Geometric understanding of what's planar about plane waves.

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it. Each group is given a different two-dimensional vector $\Vec k$ and is asked to calculate the value of $\Vec k \cdot \Vec r$ for each point on the grid and to draw the set of points with constant value of $\Vec k \cdot \Vec r$ using rainbow colors to indicate increasing value.

Students should know both the rectangular component and the geometric definitions of the dot product. Some of this activity can only be used if students know a little about complex number algebra, including Euler's formula.

• Tabletop Whiteboard with markers
• A handout for each student

The group part of this activity should be quite quick, 5-10 minutes.

• Go around to each group and mark an origin on the grid and a vector from the origin $\Vec k$. Choose simple vectors, not longer than a few grid points. Choose a variety of vectors so that the compare and contrast wrap-up will work well.
• Some groups will have trouble getting started, particularly if they try to use the geometric definition of the dot product. Ask them if they know anything else about the dot product. This activity works well if preceded by Recall the Dot Product which prompts students to think about various definitions and representations of the dot product.
• Some groups will have trouble understanding what is meant by the position vector. Do not leave them stuck here. It is usually sufficient to draw one. If necessary, ask what the components are of the vector you have drawn.
• Prompt the students to connect the points with constant value of $\Vec k \cdot \Vec r$. Remind them to color code these connections with rainbow colors. Don't give away the punchline by asking the to draw the “lines” of constant $\Vec k \cdot \Vec r$.

This is a compare and contrast activity. Ask each group to present. They should show their white board, show their vector $\Vec k$, and their curves of constant $k$.

Points that should arise:

1. The points of constant $\Vec k \cdot \Vec r$ are straight lines.
2. These lines are perpendicular to $\Vec k$.
3. The spacing between the lines is smaller when $\Vec k$ is longer.

Ask the students WHY the lines of constant $\Vec k \cdot \Vec r$ are perpendicular to $\Vec k$. Usually someone in the class can come up with the explanation that all the position vectors $\Vec r$ that have the same projection onto $\Vec k$ have the same value of $\Vec k \cdot \Vec r$. This is a good time to remind the students that they have had to use two different representations of the dot product to completely understand this problem.

The class discussion that follows asks students successively to describe the set of points of constant $\Vec k \cdot \Vec r$ in three dimensions, the set of points of constant $\cos(\Vec k \cdot \Vec r)$, and the set of points of constant $\cos(\Vec k \cdot \Vec r-\omega t)$. For students that have studied complex numbers, a similar set of questions involving the complex exponential versions of these same expressions is also appropriate.

This activity can be used very effectively in a sequence of activities (Plane Wave Sequence). Other activities included within this sequence are as follows:

• Preceding activity:

• Recall the Dot Product: A small whiteboard question where students write something they can recall about the dot product which prompts a whole class discussion of various representations and interpretations of the dot product.

• Follow-up activities:

• Scalar Plane Waves: A Mathematica or Maple worksheet showing these rainbow surfaces in three-dimensions.

• Visualizing Electromagnetic Plane Waves: A Mathematica or Maple worksheet that extends these ideas to the vector fields of electromagnetic plane waves.

• Comparing Representations of Electromagnetic Waves: A small group or homework activity where students to compare and contrast several common textbook representations of plane waves.