Notes on & illustrations of reflection and transmission & impedance: reftrans_wiki.ppt

Set up the problem of an abrupt boundary in a rope, e.g. a seamless transition from a rope of one density to another, that cause the wave speed to change at a position called $x = 0$. Ask the students if they can think of a situation where they have encountered something analogous. The previous day's discussion usually prompts a response of a light wave incident on glass or water. If not, you can elicit that.

Set up the problem so that there is single frequency of propagation (unfortunately, the problem is mathematically more tractable with a single harmonics wave, but much easier to visulaise with a non-dispersive pulse). Students know that “waves are reflected and transmitted” at a boundary, but they don't have a clear idea of the waveforms on either side, or of the phases and magnitudes of the resulting waves. Set all the parameters to be (in principle) different on each side of the boundary, $\omega _{left} $, $\omega _{right} $, $\lambda _{left} $, $\lambda _{right} $, $v _{left} $, $v _{right} $, $k _{left} $, $k _{right} $, $etc$. Discuss which quantities are the same on both sides, and why we know, or don't know. The most important one is that $\omega _{left} = \omega _{right} $, which is a consequence of the fact that the rope doesn't break at the junction.

There should be right-moving and left-moving waves on both sides of the boundary, in principle, and convention dictates that we have only one source, from the left. Discuss this explicitly - thus the disturbance on the right side is a sum of left- and right-moving waves, but the disturbance on the right is right moving only. Elicit from the students that we don't know the amplitudes of the constituent waves - our purpose is to find them.

Discuss the consequences of the boundary conditions:

1. “The rope doesn't break at the junction” is a piece of physics that has to be reflected in the mathematics. How are they to accomplish this? They've never thought about it from this point of view. It's $\psi _{left} \left( {0,t} \right) = \psi _{right} \left( {0,t} \right)$.
2. “The transverse force at a single point in the rope must be zero”. A point on the rope is massless, but it experiences finite (transverse) acceleration, so the (transverse) force must be zero. This is a harder concept, especially since the force equation was set up with infinitesimal bits of the rope with finite mass $\mu dx$, two situations that are not immediately distinct to the students. Zero transverse force leads to the continuity of the slope at the junction (but only because the tension is assumed constant throughout the rope and hence cancels from the constraint equation). $\left. \frac{\partial \psi _{left}}{\partial x} \right|_{x=0,t}=\left. \frac{\partial \psi _{right}}{\partial x} \right|_{x=0,t}$

It's important that condition 2 be stated as a force constraint rather than as “continuity of the slope”. There's no reason $a$ $priori$, for a continuous slope of the wave-like quantity $\psi$. In this case, the continuous slope is a consequence of the physical requirement of Newton's law and the fact that tension os constant. (This comes in handy later on when electromagnetic waves in cables are discussed. In that case, capacitance has an analogous role to tension, and it is not the same from one cable to the next.)

Finally, perform, or have the students perform, depending on time, the mathematical manipulations that lead to the reflection and transmission coefficients (for the displacement wave)

• $R_{\psi }=\frac{k_{left}-k_{right}}{k_{left}+k_{right}}; T_{\psi }=\frac{2k_{left}}{k_{left}+k_{right}}$

Impedance: To make the result more general, it is useful to introduce the concept of impedance, which is unfamiliar to students in the context of waves on ropes, but is of course familiar in the context of ac circuits. This serves as motivation to introduce impedance because the activity following is to study the propagation of voltage waves in a coaxial cable.

The simplest example is of a rope at the end of a piston. Define impedance as the constant that relates the applied force to $\frac{\partial \psi }{\partial t}$. The electrical analog, where the voltage corresponds to the applied force and the current to $\frac{\partial \psi }{\partial t}$ ($\psi$ corresponds to charge) is comforting! Show that for a traveling wave, the impedance is the ratio of the tension in the string to the phase velocity, $Z = \tau /v_{ph} = \tau k / \omega $. The proportionality to $k$ yields

• $R_{\psi }=\frac{Z_{left}-Z_{right}}{Z_{left}+Z_{right}}$; $T_{\psi }=\frac{2Z_{left}}{Z_{left}+Z_{right}}$

“Force” reflection & transmission coefficients: It is unconventional to think of a “force” wave or a “force” reflection coefficient, but $\frac{\partial \psi }{\partial x}$ measures the force that one side of of the rope exerts on the other, and $\frac{\partial \psi }{\partial x}$ is a harmonic disturbance that is 90 degrees out of phase with $\psi$. This is in analogy with the very commonly encountered pressure and displacement in a sound wave or voltage and current charge oscillations in a coaxial cable.

Show that the ratios of the reflected to incident and transmitted to incident amplitudes (including sign) of the 90-degree-out-of-phase wave are

• $R_{F}=-\frac{Z_{left}-Z_{right}}{Z_{left}+Z_{right}}$; $T_{F }=\frac{2Z_{right}}{Z_{left}+Z_{right}}$

The coaxial lab that follows measures a voltage wave reflection coefficient, which is of the latter type.

It's not necessary to do all of this in one lecture. The last part can come out in the course of the lab activity.