Oscillations and Waves: NoTerm-2023
HW 5b : Due W5 D5

  1. Light Propagation in a Vacuum S0 4628S Light propagates in vacuum with speed \(c\) and in a medium with speed \(v = \frac{c}{n}\) where \(n\) is the refractive index (\(n>1\)). (The refractive index is not an integer!) Show that when light is incident from vacuum (\(n=1\)) onto glass (\(n=1.5\)), about 4% of the energy (which is proportional to \(|\psi|^2\) is reflected. Also show that the light changes its phase angle by \(\pi\) when it is reflected.
  2. Square Wave in a Rope S0 4628S A rectangular traveling pulse is launched with speed \(v_1\) from the left into a very long rope. At the boundary at \(x=0\), the pulse is partially reflected and partially transmitted into a second rope where the transmitted pulse moves with velocity \(v_2\). The reflected (black) and transmitted (red) pulses are depicted some time after the original pulse encounters the boundary. The system obeys the NDWE.

    Figure: A point in time of the graph of a square wave after reflecting and transmitting at \(x=0\).

    1. Use the widths of the reflected and transmitted pulses to find the ratio of \(v_1\) to \(v_2\). Explain.

    2. Calculate the relative velocities using the pulse locations and show that this is consistent with (a).

    3. Is the mass density of the red rope smaller or larger than the black rope? Why?

    4. Describe the original pulse (height, polarity, length), showing qualitative and quantitative reasoning.

    5. Can you determine the velocities \(v_1\) and \(v_2\)? If so, what are they, and if not, what information would you need?
  3. Reflection from the End of a Coaxial Cable S0 4628S Regarding the coaxial cable we investigated:

    1. What is the impedance of your cable as measured by the matching terminating resistance?

    2. What is the total measured resistance of your cable (Be sure to include the center wire AND the shield resistances)?

    3. What is the damping parameter for your system?

    4. Now add to your data plot the results (use a line to indicate a model) expected from a model (i) in which resistance in the cable is neglected and (ii) resistance in the cable is incorporated and assumed to be relatively small compared to the impedance of the cable (“light damping”).

      Make sure axes are properly labeled, color is nicely used to convey maximum information, and legends etc. are arranged so the reader is given the information in the clearest and most direct way.