• Recognizing algebraic forms of traveling and standing waves
• Specifying solutions to second order ODE's using initial conditions

Estimated time: 30 minutes

To write down the solution to the 1-D wave equation that matches specific boundary conditions using several standard algebraic forms.

1. Familiarity with general solution of ODE, $\frac{d^2 \psi \left( t \right)}{d t^2}=-\omega ^2 \psi \left( t \right)$.
2. The basic language of waves.

• Table top whiteboards and markers
• A handout for each student

The students know that the position and velocity of an oscillator at, say, $t$ = 0, determine the two arbitrary constants that appear in the equation of motion for the oscillator: $B_{p}\cos \omega t+B_{q}\sin \omega t$ or $A\cos \left( \omega t+\varphi \right)$ . They also know that this is the solution to a 2nd order ordinary differential equation.

Now that they have the equation of motion for a “collection of oscillators”, they need to understand how to determine the arbitrary constants that appear in the general form. They have also studied the PDE that leads to this general form, but they don't yet know what this has to do with physics. Keep reassuring them that Newton's law or the Maxwell equations (which are coming up soon) will lead to this form!

The mini-lecture, or introduction to the activity, is to use the example of a transverse wave on a rope, which comes closest to a collection of oscillators. The students now know that most general form of a wave is

\[\psi \left( x,t \right)=A\cos kx\cos \omega t+B\cos kx\sin \omega t+\sin kx\cos \omega t+D\sin kx\sin \omega t\]

then if they know the position and velocity of $every$ point on the rope at, say, $t$ = 0, they will be able to determine the 4 arbitrary constants. In other words, if they know $\psi \left( x,0 \right)$ and $\left. \frac{\partial \psi \left( x,t \right)}{\partial t} \right|_{t=0}$ , then $A$, $B$,$C$ and $D$ will be known.

Students divide themselves into groups of three. Having just discussed the general form of the solution to the 1-d non dispersive wave equation, they must now find some specific forms that conform to particular conditions at time $t$ = 0. It is important to understand which constants are considered unknown, and which are known. For example, when students see, “The wave form at $t$ = 0 is $$\psi \left( {x,t = 0} \right) = \Omega \sin \left( {kx} \right)”,$$ they may not recognize that it is implied that $\Omega $ is known and that it defines $A = \Omega ;B = 0$ in the general expression $$\psi \left( {x,t = 0} \right) = A\sin \left( {kx} \right) + B\cos \left( {kx} \right).$$ Such things need to be taught explicitly.

1. It is important to recognize that some students have trouble distinguishing the arbitrary constants $A$, $B$, $C$ and $D$ from the system parameters $\omega$ and $k$. They all look arbitrary to the students. And in fact it's a subtle point. There's really only one system parameter, and that's $v_{ph}=\frac{\omega }{k}$. The frequency $\omega$, at this stage in the absence of spatial boundary conditions, can be chosen at will. The students need help classifying all these different “constants”.
2. The form of the wave equation and the general form of the solution are deliberately not given. They have been discussed in class and the students should refer to notes or remember what they are. Instructor can use either superposition of standing waves ($\cos{kx}$, $\cos{\omega t}$, etc.) or traveling waves $\cos{(kx-\omega t)}$ etc.). Similarly, the significance of $\omega$, $\Omega$, (parameters of the physical problem) and $A$, $B$, $C$, $D$, (variables introduced to solve the problem) are ideas that must be reinforced. Ask the students to say, in words, what they are. The idea that the problem is “solved” when $A$ – $D$ have been expressed in terms of $\omega$, $\Omega$ is not clear to novices.
3. Students stumble when application of a boundary condition gives no information about a coefficient. If students apply $\Psi(x,t=0)=\Omega\sin{kx}$ to the general form:\[\Psi(x,t)= A\cos{kx}\cos{\omega t}+ B\cos{kx}\sin{\omega t}+C\sin{kx}\cos{\omega t}+D\sin{kx}\sin{\omega t}\] They often conclude that $D$ = 0 even though this application gives no information about $D$.
4. Some students don’t realize that they can generate traveling waves from the general form in (3), because, they say, they have written down “a sum of standing waves”. It takes actual application of the BCs to convince them that they have generated a traveling wave. This despite the fact that they will tell you that a traveling wave is an allowed solution to the differential equation, and that the form in (2) is a general solution.
5. Some students are still novices at recognizing the form of a traveling wave and a standing wave. The animation exercise is helpful in this regard.
6. Some groups get through 2 or 3 of the examples in the 15 minutes allotted; others only one. Few do them all. It is important to have the groups report on their results, so everyone can complete the table. Stress that the examples a particular group did not do in class should be studied at home. Instructor may have a master computer set up with animation capability. As students report values for coefficients, these can be entered in Maple or equivalent and the waveform animated for all to see.
7. Beware of groups using different coefficients for different terms. Be sure they have a consistent convention beforehand. If not, it makes for good discussion to have one group’s “$A$” be identified with another group’s “$C$”, but it takes time.
8. Instructor and TA circulate during group discussion, discussing stumbling points with a group, making sure groups remain on-task. Groups working easier problems must be directed to work on a second, harder problem. Particularly good groups could be assigned to write the other form of the general solution (traveling or standing as the case may be).

Groups write their values for the coefficients on the board and whether they think the wave is traveling or standing. Ask a single group to present one result, and as they do so, the instructor runs a Maple worksheet Maple 13;Maple classic, entering the group's coefficients and running the animation to see if the wave progresses as predicted. Connecting traveling waves to the combination $x-vt$ and standing waves to separate $x$ and $t$ is critical, especially if one is a superposition of the other.

Ask the groups to (re)derive, for example, the identity \[\cos \left( kx-\omega t \right)=\cos kx\cos \omega t+\sin kx\sin \omega t\] and elicit the response “a traveling wave can be written as the sum of two standing waves”. $Vice$ $versa$ for: \[2\cos kx\cos \omega t=\cos \left( kx-\omega t \right)+\sin \left( kx-\omega t \right)\] Most students are not able to pull any of these identities from memory. They no longer memorize them in earlier courses.

This is a good activity to serve as the foundation for a later discussion of Fourier Series.