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Introductory Physics Homework Help
Bandwidth Theorem: Find Min Angular Freq in Propagating Wavepacket
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[QUOTE="bananabandana, post: 5081065, member: 490819"] [h2]Homework Statement [/h2] Consider a propagating wavepacket with initial length ## L_{0}##. Use the bandwidth theorem to show that the minimum range of angular frequencies present in the wavepacket is approximately: $$ \Delta{\omega}\approx \frac{v_{g}}{L_{0}} $$ [h2]Homework Equations[/h2] Bandwidth theorem: $$ \Delta f \Delta t > 1$$ [h2]The Attempt at a Solution[/h2] Use the following approximation for the group velocity ## v_{g}## $$ v_{g}=\frac{\Delta \omega}{\Delta k} $$ Using bandwidth theorem: $$ \Delta \omega \Delta t > 2 \pi $$ $$ \Delta t =\frac{\Delta v_{g}}{\Delta l} $$ $$ \therefore \Delta \omega > \frac{2\pi \Delta v_{g}}{\Delta L} $$ If $\Delta v_{g}$ is small: $$\Delta v_{g} \approx v_{g}, \Delta L \approx L_{0} $$ Therefore: $$ \Delta \omega > \frac{2 \pi v_{g} }{L_{0}} $$ So $$ \Delta \omega \approx \frac{v_{g}}{L_{0}} $$ But this seems a bit of a fudge? Can anyone explain how I might get a more kosher version? Thanks! *EDIT: No this is completely wrong - sorry - the limit of ##\Delta L ## as ##\Delta v_{g} \rightarrow 0 ## is zero! How do I fix this? Thanks :) [/QUOTE]
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Introductory Physics Homework Help
Bandwidth Theorem: Find Min Angular Freq in Propagating Wavepacket
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