Middle division students need to understand the three different velocities that are relevant for describing wave propagation. We use a sequence of activities to introduce and build conceptual and mathematical understanding of these different velocities.

First, we introduce and distinguish between phase velocity and material velocity in an interactive demonstration of waves on a string. Our students usually come in knowing that the speed of a wave is equal to the product of the wave's frequency and wavelength. However, many do not distinguish between the propagation speed of the wave (in this case, phase velocity) and up-and-down motion of the small segments of rope. In this lab, students plot the angular frequency as a function of wavenumber for some standing waves, and identify the slope as the phase velocity. However, they do not perceive this propagation speed for the standing waves; the only motion they perceive is the up-and-down motion of the rope - the material velocity.

Next, we have the students practice mathematical descriptions of waves. Specifically, we have students practice using initial conditions to specify a wave solution. Our students have some knowledge of differential equations, and some students know that a second-order differential equation requires two integration constants to specify a solution. Therefore two initial conditions are needed, e.g. the initial spatial distribution of the wave and the initial distribution of material velocities.

Then, we further strengthen students conceptual understanding of wave velocities by considering the energy density of a wave. When asked to identify the energy of a wave, students readily suggest there is kinetic energy but are vague about the specific velocity needed to calculate the kinetic energy. During the activity, students come to see how the segments of high kinetic energy (high material velocity) of a traveling wave are transported along a rope at a rate equal to the phase velocity.

Finally, students are asked to measure the propagation speed of a pulse sent along a coaxial cable. In this lab, students send a square pulse along the cable, which has the property that the constituent waves all travel at approximately the same speed. This can lead nicely into a discussion of a situation where this might not be the case - in which case it is important to distinguish between phase velocity and group velocity. More directly, this lab allows for a nice discussion of material and phase velocities: in an electromagnetic wave, when is the material velocity?

• Dispersion Relation for Waves on a String (Interactive Demo)

• Using Initial Conditions to Specify Analytic Wave Solutions (Small Group Activity)

• Energy Density of Waves on a String (Small Group Activity)

• Wave Propagation in a Coaxial Cable (Laboratory Activity)