Analyze beats using complex exponentials

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Homework Help Overview

The discussion revolves around evaluating the expression e^(iω1t) + e^(iω2t) using complex algebra, specifically in the context of beats in sound waves produced by two sources with nearly identical frequencies.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the transformation of the complex exponentials into cosine and sine terms, with one participant attempting to express the sum as the real part of the complex expression. Questions arise regarding the proper use of identities related to the frequencies ω1 and ω2.

Discussion Status

The discussion is ongoing, with participants providing hints and guidance on how to approach the problem. There is an emphasis on using complex exponentials rather than relying solely on trigonometric identities. Multiple interpretations of the problem are being explored, particularly regarding the identities for ω1 and ω2.

Contextual Notes

Participants are encouraged to show their work and clarify where they are stuck, indicating a collaborative effort to understand the problem better. There is a focus on deriving results from complex algebra rather than looking up solutions.

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Homework Statement



Please use the complex algebra to evaluate e^(iω1t)+e^(iω2t), w2 means omega 2?


Homework Equations


Ho do I do this problem


The Attempt at a Solution


I changed this into cos and sine terms.
 
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ωelcome to PF!

Hi Lizwi! Welcome to PF! :wink:

Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 


Thaks, They said: beats occur in sound when two sources emit frequencies that are almost the same. The perceived wave is the sum of the two waves, so that at your ear, the wave is the sum of the two cosines of w1t and w2t...( my w means omega ) use complex algebra two evaluate this. The sum is the real part of e^(w1t)+e^(w2t). notice the two identities w1= (w1+w2)/2 + (w1-w2)/2. Use the complex exponentials to drive the results; dont't just look up some trig identity.


What I did is , because they said the sum is the real part of e^(w1t)+e^(w2t) I wrote this in term of course and sine: (cosw1t + i sinw1t) + (cosw2t + i sinw2t)
(cosw1t + cosw2t) + i (sinw1t + sinw2t)
the real part is cosw1t + cosw2t
Im done!
 
(try using the X2 button just above the Reply box :wink:)
Lizwi said:
The sum is the real part of e^(w1t)+e^(w2t). notice the two identities w1= (w1+w2)/2 + (w1-w2)/2. Use the complex exponentials to drive the results; dont't just look up some trig identity.
…
Im done!

noooo, you're not! :redface:

read the hint …

they want you to write the answer in terms of p and q, where p = (w1+w2)/2 and q = (w1-w2)/2

try again :smile:​
 

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