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, A_e = 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:

 

ω_e = ω_i/2.

 

The edge waves generated have the Stoke’s solution:

 

 

The standing waves oscillate with the shape cos(k_ey). They oscillate with time cos(ω_et) at a frequency ω_e.

 

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 ω_i, 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 k_e:

 

ω_i² = 4gk_e sin(β)

f_{i­ =} ω_i/(2π)

 

where g = gravity, k_e­ = wavenumber = 2π/λ and β = slope of the wedge.

 

Wavelength can be determined using: λ = 2L/n

 

where L = width of the beach.

 

 

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 f_e = f_i/2 we find the peak which occurs at half this frequency (The 1^st 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 f_i = 1.16Hz and Mode 2, were recorded and plotted as follows:

 

 

 

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 k_e 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 k_e = 2π/λ, and the angular frequency is found using ω = 2πf_e.

 

This is repeated for Modes 3 and 4 at 15°, and for Modes 1 and 2 at 7.5°:

 

 

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 η_e decreases. Also as β increases, sin(β) and e^-cos(β) increase and so η_e increases.

 

Using the results obtained in Part 1 of the laboratory we found that there was a linear relationship between ln(P) or η_e 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:

 

 

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.