of a Wave in Varying Depths of Water
Research and Paper done by: Ryan Mills
1-17-99
IB Physics II
Symposium on Research
Table of Contents
Introduction: The Ocean Wave: A Complex Beauty
What do the vast Pacific Ocean, the Willamette River, and a small pond created by the poor drainage system at Tualatin High School have in common, besides the fact that they are all water? The answer is simple, waves. Although varying greatly in size, each of these bodies of water does contain a wave or waves, which are defined as: "the resulting motion of water caused by a disturbance" (Bascom 1964). These disturbances come in three major forms which are wind, earthquakes, and the gravitational pull of the moon on the water. Wind waves are the most common, but "they are also the most variable" due to infinite number of winds on the earth (Bascom 1964). The wind creates the wave by "air flowing over water, transmitting energy through the water" (Hidy 1971). Also, the rapid motion of earthquakes can cause the water to oscillate underground, therefore causing great seismic sea waves, known as tsunamis. Finally, the gravitational pull of the moon causes the longest type of wave, the tides, which can "last up to 17 hours" (Mogean & Hamsley 1993).
Quite similar to the regular waves studied in classrooms, ocean waves can be broken down into separate components. At the high point of the wave lies the crest, while the bottom is referred to as the trough in the wave. The distance between the crest and the trough is known as the wave height. The wavelength is the horizontal distance between two crests, "which is measured in feet" (Smith 1973). The last component of waves is the wave period, which measures the "time in seconds for a wave crest to transverse a distance equal to one wavelength" (Mogean & Hamsley 1993).
This last component, the wave period determines how waves are measured. Tides measure in as the longest waves at anywhere from three to seventeen hours. Next are the enormous, earthquake-caused tsunamis which can last from 15-20 minutes. After this comes surf beat which clocks in at approximately 100 seconds. Following is the swell, "which might be the most common at beaches," at 10 seconds (Zhang 1998). The chop is the next fastest at one to five seconds. Finally, the ripples in a pond are usually less than one second. Each of these classifications are extremely loose. Due to this, and a number of other influences, most ocean waves are awesomely complex. Several reasons for this include: the wind source is often quite broad, waves change in character over time, several sets of waves with different periods and directions are present at the same time, and water waves are influenced by the underground topography. Thousands of years of research has only shown the concrete conclusion of more wind equates to bigger waves.
What about the depth of the water though? How would this affect the wave period of waves? If waves could be taken out of the huge ocean and isolated in a separate environment at varying depths of water, could this help to explain the varying wave periods? This is what my experiment will attempt to answer. By taking waves out of a completely random environment such as the ocean and placing them in an isolated area, these questions could be answered.
I believe that as the depth of the water increases, so will the velocity of the waves. This might be due to the momentum of the water because there is simply more of it. Also, the waves might travel faster in deeper water because it is simply traveling over more water, which has a much smaller frictive force than does the ground.
1 10 ft. rain gutter with attaching ends 10,000 ml of water
Caulking 500 ml measuring cup
A piece of wood able to touch the bottom of the rain gutter
A stopwatch A pen and paper
A ruler A calculator
Blocks of wood
1. I attached the ends of the rain gutter using water-proof caulking and allowed to dry over a period of 24 hours, then I put the gutter on the blocks of wood to make it level.
2. After this, I measured exactly one foot from the end of the gutter in order to know the distance which the wave would travel, nine feet.
VIEW of rain gutter on wood from the top
3. Next, I began my research by initially measuring out approximately 2,000 ml of water with the measuring cup and placing this water into the rain gutter.
4. Then, I placed the piece of wood at the end of the gutter and gently moved it to the one foot mark, which subsequently created a wave.
5. Using the stopwatch, I measured how long it took for the wave to travel the measured distance.
VIEW of me measuring the time of the wave.
6. I repeated this process two times, allowing the water to settle each time, and took the average of the three times.
7. I then measured the depth of the water using the ruler and recorded this as well.
8. After gathering this data, I added 500 ml of water with the measuring cup, and I again
recorded the depth, took three time trials using the wood block, and averaged these times.
9. I repeated this process at 500 ml increments until I had reached a total of 9,000 ml of water in the rain gutter.
10. Finally, I took the averaged times and used them to calculate the average velocity for each measured depth and began my paper.
| Amount of Water | Time 1 | Time 2 | Time 3 | Average |
|
2,000 ml
|
7.95 s
|
7.76 s
|
7.71 s
|
7.806 s
|
|
2,500
|
6.64
|
6.92
|
6.92
|
6.826
|
|
3,000
|
6.03
|
6.46
|
6.49
|
6.31
|
|
3,500
|
6.06
|
6.27
|
6.25
|
6.193
|
|
4,000
|
5.78
|
5.74
|
5.85
|
5.79
|
|
4,500
|
5.49
|
5.56
|
5.5
|
5.516
|
|
5,000
|
5.28
|
5.16
|
5.32
|
5.253
|
|
5,500
|
4.96
|
5.1
|
5.08
|
5.046
|
|
6,000
|
5.01
|
4.86
|
4.89
|
4.92
|
|
6,500
|
4.68
|
4.71
|
4.64
|
4.676
|
|
7,000
|
4.44
|
4.57
|
4.55
|
4.52
|
|
7,500
|
4.28
|
4.34
|
4.36
|
4.326
|
|
8,000
|
4.11
|
4.16
|
4.21
|
4.16
|
|
8,500
|
4.06
|
4.01
|
3.98
|
4.016
|
|
9,000
|
3.93
|
3.79
|
3.85
|
3.856
|
Data File Data1.txt
This data table shows the results which I obtained
from the original research, except for the depth of water. I assumed that
an average time would be much more accurate than one single time for each
depth. The greatest uncertainty which I obtained was from the 3000 ml depth
level. (6.49-6.03)/2 = .23 seconds. Using this uncertainty, I can obtain
the highest and lowest wave speed ratings.
| Amount of Water | Depth of Water | Distance Traveled | Velocity |
|
2,000 ml
|
0.7 cm
|
9 ft.
|
1.153 ft/s
|
|
2,500
|
1.1
|
9
|
1.318
|
|
3,000
|
1.5
|
9
|
1.426
|
|
3,500
|
1.8
|
9
|
1.453
|
|
4,000
|
2
|
9
|
1.554
|
|
4,500
|
2.1
|
9
|
1.632
|
|
5,000
|
2.3
|
9
|
1.713
|
|
5,500
|
2.5
|
9
|
1.784
|
|
6,000
|
2.7
|
9
|
1.829
|
|
6,500
|
2.8
|
9
|
1.925
|
|
7,000
|
3
|
9
|
1.991
|
|
7,500
|
3.2
|
9
|
2.081
|
|
8,000
|
3.4
|
9
|
2.163
|
|
8,500
|
3.6
|
9
|
2.241
|
|
9,000
|
3.7
|
9
|
2.334
|
Data File Data2.txt
To determine the velocities from the previous table,
I simply divided the average time over the distance traveled.
This graph displays the relationship between the depth of water and the velocity of the wave as it traveled. The line of best fit shows the slope of the line, which confers with my hypothesis. Wave velocity increases as the depth of the water increases. I took the velocity and depth of water data from the 5,500 ml and 6,500 ml quantities to determine the slope of this line: (y2-y1)/(x2-x1) or (1.925-1.784)/(2.8-2.5) = 0.47 ft/s/cm. I chose these two amounts because their data points lie directly on the line of best fit. By extrapolating data, this product could be used to determine ocean wave speed, discounting the variables of the environment and the type of water being used. If a wave formed in an area of the ocean 100 meters deep, the wave speed could be determined using the equation: 10,000 cm * 0.47 ft/s/cm = 4700 ft/s. Obviously, ocean waves could not travel this fast. Again, this is because they are limited by their size, and the environmental factors.
Using my time trial uncertainties, I can wiggle the average velocities up or down. The result of this is a change in the slope of the graph. To do this, I simply went back to my original average time data for the 5,500 ml and 6,500 ml amounts of water. Then, I added 0.23 seconds to each time: 4.676+0.23 = 4.906 seconds
5.046+0.23 = 5.276 seconds
Then, I used these two new times to find the velocities: 9.00/4.906 = 1.834 ft/s
9.00/5.276 = 1.706 ft/s
Finally, I used the original slope equation to determine the slope. This slope will be less because the velocities are slower: (1.834-1.706)/(2.8-2.5) = 0.426 ft/s/cm.
I can also repeat this process, but instead increase the velocities by subtracting the uncertainty from the average times. After following the same process, this yields a much faster slope of 0.516 ft/s/cm.
Conclusion:
This experiment yielded a slope useful in determining
the velocity of a wave over a set depth of water. However, using this data
to find the velocity of an ocean wave would be extremely foolhardy. I controlled
this experiment rigidly, taking data from waves created out of a rain gutter.
The environment consisted of my garage, which has no wind to effect the
speed of the wave. I used fresh water instead of salt water. The magnitude
of my waves were many times smaller than those of the ocean. Perhaps the
only suitable place for my data would be measuring the velocity of waves
in a small, fresh-water pond on a clear, windless day in the fall.
The data which I have gathered may not be of much use for measuring actual water wave speeds, but it does help to explain other phenomenon. For example, this answers the question of why rowing a boat in shallow water is more difficult. The waves which the oar creates are slow and hard to start. Thus, it requires more energy to move. As stated in my hypothesis, I think that this may occur because water sitting on top of water is much easier to move than water sitting on top of a solid surface.
My experiment could also be improved in a variety of ways. If I had the electronic equipment available, I might have been able to regulate the creation of the waves, as well as the timing of each wave. Interestingly enough, I found that no matter how fast I pushed the rock to create the waves, the time only varied by a few hundredths of a second, or maybe a tenth of a second at most. I think that this might be due to the friction slowing the wave down no matter how much force is put behind the creation of the wave. Another improvement which I could make is take even more time trials for each amount of water. Perhaps five times would yield more accurate results than three.
This research could also be expanded in a number of ways. Instead of simply looking at water, other liquids, such as oils or actual salt water could be measured. Not only could I take data from a simple 10-ft. rain gutter, but also a 5-ft. or 20-ft. rain gutter. Perhaps a wider gutter would produce different results because the wave would travel over a greater amount of area.
I doubt that this research will win any Nobel Prizes,
but it lays a foundation for wave velocity in a controlled environment
and for my extended essay.
Bascom, William. Waves and Beaches: the dynamics, 1964,
New York, Anchor Books.
Hidy, George. The Waves the Nature of Sea Motion, 1971,
Toronto, Litton Educational Publishing.
Ocean Wave Measurement and Physics, eds. Mogean, Orville and Hamsley, Michael, 1993, New Orleans, Litton Educational Publishing.
Ocean Wave Kinematics, Dynamics, and Loads on Structures, ed. Jun Zhang, 1998,
Houston, American Society of Civil Engineers.
Smith, F.G. Walton. The Seas in Motion, 1973,
New York, Thomas Y. Crowell Company.
This site takes you to the Applied Maths Group Water Waves, an interesting
site!
http://http1.brunel.ac.uk:8080/depts/ma/resgrp/apmaths/water.html
An addendum of research papers on hydraulics and waves, another interesting
site!
http://www.cv.ic.ac.uk/kaa/section/researchka.html
Oregon State's Ocean Engineering site, yet again an interesting site!
http://www.ccee.orst.edu/hinsdale.htm
Professor S.J. Hogan's site, including research on waves, wow this man
is interesting!
http://www.orst.edu/dept/ccee/ocean/ocenpro.htm
A list of linear wave equations, absolutely the MOST interesting site
i've visited!
http://www.hotbot.com/director.asp?target=http%3A%2F%2Fcadig1%2Eusna%2Enavy%2Emil%2F%7Enaomeweb%2Fcourses%2Fen475%2Fwavesum2%2Ehtml&id=0&userid=28vbweZsMFAI&query=MT=Water+and+Wave+Research&RG=%2Ecom&rsource=INK