PVC bed and 0.52 m high
glass sidewalls. The initially steady flow was supplied by an upstream water tank leading to the 19 m long test section
through a series of flow straighteners followed by a smooth bed and sidewall convergent. The water discharge was
measured by a magneto flow meter with an accuracy of 10
-5
m
3
/s, carefully checked against brink depth data. A fast-
closing Tainter gate was located next to the downstream end of the channel at x = 18.1 m, where x is measured from the
upstream end of the flume. A radial gate was located further downstream at x = 18.88 m and was followed by a free
overfall at x = 19 m.
Video observations were conducted using a HD video camera Sony
TM
HDR-XR160, operating at 25 fps or 50 fps, with
a resolution of 1920×1080 pixels, a digital camera Casio
TM
Exlim EX-10, set at 120 fps (640×480 pixels), 240 fps
(512×384 pixels) or 480 fps (224×160 pixels), and a dSLR camera Pentax
TM
K-3 (movie resolution 1920×1080 pixels)
equipped with Carl Zeiss
TM
Distagon 28 mm f2 lens, producing photographs with a low degree (< 1%) of barrel
distortion. Photographic sequences in high-speed continuously shooting mode (8.3 fps) were also taken with the dSLR
camera Pentax
TM
K-3 (6016×4000 pixels). In steady flows, the water depths were measured using pointer gauges. The
accuracy of the sharp pointer gauges was 0.5 mm. The unsteady water depths were recorded with a series of acoustic
displacement meters. A Microsonic
TM
Mic+35/IU/TC unit was located at x = 18.17 m immediately downstream of the
Tainter gate. Further nine acoustic displacement meters Microsonic
TM
Mic+25/IU/TC were spaced along the channel at
x = 17.81 m, 17.41 m, 14.96 m, 12.46 m, 9.96 m, 8.5 m, 6.96 m, 3.96 m and 0.96 m above the centreline. All acoustic
displacement meters (ADMs) were calibrated against the pointer gauge in steady flows and sampled at 200 Hz. Further
details on the experimental facility and instrumentation were reported in LENG and CHANSON (2015c).
2.2 Experimental flow conditions and surge generation
Four initially-steady discharges (Q = 0.101, 0.085, 0.071 and 0.055 m
3
/s) were tested for the instantaneous free-surface
measurements, with the highest and lowest discharges being used for the ensemble-average measurements. The tidal
bore was generated by the rapid closure of the Tainter gate and the surge propagated upstream against the initially-
steady flow. The Tainter gate closure time was less than 0.15 - 0.2 s, and such a closure time was small enough to have
a negligible effect on the bore propagation. Appendix I presents some movies of the bore generation and propagation.
For a given discharge, the bore Froude number was controlled by the gate opening h after closure, the initial flow depth
d
1
and bed slope S
o
. While the bulk of experiments were performed with a horizontal slope (S
o
= 0), a steeper bed slope
was used to generate larger bore Froude numbers (App. II). For the generation of undular bores, the radial gate was
initially closed partially to raise the initial water depth d
1
. Figure 2C shows a schematic of the experimental facility with
a partially-closed radial gate. The bores were generated by the rapid closure of the Tainter gate, with the radial gate
LENG, X., and CHANSON, H. (2017). "Upstream Propagation of Surges and Bores: Free-Surface Observations."
Coastal Engineering Journal, Vol. 59, No. 1, paper 1750003, 32 pages & 4 videos (DOI:
10.1142/S0578563417500036) (ISSN 0578-5634).
3
position remaining unchanged during an experiment. For all breaking bore experiments, the radial gate was fully
opened; the bore was generated by the rapid closure of the Tainter gate
Both instantaneous and ensemble-average free-surface measurements were performed herein. For all experiments, the
instruments were started 60 s before gate closure, and sampling stopped when the bore reached the upstream intake.
During the ensemble-average experiments, a total of 25 runs were repeated for each set of controlled flow conditions;
the median free-surface elevations and instantaneous free-surface fluctuations were calculated from the total ensemble.
The experimental flow conditions are summarised in Table 1, where they are compared to past studies, and in Appendix
II, together with the experimental observations at x = 8.5 m. Note that the present study was conducted a large facility
with large flows (Table 1). Earlier dimensional analyses suggested that present results may be extrapolated to full-scale
without adverse scale effects (DOCHERTY and CHANSON 2012). This will be confirmed in comparative
presentations regrouping present results and prototype observations.
3. FLOW PATTERNS
3.1 Presentation
Visual, video and photographic observations were conducted to document the basic flow patterns of the upstream
propagation of tidal bores. Both breaking and undular bores were investigated. Figure 2 presents a definition sketch and
Figure 3 shows typical side views of the propagation of breaking and undular bores. For the present investigation, no
bore was visible for a Froude number less than unity. For 1 < Fr
1
< 1.1 to 1.3, the bore was undular. The bore was
characterised by a gentle upward free-surface rise and a series of quasi two-dimensional secondary undulations (Fig.
2B, 3B & movie CIMG0078.mov, Appendix I). For Fr
1
1.2 there was no breaking, and small shock waves initiated
from the sidewalls upstream of the first wave crest, intersecting at the first wave crest on the centreline (Fig. 3C).
Breaking bores with secondary waves developing behind the breaking roller were observed for 1.2 to 1.3 < Fr
1
< 1.4 to
1.5. These bores were characterised by a thin layer of breaking developing at the bore front across most of the channel
width, followed by a train of smooth, three-dimensional secondary waves. Herein the expression "breaking bore with
secondary waves" is used in line with PEREGRINE (1966). Other researchers used the expression "undular bore with
some breaking" to denote the same flow pattern (KOCH and CHANSON 2009, CHANSON 2010b, KHEZRI and
CHANSON 2012).
For Fr
1
> 1.4 to 1.5, the secondary wave motion disappeared and the breaking bore was characterised by a steep wall of
water with a sharp breaking front (Fig. 2A, 3A & movie CIMG0007.mov, App. I). The propagation process was highly
unsteady turbulent, with an abrupt rise in free-surface elevation and a rapidly fluctuating breaking roller (LENG and
CHANSON 2015a). The initially steady free-surface curved upwards slightly before the arrival of the breaking roller
toe for Froude numbers smaller than 2, as illustrated in Figures 2A and 3A. Such an upward streamline curvature may
be derived from theoretical considerations and was previously reported (VALIANI 1997, CHANSON 2010b,
DOCHERTY and CHANSON 2012). For Froude numbers greater than 2, the upward streamline curvature was not
seen. The breaking roller was characterised by a two-phase air-water flow region and strong turbulent interactions, with
free-surface splashes and droplet ejection (Fig. 3A). The free-surface was nearly horizontal behind the roller, although
with large fluctuations.
The visual observations were consistent with earlier findings (HORNUNG et al. 1995, KOCH and CHANSON 2009,
CHANSON 2010b, CHANSON and DOCHERTY 2012, KHEZRI and CHANSON 2012).
3.2 Instantaneous free-surface measurements
Instantaneous free-surface measurements were recorded non-intrusively using the acoustic displacement meters
(ADMs) installed above the flume centreline. Figure 4 presents typical instantaneous free-surface measurements for
two types of tidal bores: breaking (Fig. 4A) and undular (Fig. 4B). In Figure 4, t is the time since gate closure; and the
thin red solid line is the ADM sensor located immediately downstream of the Tainter gate at x = 18.17 m. At that
location, the sudden gate closure induced a negative surge associated with a drop in free-surface elevation. All other
ADM sensors showed a marked rise in free-surface elevation associated with the passage of the bore, although some
complicated transient flow pattern was observed immediately upstream of the gate (x = 17.81 m), as documented by
SUN et al. (2016) in a smaller facility. The movies CIMG0006.mp4 and CIMG0080.mp4 show high-speed movies of
the bore generation induced by the Tainter gate closure (Appendix I). The propagation of the breaking bore was
characterised by a sharp increase in water depths, followed by a fluctuating motion with nearly horizontal free-surface
behind the marked roller (Fig. 4A). The conjugate depth of the breaking bores was slightly lower than the peak
elevation of the breaking roller and was highly fluctuating, as sketched in Figure 2A. The propagation of undular bores
was associated with a smoother rise in water level, followed by a train of secondary undulations (Fig. 4B). Overall the
instantaneous ADM data were consistent with the photographic and video observations (Appendix I).
LENG, X., and CHANSON, H. (2017). "Upstream Propagation of Surges and Bores: Free-Surface Observations."
Coastal Engineering Journal, Vol. 59, No. 1, paper 1750003, 32 pages & 4 videos (DOI:
10.1142/S0578563417500036) (ISSN 0578-5634).
4
4. FREE-SURFACE PROPERTIES
4.1 Ensemble-averaged measurements
The propagation of surges and bores is a highly turbulent and unsteady process, as illustrated by the high-speed movies
(App. I). A time average would be meaningless and a series of ensemble-average measurements were conducted for
two different discharges: that is, Q = 0.101 m
3
/s and 0.055 m
3
/s. Both breaking and undular bores were generated for
each discharge. Further identical bore Froude number Fr
1
were achieved with different discharges (App. II). For each
set of flow conditions, the experiments were repeated 25 times and the results were ensemble-averaged to obtain the
median free-surface elevation d
median
and the difference between the third and first quartiles (d
75
-d
25
). The difference
between the third and first quartiles (d
75
-d
25
) characterised the instantaneous free-surface fluctuations. For a Gaussian
distribution of the data around its mean, (d
75
-d
25
) would be equal to 1.3 times the standard deviation of the total
ensemble (SPIEGEL 1972). Figure 5 presents some typical ensemble-averaged data with the time variations of the free-
surface elevation and fluctuations, where the time t = 0 corresponded to the Tainter gate closure. In each graph, the
solid black line denotes the ensemble-averaged median free-surface elevation at x = 8.5 m, where the bore was fully-
developed. Figure 5A shows breaking bore data, while Figure 5B presents undular bore data.
For all breaking bore experiments, the ensemble-averaged free-surface data highlighted the abrupt increase in water
level associated with the passage of the roller. After the roller, the free-surface increased very gradually. For the data at
x = 8.5 m seen in Figure 4A and 5A, the dimensionless rate of increase in free-surface elevation was on average
(d/t)/(g×d
1
)
1/2
~ 10
-3
after the bore. The propagation of a breaking bore was typically associated with higher
maximum free-surface fluctuations, than with undular bores, and these were caused by the highly turbulent breaking
roller (Fig. 3A and 5), the free-surface fluctuations being quantified in terms of (d
75
-d
25
) herein. With breaking tidal
bores, the free-surface fluctuations showed a marked maximum (d
75
-d
25
)
max
shortly after the passage of the bore
breaking roller (Figure 5A).
With undular bores, a key feature was the upward free-surface curvature ahead of the first wave crest, followed by a
train of secondary undulations. The free-surface fluctuation data showed a sharp increase in free-surface fluctuations
with the propagation of an undular tidal bore. A first local maximum free-surface fluctuation occurred shortly after the
passage of the first wave crest, followed by a series of local maximum fluctuations appearing in a quasi-periodic
manner during the secondary wave motion (Fig. 5B). The time-variations of free-surface fluctuations in undular bores
oscillated approximately in phase with the oscillations of the free-surface elevation.
The maximum free-surface fluctuation (d
75
-d
25
)
max
of breaking bores, the first maximum free-surface fluctuation (d
75
-
d
25
)
max
of undular bores and the time lag ∆t between the maximum fluctuation and bore front passage were analysed for
all flow conditions. Herein the time of the bore front passage was defined as the instant at which the free-surface
elevation started to rise. Mathematically this corresponded to the time when the first derivative of the free-surface
variation with respect to time was non-zero and positive. Figure 6 presents the experimental results as functions of
longitudinal distance from the gate, where x
gate
is the position of the Tainter gate. The data showed large maximum
free-surface fluctuations relatively close the gate: (x
gate
-x)/x
gate
< 0.1; further upstream large free-surface fluctuations
were also observed over the entire channel length (Fig.6A). The time lag ∆t increased rapidly with increasing distance
from the gate and tended to reach a plateau ∆t/(g/d
1
)
0.5
5 at about (x
gate
-x)/x
gate
0.1, before gradually increasing with
increasing distance for (x
gate
-x)/x
gate
> 0.4 (Fig. 6B). Importantly the largest maximum free-surface fluctuations were
observed for the breaking bore with the highest Froude number (Fr
1
= 2.2) at almost all longitudinal locations. The
dimensionless time lag was larger for the smallest water discharge (Q = 0.055 m
3
/s). Since the dimensionless results
were presented assuming implicitly a Froude similitude, the finding might hint potential scale effects in terms of free-
surface fluctuations.
4.2 Bore propagation and celerity
The position of the bore front and its celerity were deduced from the acoustic displacement meter data. Figure 7
presents typical data sets, with the bore front location and celerity presented as functions of the distance from the
Tainter gate. Both single-run (Single) and ensemble-averaged (EA) data are shown. Overall the data showed the same
distinctive trend for all Froude numbers and flow rates. Immediately after the gate closure, the positive surge formed
very rapidly and the process was associated with some strong disturbance immediately upstream of the gate (movies
CIMG0006.MP4 and CIMG0080.MP4, Appendix I). The surge celerity increased very rapidly a very short distance,
reaching maximum dimensionless value (U+V
1
)/(g×d
1
)
1/2
well excess of the fully-developed bore properties observed
for (x
gate
-x)/x
gate
> 10 (Fig. 7B). For example, for (x
gate
-x)/x
gate
< 10, the dimensionless bore celerity reached values up
to 2.3 for a breaking bore, and values in excess of 5 for undular bores. Further upstream, the surge decelerated and
propagated in a more gradual manner, reaching its asymptotical value for (x
gate
-x)/x
gate
> 10. This asymptotical limit
(U+V
1
)/(g×d
1
)
1/2
was equal to the Froude number observed at x = 8.5 m. The bore front location was relatively well
predicted by the Saint-Venant equations as shown in Figure 7A. The present results differed from the observations of
LENG, X., and CHANSON, H. (2017). "Upstream Propagation of Surges and Bores: Free-Surface Observations."
Coastal Engineering Journal, Vol. 59, No. 1, paper 1750003, 32 pages & 4 videos (DOI:
10.1142/S0578563417500036) (ISSN 0578-5634).
5
REICHSTETTER (2011), albeit her experiments were conducted for much smaller initially-steady discharges (Q = 0.02
& 0.03 m
3
/s).
The dimensionless maximum water depth (d
max
-d
1
)/(d
2
-d
1
) is shown in Figure 8 as function of the dimensionless
distance from the gate. For undular bores, the maximum wave height was that of the first wave crest.. Basically the
dimensionless maximum bore height was independent of the distance from the gate (x
gate
-x)/d
1
> 10.
Altogether both visual observations and longitudinal measurements for a wide range of flow conditions indicated that
the bore become fully-developed, that is a translating jump, for (x
gate
-x)/d
1
> 30. Further upstream the free-surface
properties varied little with upstream distance.
4.3 Unsteady free-surface analysis
The unsteady free-surface properties of fully-developed bores were analysed based upon the free-surface measurements
data at x = 8.5 m. The results were compared to theoretical developments and past experimental studies (field and
laboratory). The key features of the bore front included the maximum water depth d
max
and conjugate water depth d
2
for
both breaking and undular bores (Fig. 2). For breaking bores, the roller length L
r
, height and length of the rise in free-
surface immediately upstream of the breaking roller toe h
s
and L
s
were specifically studied, as well as the distance L
max
between the roller toe and the highest roller surface elevation. The wave amplitude a
w
and wave length L
w
were studied
for undular bores. The definition sketch of these parameters is presented in Figure 2. The full data are reported in
tabular form in Appendix II.
The dimensionless conjugate water depth d
2
/d
1
may be expressed as a function of the Froude number Fr
1
, as shown in
Equations (2) and (4) for a smooth sloping and horizontal rectangular channel respectively. Equations (2) and (4) are
compared to experimental observations in Figure 9. All present data are presented with coloured symbols, including
both breaking and undular bores analysed from video and ADM data (instantaneous and ensemble-averaged). All data
showed a monotonic increase in conjugate depth ratio with increasing Froude number. The present data were compared
to previous experimental works. Overall the experimental data with a horizontal bed slope showed a good fit with the
Bélanger equation (Eq. (4)). For 1.6 < Fr
1
< 2.4, the present data deviated from Equation (4) because of the non-
horizontal bed setup; these data matched well Equation (2). The present data also compared well with previous
experimental results (Fig. 9).
In a breaking bore with Fr
1
< 2, the free-surface ahead of the roller toe was curved upwards, as sketched in Figure 2A
and illustrated in Figure 3A, and discussed earlier. The longitudinal length and vertical height of this smooth curved
surface are presented in Figure 10 as functions of the Froude number Fr
1
. The data include both instantaneous and
ensemble-averaged measurements. The vertical height h
s
of the roller toe above the initial water surface was best
correlated by:
s
1.93
11
h
0.37
dFr
1.22 < Fr
1
< 2.3 (5)
with a normalised correlation coefficient R = 0.52. Equation (5) is presented in Figure 10A, where it is compared with
the present data as well as earlier experimental results. Altogether the observations indicated that both h
s
and L
s
decreased with increasing Froude number, tending asympotically towards zero for Fr
1
> 2.5.
For breaking tidal bores, the length L
r
of the roller was defined as the distance between the roller toe and the end of the
breaking roller, where the water depth reached the conjugate depth d
2
(Fig. 2A). The dimensionless roller length data
are plotted as a function of the Froude number in Figure 11. In Figure 11, both instantaneous and ensemble-averaged
measurements and the present data are compared with past studies of stationary hydraulic jumps. Figure 11 shows that,
although the majority of Froude numbers tested herein were lower than those in stationary jump experiments, the
present data with Fr
1
> 2 matched relatively closely stationary hydraulic jump data. Furthermore the present data trend
showed a consistent decrease of roller length with decreasing Fr
1
, in line stationary hydraulic jump data trend.
For an undular bore, two basic properties are the secondary wave amplitude and wave length, a
w
and L
w
respectively
(Fig. 2B). Figure 12 shows both the dimensionless wave amplitude and wave length as functions of Froude number.
The present data are compared to previous experimental data, a cnoidal wave solution (ANDERSEN 1978) and the
linear wave theory of LEMOINE (1948). The former solution was based upon the Boussinesq equation and the
asymptotical results for a rectangular channel are (BENJAMIN and LIGHTHILL 1954):
1
d
d
3
1
d
a
1
2
2
w
(6)
2/1
1
2
2
w
1
d
d
3
22
d
L
(7)
The present data were analysed in terms of both instantaneous and ensemble-averaged measurements. Altogether
the results followed closely previous studies, with an increase in wave amplitude and monotonic decrease in wave
