Homework 2

Q1. Standing Waves in a rope

Results of the standing wave experiment we did in class:

number of nodes01234567
frequency (Hz)8.318.926.340.445647878

( a ) Tabulate the raw data (# nodes, frequency, wavelength, for several standing waves) and also do the necessary conversions (explain, please!) to plot the “dispersion relation” (ω vs. k) for the rope and obtain the phase velocity of the waves in the rope from the graph.

( b ) Explain how the application of Newton’s 2nd law to the system predicts a value for the velocity in terms of the physical parameters of the vibrating string system. Measure those parameters to predict the phase velocity and compare with the result from (a).

( c ) Why are the waves you observed not traveling?

( d ) What is the significance of a linear dispersion relation? Give an example of a system where the dispersion is not linear.

Q2. Traveling and Standing Waves

( a ) “A standing wave is the sum of two traveling waves propagating at the same speed in opposite directions”. Prove analytically that this statement is true or untrue by explicitly adding the waves ψ1 = A1sin(kx − ωt) and ψ2 = A2sin(−kx − ωt). You could look at an animated function to try some possibilities, but you also need an analytical proof.

( b ) “A traveling wave is the sum of standing waves”. Demonstrate analytically the truth, falsehood, or partial truth of this statement.

Q3. Main 9.9

Q4. Propagation of electromagnetic waves in vacuum

(This is relevant for the lab exercise. This is really a 3-dimensional example because the wave has components in two spatial dimensions and propagates in the third. However, the components are described by the 1-d wave equation. You will meet this particular example again in the E&M and optics courses.)

Show that in a medium in which there is no free charge and no free current, electromagnetic waves propagate with a velocity of propagation

v = 1/√(µε),

where ε is the permittivity and µ the permeability of the vacuum. (In vacuum, ε = ε0 and µ = µ0, in which case the velocity has the special symbol c.)

The following guides you through the problem. The basic approach is to show that the electric and magnetic field vectors separately obey the 1-d non-dispersive wave equation, and identify the velocity.

To begin this problem, think back to previous courses – all electric and magnetic fields obey the Maxwell equations, and you will find them in Griffiths E&M. PH320 and PH422 dealt with electrostatics and magnetostatics, but you will recall from PH213 that there are dynamic (time varying) terms in two of the equations, describing motional EMF (Faraday’s Law) and a “displacement current”, respectively. These are crucial for discussing propagation of electromagnetic waves.

Write down the 4 Maxwell equations with the source (current and charge) terms set to zero. This problem is most easily done with the equations in differential form, not integral form (you must relate derivatives to each other). The equations are coupled – E and B appear together in some. You must manipulate the equations to obtain equations in E alone and B alone that relate the second time derivative to the second space derivative. Find the constant that relates the two and hence identify the velocity. [Hint: you will have to look up a vector identity for the “curl of the curl” of a vector.]

EXAMPLES FOR EXTRA PRACTICE

Main 9.1 through 9.10; 9.13; 9.14