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Old Jan 25th 2018, 06:43 PM   #1
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The adding of 3 waves.

I have been stuck on this problem for hours and i cant seem to find any information or anything online that makes since to me.

There are 3 waves traveling in the +x direction, what is the amplitude, phase and speed of the resulting wave?
y=3cos(14x-2600t)
y=4cos(14x-2600t+1)
y=5cos(14x-2600t-pi/2)

Not to seem ignorant, but i am not sure what it is i am even suppose to do. My initial though was to add the first two together, so my amplitude would be 7, but i am not sure what to do with the phase shift. I thought maybe 7cos1 as new amplitude but nothing online seems to give me any information, everything is in sin and we are being "taught' everything in cos, so i assume things just switch a bit? would the new phase be (1-pi/2) and would the rsulting speed be (2pi/14)(2600/2pi)? much help is appreciated.
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Old Jan 26th 2018, 04:29 AM   #2
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There is an identity:

$\displaystyle a \cos x + b \sin x = R \cos (x- \alpha)$

where

$\displaystyle R = \sqrt{a^2 + b^2}$
$\displaystyle \alpha = arctan\left(\frac{b}{a}\right)$

I think that this identity together with

$\displaystyle \cos x = \sin \left(\frac{\pi}{2} - x\right)$

should be enough to solve your problem. Basically, you will want to use the above formula to decompose your cosine waves into a mixture of sine and cosine waves where the phase difference is separated out into the sine part. Then you can add up the separate contributions.
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