Nine Singhara IB Physics HL The Effect of Volume on the Period of Seiche Waves In a Rectangular TankIntroductionWhen a person rocks back and forth inside a bathtub that is partially filled with water,waves are created that can grow and overflow the tub. The same thing happens to thewater in swimming pools during earthquakes. If the oscillations from the earthquakesare at the right frequency, the waves reach its resonance, resulting in standing wavesas the waves that hit the sides is reflected back to cause interference and gravity seeksto restore the horizontal surface of the body of water. This is shown in the diagram inFigure 1. In the 1964, the Alaska earthquake set swimming pools as far away as in Puerto Rico in oscillation. ‘Seiche’ is a technical term that describes effects such as these where the resonant oscillation of water or standing waves occur inside an enclosed basin.(http://www.soest.hawaii.edu/GG/ASK/seic he.html)(http://en.wikipedia.org/wiki/Seiche) The velocity of the Seiche waves is dependent on the depth of the water; therefore the frequency is also affected by depth among other variables. Since period is the inverse of frequency, the depth of water also affects it. The fact that the dept of the water is also a factor that makes up the volume brings up the question: “What is the relationship between the volume of water within a rectangular tank and the period of the Seiche waves? Figure 1: Sample motions of Seiches with different The period of a Seiche wave is represented by the harmonics. Merian formula: (http://www.pac.dfo- mpo.gc.ca/science/oceans/t 2L T= (Eq. 1) sunamis/documents/Handb n gd ook_Chapter9.pdf)The variable Lrepresents the length of the basin that contains the water andthevariable n represents the number of harmonics. The variable g stands for gravitationalforce and d stands for depth of the water.(http://homepages.cae.wisc.edu/~chinwu/CEE618_Impacts_of_Changing_Climate/John/Seiche.htm)
Knowing that for a rectangular space, the volume is calculated by multiplyingtogether the length, width, and height, the formula can be manipulated so that thevariable d could be replaced. V d= (Eq. 2) LwV is volume of water, L is length of the basin, and w is width of the basin, giving aformula that could be used in this investigation: 2L T= (Eq. 3) gV n LwOnce manipulated into a y = mx form to show the relationship between the twovariables, which are period and volume, the resulting formula becomes, 4L3w 1 T2 = ´ (Eq. 4) n2 g VDue to this theory, it is predicted that the period squared will be inverselyproportional to the volume. The factors that will affect the constant will be the lengthand with of the basin, the number of harmonics, and gravity.This experiment will be conducted through the simulation of the Seiche waves effectsin a rectangular tank. The motion of the waves will be recorded and position-timegraphs of the motion will be plotted on LoggerPro using the videos. A sine fit will beused to figure out the mathematical equation of the motion. The general equation ofthis is: y = A sin (Bx + C) + D (Eq. 5)The B value is taken from this equation to find the period using the relationshipbetween this B value and the period of the wave, which is given by the equation: 2p T= (Eq. 6) B
DesignResearch Question: What is the relationship between the volume of water within arectangular tank and the period of the Seiche wave?Variables:The independent variable for this experiment was the volume of the water in the tank.This was calculated using the depth of the water, the length, and the width of the tank.The dependent variable was the period of the Seiche waves calculated by the value ofB taken from the graph of the motion of the waves that were fitted with a sine curve.There were several factors that were kept constant throughout the experiment.Materials and Procedure:A medium sized rectangular glass tankwas placed on a tablewith locked wheels.The dimensions of the tank used in this experiment are stated in the data collectionsection. A clear ruler was then placed vertically to the outside of the tank so that thezero is aligned with the bottom of the tank where the water will come in contact withwhen it was poured into the tank. Taped to the left along the corner line of the longerside of the tank, the ruler was used as a tool of reference. A big piece of white paperwas also taped to the side of the tank opposite of the ruler so the waves can be seenclearly.A large beaker was used to transfer some water from the sink into the tank. Thedepthwas measured by looking at the ruler taped onto the tank and recorded on anappropriate table. A stand is clamped to the same table that the tank is on. A clampthat has a digital camera on it is attached to this stand and adjusted so that it is on thesame line of vision as the surface of the water and the frame encompasses the entiretank. This set up is shown in Figure 2. Figure 2: The Experimental Set-Up
After the experimental set up wasprepared and ready, the videocamera was switched on and put onrecord. The table is then shakenfrom side to side with its wheels stilllocked until standing waves occur,allowing the motion of the standingwaves to continue on by its own andeventually subside. The rockingback and forth of the table simulatesoscillations that occur during naturalphenomenon such as an earthquake. Figure 3: A frame from the video thatThe force that was input into making captured the motion of the standing wavesthe table shake and putting the bodyof water into motion would vary with the different depths of water in the tank due tothe varying natural resonance frequency, which is dependent on the depth of the wateritself.The video was switched off after a reasonable number of clean standing waves havebeen recorded. The same procedure is repeated two more times with the same waterdepth. Everything is then repeated with new volumes of water until data for sixdifferent volumes are recorded.Data Collection and ProcessingFigure 4: Position-Time Graph For Waves at 8,610 cm³ For Trail 3This is an example of the graphs plotted from the videos. The Sine curve of y =1.819Sin(14.67t + 4.286) + 15.25is fitted through the points in the graph. The B valueis 14.67. Implications of the graph’s quality are discussed in the evaluation.
Table 1: Raw Data For the Depth of the Water and The ‘B’ Value Including theCalculated Average and Uncertainty Depth of Value of ‘B’ Water (± 0.1 units) (± 1cm) Trial 1 Trial 2 Trial 3 Average Uncertainty 6 14.3 14.2 14.4 14.3 0.10 7 14.4 14.6 14.9 14.6 0.10 9 15.3 14.9 14.9 15.0 0.20 11 14.8 14.9 14.7 14.8 0.10 12 14.7 14.6 14.6 14.6 0.02 13 14.5 14.6 14.6 14.6 0.05The depth of the water measured for six different values have the uncertainty of ±0.01 m. The ‘B’ values have the uncertainty of ± 0.1 units, calculated by taking therange of the trials and dividing it by two.Observations:Dimensions of the TankWidth: 21 (± 1 cm)Height: 27(± 1 cm)Length: 40 (± 1 cm)Qualitative Observations Some waves were sloshing diagonally in the tank instead of in linear back and forth motion between the two sides. At some points, some water spilled out of the tank when the waves got too large. All volumes have standing waves of 3 harmonics; therefore the value of n is 3 for all volumes.Table 2: Processed Data For Volume of Water and the Period Squared Volume of Period Relative Period2 Relative Water (± 0.1s) Uncertainty (± 0.09 s) Uncertainty (± 3 cm3) (%) (%) 4674 0.44 23 0.19 47 5740 0.43 23 0.18 49 7462 0.42 24 0.17 51 8610 0.42 24 0.18 50 9512 0.43 23 0.18 49 10578 0.43 23 0.19 48This table shows the independent variable, volume of water, and the dependentvariable, period, manipulated into period2in order to be graphed for a recognizablerelationship. The uncertainty of volume is ± 3 cm3 and the highest error value forperiod is 24%, making the highest error value for period2 51%. The calculation ofthese uncertainties is shown in the sample calculations.
Figure 5: The Final Graph of Volume of Water vs. Period2The volume’s uncertainty of ± 3 cm3 is negligible in this graph, whereas the error barfor the period2 is very large 51 percent error. When a curve that goes through all thedata points is fitted, the relationship turns out to be quadratic with the equation of y=1.392x2-2.204x+0.2652. This relationship however, does not fit the theory that thevolume will be inversely proportional to period2.Figure 6: The Final Graph of Volume of Water vs. Period2 Inverse CurveWhen an inverse curve is fitted through the graph, due to the large error bars, thecurve still goes through all of them. Therefore, this relationship given by the equationy= can be considered valid, although inconclusive. Theoretically, the A value of1214 should be equal to the constantrepresented by 4L w . 3 n2g
Sample CalculationsUncertainty of Value ‘B’Using Data From Depth of 6 cm Maximum - Minimum 14.4 -14.2UB = = = 0.1 2 2Volume of WaterUsing Dimensions of Tank and Average Depth of Water From Depth of 6 cmV = Width ´ Length ´ Depth = (21±1)´(40 ±1)´(6 ±1) = 4674 ± 3UncertaintyUV = Uw +UL +UD = 1+1+1 = 3Period of WavesUsing Average ‘B’ Value from Depth of 6 cm 2p 2pT= = = 0.439 ± 0.1 B 14.3± 0.1 UT 0.1Relative Uncertainty = = = 23% T 0.439Uncertainty of Period2 of WavesUsing Highest Uncertainty of Period for Volume of 4674 cm3UT 2 = % Uncertainty of T + % Uncertainty of T = 24+24 = 48% PeriodAbsolute Uncertainty = ´ 48 = 0.09 100 UT 0.09Relative Uncertainty of Period2 = = = 47% T 0.19Expected Constant (Value of ‘A’) For Graph In Figure 6Using the Known Measured Values4L3w = 4(40)3 (21) = 60,890 n2g (3)2 (9.81) Theoretical - ExperimentalPercent Error For ‘A’ Value = ´100 = 98% Theoretical
ConclusionThe aim of this experiment was to investigate the relationship between the volume ofthe water within a rectangular tank and the period of the standing Seiche wavescreated by the simulation of oscillations. The results for this experiment areinconclusive. According to the Volume vs. Period2graph in Figure 5, the curve thatfits all the points best is of equation y= 1.392×10-9x2-2.204×10-5x+0.2652. Thismakes the relationship between the two variables a quadratic one where in shallowdepths; the period2 decreases exponentially as volume increases, and in greaterdepths, the period2 increases exponentially as volume increases. The vertex, whichlies somewhere between 7,000 and 8,000 cm2, suggests the turning point where thischange in the trend of their relationship occurs. This relationship very much possible,however, with the large window of error of 51%, this data is unreliable. Other curvescould easily be drawn to fit within the span of the error bars, making otherpossibilities valid as well. The graph in Figure 6 is fitted with an inverse curve, thetheoretically predicted relationship between the volume and the period2. The curveequation of y= , with 1214 as the value of ‘A’ is easily contained within therange of the error bars, which although does not to any extent conclude the theory thatthe relationship between the two variables would be that of an inverse proportion,does not disprove it. However, the expected value for this constant A is 60,890,making the experimental error for the constant a solid 98%. This extremely high valueindicates the unreliability of the outcome in regards to the theory. This makes the dataacquired noticeably inaccurate, and also makes it almost impossible to identify itsprecision.EvaluationThere are several weaknesses and limitations to the procedure of this experiment. Thefirst weakness was due to the method of the shaking of the table. Although thedirection at which force was applied to put the water waves into motion was along thelong side of the tank, it did not completely go toward that direction. The table’slocked wheels sometimes rotate around due to the asserted force and the shakingended up going in a diagonal direction rather than straight back and forth. This causedthe waves to slosh diagonally inside the tank, interfering with the initial standingwaves that have formed a visible line of motion along the glass tank. The resultingwaves are at times much higher and at times much lower than expected as seen on thesample graph in Figure 4.A way to reduce this error is to put one side of the table upagainst a stable flat surface such as the wall and shake the table from side to side,keeping it aligned with the surface.Another weakness within this experiment is the plotting of the graph made difficult bythe aforementioned weakness as well as the some waves that did not form adistinguished line of its motion on the glass. With no clear wave line, it is verydifficult to know where on the spread out space to plot. The method used to determinewhere to plot the graph is by eye and is naturally inaccurate. This causes the quality ofthe graph plotted to be low due to the inconsistencies as seen in Figure 4. Onepossible method to eliminate this error is to pick a period of time where theSeichewaves made create more or less clean waves that form clean lines on the glass tank toplot a graph from.