Wave Dispersion and Mathematica


Elizabeth Petrik, a graduate teaching fellow for Physics 15c, created this Mathematica activity to help students build physical and quantitative intuition about wave dispersion. The usage of Mathematica in this activity allows for students to not only solidify the concepts they learned in class, but also create a working program that helps them understand wave dispersion in another medium.

Students were supposed to know how to use a wave equation to find the dispersion relation (i.e., the velocity of waves of different wavelengths) and the phase and group velocity. In addition, the instructor asked them to bring a laptop with Mathematica installed to discussion section. The instructor prepared a handout reviewing dispersion concepts, and wrote prompts on the board. She then proceeded to demonstrate the Mathematica exercise on her own computer on the overhead projector.

Students could work either individually or in groups on this dispersion review exercise. Then, they regrouped as a section, and students answered the questions on the handout out loud and came up to the board to fill in a plot drawn by the instructor. 

After they worked through the dispersion review exercise in the handout, the instructor wrote a wave equation on the board along with three questions: (a) What is the dispersion relation? (b) What is the phase velocity? (c) What is the group velocity? Students solved these problems individually and then conferred with each other in small groups to check answers. Students then volunteered to write their answers to these questions on the board (Volunteers were given a small prize, like a piece of candy or a physics toy).

Next, students got out their computers and plotted the dispersion relation they had just found in Mathematica for a two different sets of different parameters, following the instructor’s example on the overhead projector. Then they plotted a Gaussian wave pulse, which was a waveform that they had studied in the previous homework set. The instructor then prompted them for the Fourier transform of the Gaussian pulse, which they had found in their homework, and wrote it on the board along with the dispersive propagation behavior that they had just derived. The instructor then announced that they were going to find how the pulse evolved in time using the dispersive propagation equation she had just written. This equation involved a complicated integral that we could evaluate numerically in Mathematica for case (i) and (ii) as well as for the non-dispersive case [case (iii)]. Before doing so, she asked them to come up with a prediction for how they thought the wave pulse would evolve in each of the three cases and invited them to give their predictions out loud. Finally, they calculated and plotted the evolution of the wave pulse for the three cases and discussed the behavior as a group, examining how dispersion affected the shape of the wave pulse over time and whether the predictions we had made were borne out. 

In the end, each student had a working Mathematica program with examples of defining functions and variables, numerically evaluating integrals, and plotting equations. This was useful to the many students in the class who had not worked with Mathematica before and who found it beneficial for visualization and aiding with their homework. In addition, the students had a visual example of wave dispersion, which they had previously examined only in the abstract, as well as a procedure for propagating dispersive waves in the general case.

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