Calculating the speed of wave in water Made by Ilya Introduction For this physics report, we have to calculate the speed of a waterwave inside a ripple tank (fig.1). A ripple tank is a device that uses light to project water waves onto a screen. By measuring the distance between these white wavefronts, we can find the wavelength of the waves, and since we already have the frequency from the machine, we can easily calculate the speed of the water waves. To make the result of the experiment for accurate, we took multiple wavelengths from different frequences to ensure the preciseness of our experiment. For this specific experiment, we used 10, 12.5, 15, 17,5 and 20 hertz of frequency. Fig. 1 Apparatus For this experiment we used a ripple tank, which reflects the wavefronts of the water onto the screen Frequency meter and source of vibration Ruler Center of water waves A waterfront Reference object Screen The formula In order to find the speed of any wave, we need to use the formula: ๐ฃ๐๐๐๐๐๐ก๐ฆ = ๐๐๐๐๐ข๐๐๐๐ฆ ∗ ๐ค๐๐ฃ๐๐๐๐๐๐กโ In our case, we can use this formula to calculate the speed for water, but since we have multiple data points, the result will be more accurate if the find the velocity through a graph. So, we need to adapt the formula: 1 ๐ฃ๐๐๐๐๐๐ก๐ฆ = ๐๐๐๐๐ข๐๐๐๐ฆ ÷ ๐ค๐๐ฃ๐๐๐๐๐๐กโ Using a different form of the original formula, we can use technology to graph the ! frequency against "#$%&%'()*, the gradient of the best-fit line will be the velocity of the water waves. Procedure • Select 5 different frequencies under 20Hz. • After selecting 5 different frequencies, set up a camera so that the entire ripple tank is in the frame. • Use tape to stick a ruler onto the screen, this can be used as caliboration stick in the tracker software. • Set the vibrator to the first frequency and allow the camera to record the wavefronts. • After about 10 seconds, set the vibrator to the next frequency, and repeat the process. • At the end, put in a reference object, and measure the diameter of the reference object in real life. • After recording all the different frequencies, upload the video into the tracker application and record 3 different wavelengths for each frequency, write these down. • Using the size of the reference object in real life and on the screen, calculate the size ratio. • Change all the wavelengths to the real size and then graph 1/wavelength and frequency. • The gradient of this graph should be the velocity of the water waves. Where is the wavefront? For this experiment, it is very important for us to fully understand where the wavefronts are. Since we used a ripple tank, the light will be projected onto the water and reflect off a mirror onto the screen. Since waterwaves consist of crests and troughs, when the light hits the wave, there will be convex or concave. If the light hits the convex part of the wave, the light will refract and all the light will converge at a single point. If the light hits the concave part of the wave, the light will reflected and thus no light will be shined onto the screen. So we can conclude that the white circle on the screen are the crests of the waterwaves. Method of collecting data In this experiment, we used an online application called tracker to help us find the distance between the different wavefronts. This helps us improve the accuracy and preciseness of the results of the experiment. We used a video to record the wavefronts of all the different frequencies. We took screenshots of when the wavefronts were most visible and then put them into the tracker application. For each frequency, we would record 3 different wavelengths on the same plain, and then take the average. We did this to make sure the value of the wavelength was as accurate as possible. Using the tape measure feature in tracker, we tried to make the wavelengths as accurate as possible. We also used axes to make sure the points between 2 wavefronts were straight, this also increases the accuracy. Raw data table Wavlength and frequency: Reference object: • Please note that all the values in the table are written in 3 significant figures Processed data table Processed data table for wavelenghts on screen, including the average and the uncertainty: 3 significant figures Table for real water wavelength and its uncertainty: Table for 1/wavelength and its uncertainty, this is the data that we will use to graph the velocity: Table for the ratio between real size and screen size: 3 significant figures 3 significant figures 3 significant figures Calculation samples The process to get our final value of velocity is quite complicated, so I will use a calculation sample to show how I did it. • First, I measured the value of wavelength, in my example I will use the 3 wavelengths of the 10 Hz frequency. • After this, we want to find the average of these wavelengths: 0.0456 + 0.045 + 0.045 = 0.0452 ๐๐๐ก๐๐ 3 • After getting this number, we can use the ratio (1.74) to find the real-life size of the wavelength: 0.0452 = 0.0260 ๐๐๐ก๐๐ 1.74 • After obtaining this value, we can calculate the value of 1/wavelength: 1 = 38.5 ๐๐๐ก๐๐ !" 0.0260 • This is the value we use to graph the value of velocity; we need to do these calculations for every frequency to be able to graph to value of velocity. • We should also calculate the values of uncertainty to help us determine the value of uncertainty for the velocity. Graphing the velocity • This graph displays the frequency over 1/wavlength which is the velocity of the water, it also displays the maximum gradient and minimum gradient • The graph includes error bars which show the uncertainty of the velocity. • Velocity of water waves = 0.263 m/s • The max. gradient is 0.324 and the min. gradient is 0.226 • (0.324 – 0.226)/2 = 0.049 m/s • So final value of velocity is 0.263 ± 0.049 m/s Precision analysis (conclusion) • Our final velocity for the water waves is 0.263±0.049 m/s 1.123 • This mean that our percentage uncertainty is 1.456 = 18.7% • This is a very high number, which means there were many errors in our experiment. Error analysis False measurement in tracker • Inaccurate measuring in the tracker application can cause error in our final velocity of the water waves, inaccurate measurements in the tracker software could include: • Inaccurate distance between wavefronts. • The calibration stick may not have been entered accurately. • The center of the wavefronts may have been marked incorrectly, causing incorrect axes. • Inaccurate measurement of the diameter of the reference object. All these human errors will decrease the accuracy and precision of the result, an increase in these measurements mean that the velocity of this wave increase, and vice versa. Error analysis False ratio • The measurement of the diameter of the reference object on the screen is a crucial step of the process, as it determines the ratio between the wavelengths on the screen and the real lengths of the wavelengths • The diameter might have been measured when the reference object was half floating, causing an incorrect measurement and ratio. • The reference object should be directly under the light source so that its real diameter is projected onto the screen. These errors will also decrease the accuracy and precision of the final result, an increase in the diameter will decrease the velocity of the wave, and vice versa. Error analysis Waves dispearse • As the wavefronts travel through the water, they will start to decelerate because of friction caused in the water. • This means that the wavelength may begin to change as the wave goes on. • This type of error is not avoidable. This error will also reduce the accuracy and precision of the final value of velocity. This type of error is not avoidable and so we don’t know the effect of this error on the velocity. Error analysis • Air conditioning If the air conditioning was on, it may have affected the way the wavefronts were moving, causing an inconsistent wavelength and thus decreasing the accuracy of the experiment. This error can decrease and increase the velocity. • The light source The light source is not directly above the entire water surface, the effect of this is that as the waves travel further away from the light source, the angel at which the light hits the water will be different, causing a change in the shadow of the wavefront projected on the screen. This error can also decrease or increase the velocity. • Amount of water The amount of water in the ripple tank can also be a source of error. This is because if there is less water in the tank, the amplitude of the water waves will be lower, and thus it will be less visible on the screen. This will make it harder to measure wavelength and thus the accuracy of the experiment will decrease. This error makes it harder for us to measure the wavelengths, and thus the precision goes down.
