Sarahisme Posted May 8, 2006 Share Posted May 8, 2006 hey everyone hmm...can't seem to work out how to attack this question.... any suggestions guys? Cheers Sarah Link to comment Share on other sites More sharing options...
ydoaPs Posted May 8, 2006 Share Posted May 8, 2006 use the euler relation i'm completely guessing, but [math]e^{i\pi}=-1[/math], and it said to use that, so maybe [math]k_1{x}-w_1{t}=\pi=k_2{x}-w_2{t}[/math] Link to comment Share on other sites More sharing options...
Sarahisme Posted May 8, 2006 Author Share Posted May 8, 2006 why set [math] k_1{x}-w_1{t}=\pi=k_2{x}-w_2{t} [/math] ? why is it to pi? i'm not quite following... Link to comment Share on other sites More sharing options...
ydoaPs Posted May 8, 2006 Share Posted May 8, 2006 i just set each = to [math]e^{i\pi}[/math]......i kinda just BSed the answer. i'm only in AP Physics B. [math]{\Delta}kx-{\Delta}wt=\pi[/math] looks like a good answer to me, except it is missing the "time-dependant modulating factor." if that is A, then it might be [math]A({\Delta}kx-{\Delta}wt)=\pi[/math]. keep in mind i have no idea what i'm doing. Link to comment Share on other sites More sharing options...
Sarahisme Posted May 8, 2006 Author Share Posted May 8, 2006 hmmm ok so i get: [math] 2A e^{i(kx-wt)} cos( \frac{k_1 x-k_2 x-w_1 t+w_2 t}{2}) [/math] i think thats ok.... but as for the phase and group velocities...??? Link to comment Share on other sites More sharing options...
swansont Posted May 8, 2006 Share Posted May 8, 2006 What are the equations for phase and group velocities? Link to comment Share on other sites More sharing options...
Sarahisme Posted May 8, 2006 Author Share Posted May 8, 2006 i think they are: [math] v_{phase} = \frac{w}{k} [/math] and [math] v_{group} = \frac{dw}{dk} [/math] Link to comment Share on other sites More sharing options...
swansont Posted May 8, 2006 Share Posted May 8, 2006 Does the frequency vary with k? (You haven't described any such relationship, so if this is an EM wave in free space the answer is no, and [math]\frac {w}{k} = c[/math]) Link to comment Share on other sites More sharing options...
Sarahisme Posted May 9, 2006 Author Share Posted May 9, 2006 you've lost me i think... to get my above answer i set [math] w = \frac{w_1+w_2}{2} [/math] and [math] k = \frac{k_1+k_2}{2} [/math] and its a plane wave? Link to comment Share on other sites More sharing options...
swansont Posted May 9, 2006 Share Posted May 9, 2006 You get beats, which are the wave packets. look here for a visual, and see here for the explanation. If the phase and group velocities are equal, the individual oscillations don't move with respect to the packet. If you go away from the plane wave description, so you have only one wave packet, then when the velocities aren't equal you get dispersion, and the wave packet spreads out. (with the plane waves, all dispersion does is move an oscillation from one beat to the next) Link to comment Share on other sites More sharing options...
Sarahisme Posted May 10, 2006 Author Share Posted May 10, 2006 oh ok i get it now! thanks swantsont -Sarah Link to comment Share on other sites More sharing options...
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