Analyze beats using complex exponentials

[Lizwi]

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.

 

[tiny-tim]

ωelcome to PF!

Hi Lizwi! Welcome to PF!

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

 

[Lizwi]

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!

 

[tiny-tim]

(try using the X₂ button just above the Reply box )

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!

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 ​

 

[asteriatic]

To analyze beats using complex exponentials, we first need to understand that complex exponentials can be written in the form e^(iωt), where ω represents the angular frequency and t represents time. In this case, we are given two complex exponentials, e^(iω1t) and e^(iω2t), where ω1 and ω2 are two different angular frequencies.

To evaluate the sum of these two complex exponentials, we can use the properties of complex algebra. First, we can use Euler's formula, which states that e^(ix) = cos(x) + i sin(x). Using this formula, we can rewrite the two complex exponentials as e^(iω1t) = cos(ω1t) + i sin(ω1t) and e^(iω2t) = cos(ω2t) + i sin(ω2t).

Next, we can use the fact that the sum of two complex numbers is equal to the sum of their real parts plus the sum of their imaginary parts. Applying this to our two complex exponentials, we get e^(iω1t) + e^(iω2t) = (cos(ω1t) + cos(ω2t)) + i (sin(ω1t) + sin(ω2t)).

This is now in the form of a complex number, with a real part and an imaginary part. To evaluate this, we can use the properties of trigonometric functions. For example, we know that cos(x) + cos(y) = 2cos((x+y)/2)cos((x-y)/2). Applying this to our expression, we get e^(iω1t) + e^(iω2t) = 2cos((ω1t+ω2t)/2)cos((ω1t-ω2t)/2) + i2sin((ω1t+ω2t)/2)cos((ω1t-ω2t)/2).

Using similar properties for the sine function, we can further simplify this expression. This will give us a final result in terms of cosines and sines, which can be evaluated for specific values of ω1 and ω2.

In summary, to analyze beats using complex exponentials, we can use the properties of complex algebra and trigonometric functions to simplify the expression and evaluate it for