FREQUENTLY ASKED QUESTIONS

August 29, 2007

Administrative Questions

Can we get handouts/notes for each class?

I don't give out class notes, but there will occasionally be handouts for specific topics. If anything in your notes is incomplete, you can ask on the minute questionnaire (or ask me later).

Content Questions

Can waves be both longitudinal and transverse?

Actually yes. When looking for examples I found some nice animations on this web page: indeed some kinds of water surface waves are both longitudinal and transverse. Note that water waves are not simple harmonic traveling waves; wave equations describing different kinds of surface waves are more complicated than the basic wave equation we've been considering. Here's a Wikipedia link on water waves for those who might be interested.

What exactly does power of a wave represent?

Power is energy/time. Waves transport energy, so the power of a wave is the energy per time that it transports.

An analogue for power's relation to displacement and frequency would be nice. Perhaps a dimensional analysis?

I'm not sure I can write a simple analogue, but here is a link with a derivation of power transported by a wave for the specific example for a traveling wave on a string:
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/powstr.html.
(Click on ``Show'' for full derivation). In this case, the power is averaged over time.

Here is another derivation: in the figure from the wave equation derivation link, we have that the transverse ($\perp$ to the string) component of the force on the string is $F_{\rm {transverse}} = T \sin \theta \sim T \tan \theta = T \partial u/\partial x$. (Here $T$ is tension, not period.) The transverse velocity (velocity $\perp$ the string) is $v_{\rm {transverse}}= \partial u/ \partial t$ (and can be positive or negative). You may recall from mechanics that $P= \vec{F} \cdot v_{\rm {transverse}}$. Plugging in, this is then $P= - T \frac{\partial u}{\partial x}\frac{\partial u}{\partial t}$. Now plug in the harmonic traveling wave solution, $u=H \sin (k x - \omega t)$, and you get $P = Tk \omega H^2 \cos^2(k x - \omega t)$, and since for this case $T= \mu v^2$ and $v=\omega/k$, we get $P = \mu v \omega^2 H^2 \cos^2(k x - \omega t)$. If you average over time, the cosine$^2$ term becomes $\frac{1}{2}$.

For this particular example, you can see that $P \propto \rm {amplitude}^2$, and $P \propto \rm {frequency}^2$. It turns out that these relations are general features for all waves, not just waves on a string.

The basic idea is this: you figure out the force and transverse velocity in terms of the displacement, use these to get the power, and plug in the wave solution.

You can also check that the dimensions work out: dimensions of $\mu$ are mass/length, dimensions of $v$ are length/time, dimensions of $\omega^2$ are 1/time$^2$, and dimensions of squared amplitude are length$^2$: this works out to $ML^2/t^3$; since $P=F v$, dimensions of power are $(M L/t^2) * L/t = ML^2/t^3$, as required.

What's the difference between hard and soft boundaries?

The difference has to do with the internal forces of the medium at the boundary. Suppose the boundary can't move when a wave hits it, and the wave (say) pushes upward on the boundary when it hits, then the boundary will (by Newton's third law) push down on the piece of string: this will make a reflected wave with a phase change of $\pi$. In contrast, if the string is allowed to move at the boundary (a ``soft'' boundary), a reflected pulse with no phase change results.

Here is the link I showed in class with some more info on this and some animations which may make this more clear:
http://www.kettering.edu/$\sim$drussell/Demos/reflect/reflect.html.

Different medium densities can affect the boundary conditions, $e.g.$ for a string, high density (mass/length) means low wave speed, and a wave reflected from string of a higher density behaves like one reflected from a fixed end (see animations at bottom of above link).

You said the boundary condition for a soft boundary at $x^*$ is $\partial h/\partial t (x^*, t) = 0$? Huh? That doesn't seem right.

It's the first derivative with respect to $x$, not $t$, that must be zero at a soft boundary. If you look carefully at the soft-boundary-reflection animation, http://www.kettering.edu/$\sim$drussell/Demos/reflect/reflect.html. you will see that this is always true: the slope at the boundary is always horizontal.

What is the effect of boundary conditions on reflections? Also, if one were to have an interface between two different wave-transmitting media: how does the math work for the part that reflects back or the part that gets transmitted?

See link for questions above. It is possible to have some reflection and some transmission, and at the interface the waves and their first derivatives must match. We're actually going to see examples of this sort of thing, in a different context, later.

Can you explain coherence again?

The condition for coherence is that the $\omega$'s and $\phi$'s of two waves must have a definite, known (or ``figurable-outable'') relation. The two waves don't have to have the same frequency or the same phase, nor do these necessarily have to be constant (they can be varying in time in a known way), but they do have to have known frequencies and phases. By ``known'', I mean that at any point in space and time we can figure out the phase and frequency of each wave, so that we know what the summed amplitude is that point in space. The phases and frequencies can't be random.

For pretty much all examples we'll see there will be a fairly simple relations, e.g. constant phase differences and frequencies.

What does it actually mean for waves to be coherent?

See above question.

When we talk about coherent waves and say there is a relation between $\omega$, $\phi$, and what constitutes a relation? Just when we know the frequencies or that they are the same?

See above questions: frequencies and phases need not be the same, but they must be known. If combining waves have phases and/or frequencies which are random, or change randomly, with respect to each other, then there will be no regular or calculable interference: everything just gets all smeared out. We say that the waves are incoherent. If, in contrast, the relative phase of the combining waves can be known (and, presumbly, the relative phase shift follows some regular pattern), then there will be interference pattern that can be calculated.

Does coherence mean that $\omega$ and $\phi$ are related, or coupled, or just that we know them for some particular wave?

They need not be related or coupled to each other; but we need to know them and be able to figure out the relative phase difference at any point in order for the waves to be called coherent.

Does polarization matter for light coherency?

Well, it can matter for interference. Coherence of phase and frequency is independent of polarization, but the electric and magnetic field directions do matter when waves are added.

What do Maxwell's equations mean?

Maxwell's equations describe electricity and magnetism; all electromagnetic phenomena can be represented by solutions to Maxwell's equations under different boundary conditions. Electromagnetic waves are one possible solution.

Here's a link I found from googling around that may be helpful for those who want a quick review: www3.baylor.edu/ Walter_Wilcox/courses/phy2435/chap32xx.pdf. Generally you can find this sort of stuff in any intro E&M text.

You need to have a general understanding of these concepts and we will be drawing on this material at various times, so you may want to brush up if you are rusty, or do some serious reading if you haven't seen this before. But this is not the focus of the course and you will not be asked direct questions about Maxwell's equations.

How do Maxwell's equations give wave equations under some conditions?

Solutions to Maxwell's equations need not be waves. For instance, a single point charge that's just sitting there will just have a static electric field associated with it that's the solution to Maxwell's equations; it won't make an electromagnetic wave.

But a common physical situation is an oscillating charge (or oscillating magnetic field); Maxwell's equations applied to this case do lead to wave equations for electric and magnetic fields. You can very likely find a derivation in your intro E&M text.

The derivation of an electromagnetic wave propagating in vacuum makes use of Ampère's and Faraday's Laws. Here is an example link:

http://musr.physics.ubc.ca/$\sim$jess/hr/skept/Maxwell/node5.html

You can think of an electromagnetic wave qualitatively in this way: moving charges are equivalent to electric fields; for an oscillating charge the electric fields wiggle up and down. Changing electric fields (currents) create magnetic fields by Ampère's Law; changing magnetic fields in turn create electric fields, by Faraday's Law. The wiggling fields therefore keep propagating themselves along.

How did you get the equations for $\partial^2 E_x/\partial z^2$, $\partial^2 B_y/\partial z^2$ and $c$, and what do they mean?

The wave equations involving these quantitites come from Maxwell's equations in vacuum. See links in above questions for a derivation and more information.

Can you clarify EM waves?

See above question: electromagnetic waves result as a solution to wave equations derived from Maxwell's equations. The links in the above questions may help.

Why does $c=1/\sqrt{\mu_0 \epsilon_0}$ if the values seem to be unrelated? i.e. how are they related?

This is the cool thing: $\mu_0$ and $\epsilon_0$ are quantitities that apparently have nothing to do with light. $\mu_0$ is related to magnetic field and current; $\epsilon_0$ is related to charge and electric field. Yet these quantities can be measured to predict the speed of light! They are connected through Maxwell's equations.

The $\mu_0 \epsilon_0$ factor comes from Maxwell's equations: it shows up in the wave equation $\frac{\partial^2 E_x}{\partial z^2} = \mu_0 \epsilon_0 \frac{\partial^2 E_x} {\partial t^2}$ (and with a similar one for $B_y$). This is an equation of exactly the same form as the general wave equation, where the $x$-component of the electric field, $E_x$, takes the place of the displacement (it's the thing that's wiggling), and $1/v^2$ is $\mu_0 \epsilon_0$. Since the equation is identical to the wave equation, a traveling wave is a solution: $E_x=E_0 \cos{(kz-\omega t + \phi)}$ satisfies the equation (try plugging it in!). Furthermore, we know what the velocity of the wave solution is, from the form of the wave equation: it's $1/\sqrt{\mu_0 \epsilon_0}$. So the speed of light $c=1/\sqrt{\mu_0 \epsilon_0}$!

Can you explain about the field lines of EM waves?

Here's a a link I found which has some pictures of electric and magnetic field lines for electromagnetic waves generated by an oscillating dipole: note that the vacuum equations that I showed in class correspond to the waves far away from this dipole source (``far field'').

Also note, regarding the question about magnetic field closed loops: magnetic and electric field lines are not the same thing as electric and magnetic field vectors. A field vector has direction in the direction of the field, and length proportional to the magnitude of the field at that point. Field lines are not vectors: they are lines drawn such that at each point in space, the direction of the line is tangent to the direction of the field vector. The magnitude of the field is indicated by the density of lines in a region of space.

In the EM wave I drew, the arrows represent electric and magnetic field vectors along the $\hat{z}$ axis. They are not field lines.

Are EM waves linear?

Electromagnetic waves in a vacuum which are solutions to the wave equations that come from Maxwell's equations are linear. When you introduce matter in complicated situations, you can get nonlinear waves.

It seems that, if you had an animation of an electric wave propagating with electric field lines, the field lines would move longitudinally as the wave passed by. If we considered ``displacement'' for an electric field to be the displacement of the field lines, would we then call the wave longitudinal?

Well, the electromagnetic wave equations are for components of $\vec{E}$ and $\vec{B}$, and these wiggle transversely to the direction of wave propagation- that's why it's called a transverse wave. This distinction is really just nomenclature.

Why are the electric and magnetic fields perpendicular to each other? Must they be? And why does this generate light?

Yes, the electric and magnetic fields are perpendicular: this is a property of the solutions to Maxwell's equations. They are also both perpendicular to the direction of motion of the wave.

The fact that what we see as light (and radio waves, gamma rays, etc.) is electromagnetic waves is a connection that has has been determined experimentally. When you wiggle a moving charge at the appropriate frequency, you see light.

Why can we say the lines are parallel for the Young's slit experiment?

The farther the screen is from the slits, the more parallel the lines. In the case that the screen is very far away compared to the distance between the slits, $D»d$, the lines from the slits to the screen can be considered parallel to a very good approximation.

Will we cover phasor diagrams?

No, we don't cover them in this class, although I may post a couple of links.

So when do we get to learn that Newton was wrong?

Newton wasn't wrong. Newtonian physics is quite correct within its domain of applicability. However, a week or so from now we'll see how one can go beyond Newton.