## Plane Waves

By the time students are in their senior year, most have had some experience with planes, but many students are unable to say what's planar about plane waves. This sequence of activities helps students understand $\vec{k}\cdot\vec{r}$ and visualize different representations of plane waves commonly used in physics. Some of the activities in this sequence were the first to be designed and have even found their way in TA training sessions and teaching seminars. We often use these activities as guided whole class discussions, with the instructor controlling the computer, although they can work well as small group activities.

Overall, this sequence attempts to introduce students to what is planar about plane waves through first considering what is meant by the dot product and in particular, $\vec{k}\cdot\vec{r}$–a scalar quantity. Once students have an understanding of this dot product both algebraically and geometrically, as emphasized in Visualizing Plane Waves, they can proceed to understand $\cos{\vec{k}\cdot\vec{r}}$ as alternating through values ranging from -1 to 1. Adding in time dependence means that these planes progress in time and can be expressed as $\cos{\vec{k}\cdot\vec{r}-\omega t}$. This forms the fundamental understanding of the algebraic and geometric meanings of plane waves as used in physics.

Students begin to learn about plane waves in the context of electricity and magnetism in Visualizing Electromagnetic Plane Waves which introduces the idea of perpendicular fields, electric and magnetic, which oscillate in strength according to position which extends the algebraic notation of plane waves to include an expression of the form, $\vec{E_0}\cos{\vec{k}\cdot\vec{r}}$. Which allows students to compare and contrast many different representations of plane waves which are used in physics and analyze what information is presented in each representation and what is neglected.

### Activities

• Recall the Dot Product (Estimated time: ): This small whiteboard question prompts students to recall something they recall regarding the dot product. This can be used to remind students of the different ways in which the dot product can be represented. This small whiteboard question can lead directly into the following activity, Visualizing Plane Waves, which requires students to think about the dot product first using unit vectors and then as a projection.
• Visualizing Plane Waves (Estimated time: 5-10 minutes, Wrap-up: ): In this small group activity, students calculate $\vec{k}\cdot\vec{r}$ for a given $\vec{k}$ for each point on a grid and then connect points with constant value of $\vec{k}\cdot\vec{r}$. When students are first taught about plane waves, most simply recognize the algebraic expression of a plane wave, but most do not understand what is planar about plane waves. This is particularly troubling when students move on to study areas in physics which heavily rely on plane wave approximations, such as optics. This small group activity has students connecting the algebraic expression of a plane wave with its geometric content, allowing students to visualize what is planar about plane waves.
• Scalar Plane Waves (Estimated time: ): This computer visualization activity allows students to visualize scalar plane waves by choosing a wave vector, $\vec{k}$, and seeing the resulting orientation of plane waves. This addresses representations of plane waves as well as what is planar about plane waves.
• Visualizing Electromagnetic Plane Waves (Estimated time: ): This Mathematica (or Maple) activity is designed to allow students to visualize electromagnetic plane waves by choosing a $\vec{k}$ and exploring the electric and magnetic field vectors of a given plane wave. The first time students are introduced to plane waves is likely in a class on electrodynamics. Motivated by Visualizing Plane Waves, this small group activity has students work out the algebraic expression and geometry of an electromagnetic plane wave, allowing them to visualize the planar behavior of an electromagnetic plane wave.
• Comparing Representations of Plane Waves (Estimated time: ): This activity can be used as a homework problem or as a small group activity where students analyze multiple representations of plane waves, noting which features are present and which are ignored in each representation. This builds on the representations introduced in Visualizing Electromagnetic Plane Waves and Scalar Plane Waves. Most upper-division students learn at least one representation of plane waves, but many have trouble connecting one representation of a plane wave with another. This final activity allows students to witness the various representations of plane waves professional physicists use.

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