RipStik sine wave regression mathematical model laboratory

Summer term with online lectures and residential labs meant that the introduction to waves using a RipStik could only be presented via a video. 

The evening prior to laboratory nine I found that Desmos could regress to a sine function if given the five points seen in the image above: three centerline points and the peak of a crest and the bottom of a trough. The regression remains fairly stable for small perturbations of the data points. Prior experiments had yielded unstable results often with extremely high frequencies. This would form the gist of laboratory nine.

I started the build of the poster paper inside and then moved outside. Wind, even the slightest of breezes, is problematic. 

Scotch tape was used to assemble an eight sheet run. Each sheet is 83 cm long, thus the whole run was 664 cm. The wind held off long enough to make the run.

The run was made barefoot as notes from prior terms on runs made during a Monday lecture-demonstration suggested a larger amplitude and more control resulted from making the run barefoot.

The students determined the best way to get the sheets back to the classroom

By sheer coincidence, eight sheets just fit on the lab table.

I then gave the introduction seen above for generating a table, graph, and the function for the regression. I had wanted to use y₁~a sin(2πx₁÷𝜆) but this proved unstable in testing the night before. High frequency functions were the consistent result. Only the form a sin(bx) proved stable. This meant that the wavelength would have to be calculated post hoc from 2π÷b.

Joyleen, Melissa, Kiora, and Allison.

I deliberately omitted measurement of time and thus frequency, period, and wave velocity. With no introduction during the summer term, I thought keeping this simple would be preferable.  As might be seen in the images, I sketched the curve with with the larger amplitude using a marker. There were six wavelengths on the eight sheets. The sketched curve provided a starting point for questions such as "How many waves are on the paper?" and "What counts as a wave?"

Nicole and Angelica make measurements.

I went ahead and divided up the six sine waves using the lines that can be seen running edge-to-edge on the paper. I had intended to cut the paper and give each group one full cycle of a sine wave. Six sine waves in 664 cm means a wavelength of up around 110 cm.  Note that my board control is not sufficient to ensure that I started and ended on an integer multiple of the wavelength. 

Nicole, Ilani

The class decided to work with the paper as is. In the above image Nicole is identifying the centerline. Angelica is measuring perpendicular to the centerline to obtain the amplitude.

The students worked on making their own centerline and then choosing the five points. Three are the centerline crossing points for one cycle of a sine wave. The wave was intentionally measured from (0,0) with a positive crest at π÷2 as seen above, a centerline crossing at π, a negative amplitude for 3π÷2, and a final centerline point at 2π. 

Ilani measures the amplitude

The eight sheets with six waves could support up to six lab groups, twelve pairs of students. I had meant to cut the sheets apart and give each pair their own wave to work on, but the students chose to leave the papers connected. As one student noted, "The order will get mixed up." The order might matter if the wavelength was changing as I moved down the paper. The regression wavelengths from west to east were 109.8 cm, 116.4 cm, 109.7 cm, and 103.6 cm. That does not suggest much more than random variation in measurements. 

Angelica recorded the coordinates 

The data that was generated was fairly consistent from group to group. 

I did remind the class that the fourth data point had a negative y-value.

Angelica suggested the graph axis labels based on a diagram in the textbook that I had long ago forgotten about.

The four functions. The average of the four amplitudes was 6.9 cm, while the average of the four wavelengths was 109.7 cm.

The laboratory required two hours from set-up to finish. My sense is that adding in the element of time might overcomplicate the analysis. The time would be a total run time which would have to be divided by six, in this case, to obtain the period for one wave. Period has proved to be a difficult concept for some students to fully grasp, and that the period would be a calculated result adds to the obscurity. While the present lab is simple, there is an appealing cleanliness to only looking at the wave in space and setting aside the wave in time. This does mean that frequency and wave velocity are not introduced, but for these students the sine function is already one mystery.

I think laboratory also brings me back around to why this class does not have a mathematics prerequisite. Some colleagues have suggested that physical science should have a mathematics prerequisite. I counter that mathematics should have a physical science prerequisite. I believe physical science is the better vehicle for introducing mathematical concepts. As one student remarked while looking at their Desmos chart, "The wave goes on forever." 

I think this laboratory may be a worthy alternative to the speed of sound laboratory, and has the advantage of being less impacted by weather. I think during a regular term with the introduction on Monday to the RipStik wave and demonstrations on Wednesday, the lab could be expanded on a Thursday into frequency and period. Perhaps.