FLUID MECHANICS 302
LABORATORY 1 – EDGE
WAVES
BY GIANNI ABBONDANZA
(0028548)
AIMS
- To investigate the structure of standing
waves over a confined beach
- To test a possible mechanism for the
formation of beach cusps
- To demonstrate the use of time series
analysis via the Fourier transform and spectral analysis
BACKGROUND
In this laboratory we use a mathematical model
and a physical model to investigate a possible mechanism for the formation of
beach cusps. Beach cusps are formed due to the rhythmic nature of edge waves
acting on a beach.
Edge waves are produced from incident waves
which are normal to the beach, and resonate in an alongshore direction. When
incident waves reflect from a sloping boundary, the interaction between the
incident and reflected wave produces edge waves which travel parallel to the
shore.
Edge waves, resulting from the interaction
between shoaling waves and nearshore topography, have
the following equations of motion:
Where g = gravity, Ae
= wave amplitude, k = 2π/λ is the wavenumber,
ω = 2π/T is the angular frequency, and β = slope of the
wedge/beach.
Interference between incoming and outgoing
waves on a beach may produce resonance in an alongshore direction. This
resonance will only occur at certain frequencies. At these frequencies edge
waves are produced at an angular frequency half that of the incident waves:
ωe = ωi/2.
The edge waves generated have the Stoke’s solution:
The standing waves oscillate with the shape cos(key).
They oscillate with time cos(ωet) at a
frequency ωe.
As x, the distance from the shore, increases
the exponential term above decrease rapidly – known as offshore decay.
The dispersion relation:
predicts the only wavenumber
that can be excited for a particular frequency. From this we can determine the
frequencies which can from standing waves.
Using an incoming wave frequency ωi, determined using the dispersion
relation, we can then excite various standing waves.
Spectral analysis is used to analyse the
physical process in this laboratory.
PRE-LAB QUESTION
The incoming angular frequency of waves
required to excite mode 1, 2, 3 and 4 standing waves in the edge tank can be
determined in terms of the required wavenumber ke:
ωi2 = 4gke
sin(β)
fi = ωi/(2π)
where g = gravity, ke
= wavenumber = 2π/λ and β = slope of
the wedge.
Wavelength can be determined using: λ =
2L/n
where L = width of the beach.
β = 15 |
β = 7.5 |
||||
n |
2 |
3 |
4 |
1 |
2 |
λ |
1.2 |
0.8 |
0.6 |
2.4 |
1.2 |
ke |
5.24 |
7.85 |
10.47 |
2.62 |
5.24 |
ωi2 |
53.18 |
79.77 |
106.35 |
13.41 |
26.82 |
fi |
1.16 |
1.42 |
1.64 |
0.58 |
0.82 |
METHOD
Part 1 – Offshore decay of edge wave modes
- A mode 2 standing wave is set up in the tank,
with the gauge placed as close to the beach as possible.
- The distance of the gauge from the origin
(where the still water reaches) is measured.
- The power spectrum and timeseries
information is acquired by computer sampling.
- The data acquisitions are repeated at 4 other
distances from the origin between 0 and 1000mm.
Part 2 – Edge wave modal structure
- A mode 2 standing edge wave is initiated, and
the gauge placed at the node, on the beach at an angle of 15 degrees.
- The power spectrum and time series
information is acquired by computer sampling.
- This is repeated for modes 3 and 4.
- The beach slope is changed to 7.5 degrees,
and data taken for modes 1 and 2.
RESULTS
To find the power of each edge wave, we examine
the power spectrum plot. We locate the peak which occurs at the incident
frequency, and using the relationship fe = fi/2 we find the peak which
occurs at half this frequency (The 1st peak in the power spectrum).
The power is then read off the dbVrms axis.
The values of p (dbVrms)
and ln(p)
for various distances from shore, at fi = 1.16Hz and Mode 2, were recorded and plotted
as follows:
Distance from
shore (mm) |
p (dbVrms) |
ln(p) |
171 |
14446.68 |
9.58 |
398 |
1812.92 |
7.50 |
595 |
443.65 |
6.10 |
779 |
129.11 |
4.86 |
955 |
44.05 |
3.79 |
There is a strong correlation between the
distance from the shore and the natural log of the Power, as demonstrated by
the line of best fit.
To show how the rectangular frequency
of the edge waves vary with wavenumber ke and the slope of the beach/wedge, we first
find the angular frequency for each mode. This is then plotted against the wavenumber for each mode.
The frequency of each edge wave is found as the
frequency of the peak which occurs at half of the incident frequency. This is
demonstrated for the mode 2, slope = 15° edge wave:
The wavelength and wavenumber
for each mode is calculated using ke = 2π/λ, and the angular frequency is found using ω = 2πfe.
This is repeated for Modes 3 and 4 at 15°, and
for Modes 1 and 2 at 7.5°:
Slope |
Mode |
λ = 2L/n |
ke = 2π/λ |
fe |
ω2 = (2πfe)2 |
ω2 = gkesin(β) |
15° |
2 |
1.2 |
5.24 |
0.583 |
13.42 |
13.29 |
3 |
0.8 |
7.85 |
0.708 |
19.79 |
19.94 |
|
4 |
0.6 |
10.47 |
0.817 |
26.35 |
26.59 |
|
7.5° |
1 |
2.4 |
2.62 |
0.292 |
3.37 |
3.35 |
2 |
1.2 |
5.24 |
0.408 |
6.57 |
6.70 |
Plotting the square of angular frequency
against wavenumber for the various modes and slopes
indicates a close linear relationship between the two variables.
DISCUSSION
1. The equation:
demonstrates that the water elevation decays exponentially the further the distance from the shore (x). The equation predicts that as x increases, e-kx decreases and hence ηe decreases. Also as β increases, sin(β) and e-cos(β) increase and so ηe increases.
Using the results obtained in Part 1 of the
laboratory we found that there was a linear relationship between ln(P) or ηe
and the distance from the shore.
The line of best fit: ln(P) = -0.0073x + 10.613,
confirms that the elevation decays exponentially as x increases (i.e. P =
40660e-0.0073x).
2. The following table reports the theoretical
and experimental angular frequency for each mode and slope:
Slope |
Mode |
λ = 2L/n |
ke = 2π/λ |
fe |
ω2 = (2πfe)2 |
ω2 = gkesin(β) |
15° |
2 |
1.2 |
5.24 |
0.583 |
13.42 |
13.29 |
3 |
0.8 |
7.85 |
0.708 |
19.79 |
19.94 |
|
4 |
0.6 |
10.47 |
0.817 |
26.35 |
26.59 |
|
7.5° |
1 |
2.4 |
2.62 |
0.292 |
3.37 |
3.35 |
2 |
1.2 |
5.24 |
0.408 |
6.57 |
6.70 |
Plotting the dispersion relation:
for each slope gives the following plot:
Plotting the angular frequency squared against
the wavenumbers for the 7.5° and 15° slopes, gives
the following relationship (see above table for calculations):
The above graphs are almost identical (as are
the theoretical and experimental values of ω2 reported in the table), which confirms that the
theoretical model is an adequate model for predicting the relationship.
On a seasonal timescale the presence
of tidal variations will cause alterations in the waves approaching a
particular location. Consequently the dispersion relationship may not hold as
the beach slope and other factors may change.
3. The location of a groin can have substantial
impacts on the formation of edge waves. If a groyne is constructed at a
location of no movement parallel to the beachfront, an antinode, then the
effects of the groyne on wave propagation will be minimal.
Alternatively if the location coincides with
the position of a node then the groyne will eliminate the existence of standing
edge waves. Building a groyne anywhere
between the node and anti node will interrupt the horizontal movement caused by
standing edge wave patterns. As a result
the groyne can dampen erosion and soil movements and the creation of beach
cusps.
4. There are many
applications of harmonics in engineering practice. Sound engineers need to
consider the effects of harmonics when designing theatres and auditoriums.
Structural engineers need to be aware of the effect of resonance on buildings
and bridges. If the external frequency applied to a structure due to wind,
waves or earthquakes matches the natural frequency of the building material
then resonance can have disastrous effects on the structure
5. It is possible for wavetrains of different frequencies to arrive at a beach
simultaneously as a result of a number of different factors. Ocean floor irregularities, differences in
fetch lengths, variations in wind speed generating the trains, extreme weather
conditions such as storms and even large ocean vessels can cause the range in
frequencies.
6. The model dealt with
in this laboratory was vastly simplified as there a number of other factors
that could contribute to the formation of beach cusps. Other factors that can
cause beach cusps include wind conditions, storms, ocean floor irregularities
(sand bars, banks, reefs) and non uniform soil
conditions. Coastal patterns could also affect incoming waves can alter the
formation of cusps.
CONCLUSIONS
This laboratory investigated a
possible mechanism causing beach cusps. After comparing theoretical formulas
for standing edge waves and the dispersion relation for edge waves we can conclude
that the theoretical formulas are an adequate depiction of the real situation.
However, the dispersion relationship
has its limitations isn’t expected to stand up to a seasonal timescale, and
there are also several other factors that can lead to the formation of beach
cusps. The model of edge waves, while quite accurate, doesn’t account for all
these different factors.