Activity: Travelling wave solution

Contemporary Challenges 2022 (4 years)
Students work in a small group to write down an equation for a travelling wave.
  • Media
    • activity_media/wave-sketch_IhcmUnF.svg

This activity follows Introducing quantum mechanics.

Write a function \(y(x,t)\) that describes the motion of this string:

A wave with an amplitude of 0.1 m, it travels to the right, and has a wavelength of 8 m. In one second it travels half a meter to the right.

You may put the origin of your coordinate system wherever you want. In your final expressions, do not use any symbolic variables except for \(y\), \(x\) and \(t\). Please do remember (as always!) to include any units that are associated with constant numbers.

First try to construct this wave on your big whiteboards and then you can check your answer by putting it into Desmos.

Extra fun
Show that your answer is a solution to the wave equation: \begin{align} \frac{\partial^2 y}{\partial t^2} &= v^2 \frac{\partial^2 y}{\partial x^2} \end{align}

  • assignment Series Convergence

    assignment Homework

    Series Convergence

    Power Series Sequence (E&M)

    Static Fields 2022 (5 years)

    Recall that, if you take an infinite number of terms, the series for \(\sin z\) and the function itself \(f(z)=\sin z\) are equivalent representations of the same thing for all real numbers \(z\), (in fact, for all complex numbers \(z\)). This is not always true. More commonly, a series is only a valid, equivalent representation of a function for some more restricted values of \(z\). The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain, called the “interval” or “region of convergence.”

    Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then, using the Mathematica worksheet from class (vfpowerapprox.nb) as a model, or some other computer algebra system like Sage or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence. You may need to include a lot of terms to see the effect of the region of convergence. Keep adding terms until you see a really strong effect!

    Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).”

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    2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water \(c_p = 4.2\) Jg\(^{-1}\)K\(^{-1}\) and the density of water is 1 g cm\(^{-3}\), and both are roughly constant over this temperature range.

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Learning Outcomes