Velocity of a wave in a tank at varying depths of water

An investigation into how the velocity of a wave is affected by varying the depth of the water it travels through.

A tray was filled to a certain depth with water and one end raised to 45mm above floor level. The tray was placed on a folder 45mm high for each experiment so the height from which that end of the tray is dropped remains the same throughout the series of experiments. The sliding out of the folder from under the tray and the starting of the stopwatch was simultaneous so the delay in starting the stopwatch due to human reaction time was approximately cancelled out by the time taken for the tray to fall the 45mm from atop the folder.

A folder was selected to hold up the tray because it could be removed, thus allowing the tray the fall, without changing the height of the fall or creating small ripples across the surface of the water which could compromise the reliability of the investigation

Once the wave started it was allowed to traverse the length of the tray 4 times before the stopwatch was stopped at the point at which the wave was judged to have reached the same side of the tray from which it had commenced 4 lengths earlier.

The wave was allowed to traverse the tray 4 times as opposed to just once because the percentage error was smaller since the error in measurement is fixed and we are measuring over a larger distance.

The tray was first filled to a depth of 1cm for the first series of experiments then the depth of the water was increased by 1cm for every series of experiments thereafter.

The level of the water was increased to 6cm in depth so 6 different series of experiments were made. The independent variable was the depth of the water with the dependant variable being the speed of the resulting wave.

The equipment

1 x steel ruler: accurate to � 1mm used to measure depth of water, height of drop and length of tray

1 x plastic tray (length 355mm, 35.5cm): This acted as the wave tank

1x plastic jug: Used to control the amount of water in the tray

Potential health hazards

Minimal risks involved in this experiment apart from possibility of slipping on spillage.

Problems that arose and adaptations made to experiment

It was found that once the tray was filled with water up to a depth of more than 4cm the waves that were generated would cause water the surge over the top of the tray and onto the floor. This not only disrupted the wave, it caused the water level in the tank to fall. This meant that six different sets of results would have to be found between a depth of 0.5cm and 4cm. Therefore the experiment was altered so that a different set of results were taken every time the depth of the water was increased by 0.5cm starting at 0.5cm deep and ending at 3.5cm deep.

Inaccuracy

Six results were taken on each occasion to reduce the effect of human (random) error.

Also, because the ruler was accurate to � 1mm the percentage of (systematic) error lowered as the depth of the water increased

Below is a table of all the results recorded for the speed of the wave. In total thirty sets of results were taken as five results were recorded for each of the six depths.

Length of tray x 4 = 142cm

Depth of water in (mm)

Time taken for wave to cross tray 4 times

1

2

3

4

5

6

average

5

7.40

7.55

7.56

7.52

7.52

7.53

7.51

10

4.37

4.61

3.28

4.34

4.41

4.42

4.43

15

4.11

4.13

4.19

4.15

4.15

4.11

4.14

20

3.83

3.80

3.81

4.02

3.73

3.79

3.83

25

3.40

3.38

3.32

3.44

3.42

3.38

3.39

30

3.04

3.06

3.23

3.11

3.03

3.07

3.09

Average = time taken for wave to traverse the tray once

4

Depth of water

5mm

10mm

15mm

20mm

25mm

30mm

Average time taken to cross tray once(s)

1.88

1.11

1.04

0.96

0.85

0.77

Then to find the speed of the wave, the length of the tray was divided by the time taken for each wave to cross the tray once. This can be shown by the equation

Speed = distance / time

5mm = 35.5 = 18.88cm/s

1.88

10mm = 35.5 = 33.49cm/s

1.06

15mm = 35.5 = 34.13cm/s

1.04

20mm = 35.5 = 36.98cm/s

0.96

25mm = 35.5 = 41.76cm/s

0.85

30mm = 35.5 = 46.10cm/s

0.77

Velocity cm/s

18.9

33.5

34.1

37.0

41.8

46.1

(velocity)2 cm2/s2

357.2

1122.3

1162.8

1369

1747.2

2125.2

Depth (cm)

0.5

1.0

1.5

2.0

2.5

3.0

Research suggests that for shallow water where the depth of the water is less than the wavelength / 20 then,

V � Vgd therefore V2 = g.d so if this data was inputted into the equation

y = mx + c, a graph of v2 against d would yield a straight line graph with an intercept at the origin and a gradient equal to gravity (approx.)

Where:

V= velocity

g = acceleration due to gravity m/s2

d = depth of water

The results obtained (see graph) crudely support the equation linking velocity and depth given above.

Due to the fact that the experiment was relatively crude the results indicate that gravity is � 7.1m/s2. It should be noted however that one result was ignored when the line of best fit was drawn.

Were the investigation to be repeated, more results would be gathered from a greater range of depths in order to get a more accurate line of best fit and further lower the impact that the random error had on this investigation. Also, the wavelength would be measured, possibly by using a strobe light. This would verify the condition that this equation only works if d= wavelength / 20

Conclusion

From the results of the experiment we can state that the greater the depth of the water a wave is travelling in increases, the greater the speed of the wave.

What could have affected my results

The results could be inaccurate because the judgement of when the wave had reached the end of the tray was done by the human eye and the stopwatch was stopped by hand. Due to the human reaction time the results could be smaller or larger than what they really should have been.

The omitted result was not replaced by another so the average for that set of experiments is based on five results not six and so may be less accurate than the other averages.

At 5mm the water did not cover the entire floor of the tray when tilted so when the tray was dropped the wave had to flow over the surface of the tray which may have been why the results for 5mm seem disproportionate to the rest.

Due to the relative crudity of this experiment it has limitations. It cannot used to calculate the precise speed of the waves as this would require far more accurate sensors to detect when the wave has reach an end. Also it could not be used for very small amounts of a liquid due to the fact that when level the fluid would be too shallow to even support a wave. This means that we cannot find out whether the statement made for the conclusion continues to be accurate at very large and very small depths. The line of the graph might perhaps level off at either end.

If the investigation were to be performed again then laser sensors would be used to detect precisely when the wave reached an end and it would be linked electronically to a timer to find the precise time when the wave had had completed the 4 lengths. Using these devices would also lessen the effect of random error.