Direction of propagation of waves

I don't know how to work out the direction of propagation of waveforms and I would really appreciate some help if possible. I've attached three examples which I'm stuck on, and I've included the answers below. Thanks in advance,

1. +z direction
2. -z direction
3. -z direction

waves.jpg12.9KB

Reply 1

Phichi

For a wave to have propagated through space, the value of the given equation must be constant for two values of t, for it to have moved from one place to another. I.e, in example one, the phase inside the sine must be constant. Thus, we have ignoring the constant at the start,

0.2z-3t

. As t increases, for the equation to remain constant, the value of z must increase, and hence, its propagating in the +z direction. Similar logic can be used for the other two. E.g. in 3, for it to remain constant, as all coefficients are negative, as time increases, the value of the expression in the exp will decrease. If z decreases, due to its negative coefficient, it'll increases the value, hence counter acting the decrease from the time increases.

I.e.

-z-t

If t increases, z must decrease to make the expression constant.

To understand, for a particular part of the wave to have moved, it must have the same value at two different points in time (where time is increasing).

(edited 9 years ago)

Reply 2

Phichi

Original post by ryanwhit

Did you manage to get the answers?

Reply 3

ryanwhit2

Original post by Phichi

Did you manage to get the answers?

Yes, thank you for the help! I'd forgotten my login details.

Could I ask another similar question if you have a moment? How do you tell if a wave has right or left hand polarization? Is there a sure-fire method to do this? I know how to work out if the polarization is circular or elliptical, but I can't figure out the direction.

Thanks :smile:

Reply 4

Phichi

Original post by ryanwhit2

I do it by just studying the motion of both the x and y components. For example:

If you have a wave of the form;

E(z,t)=\hat{x}E_x(z,t) + \hat{y}E_y(z,t)

If you observe the motion of the x and y components you can determine the polarisation (clockwise or anti-clockwise).

E.g

\overrightarrow{E}=E_xcos(kz-\omega t)+E_ysin(kz-\omega t)

When t=0 and assuming its travelling in the positive z direction., E would go from the +x to +y to -x to -y, an anti clockwise rotation in its direction of motion.

However if you can visualise this coming towards you (opposite in the -z direction), it's a clockwise rotation, thus right handed.

Sorry for the bad notation, I'm exhausted. However, I'm unsure of a 'sure-fire' way to work it out, unless somebody else is, just studying the initial motion of the wave however, isn't too tricky.

(edited 9 years ago)

Reply 5

ryanwhit2

Yeah that makes sense thanks :smile:

The following two are the only ones I can't figure out now. c) is right handed and d) is left. Have you seen ones like this before?

In c) I know that Ey trails by pie/4, and in d) Ey leads by pie/2, but I don't know how that effects things really. Thanks again

waves2.jpg13.4KB

Reply 6

Phichi

Original post by ryanwhit2

Yeah that makes sense thanks :smile:

Same logic here, lets look at c).

The

-\frac{\pi}{4}

is just the phase offset of the y component. Thus at z=0 t=0 initially the y component is in the on the negative y-axis direction, if you plotted a single xy plane. x is initially at 0. As time passes, x begins to initially increase, and y also begins to increase. So E starts at -y, then as the y component reaches 0, the x component is in the positive x direction, so E goes from -y to +x. Then the y component is in the +y direction, and eventually the x-component decreases again. After some time the x-component is in the -x direction, and y is decreasing. Thus E now goes from +y to -x, and so on and so forth. So if E goes -y -> +x -> -y -> +x, this is an anti-clockwise rotation in the +z direction. Again, if we visualise the wave heading towards us (-z direction), it'd appear clockwise, and hence, right handed. Try d) and let me know how you get on. I personally draw an xy plane, and put the initial directions of both xy components on the graph, show the direction of E, and visualise the motion as time passes, and z increases.

For the phase offset, where y is initially negative, this isn't perfect, but supplies a sufficient visual representation: http://www.wolframalpha.com/input/?i=sin%28x-pi%2F4%29

I'm not awfully good at explaining these sinusoidal type relationships in words, as I primarily visualise them with a graph on paper, or in my head. Sorry :smile:

Hope it helps.

(edited 9 years ago)

Reply 7

ryanwhit2