A Questions about the energy of a wave as a Taylor series

[Chump]

I've read that, in general, the energy of a wave, as opposed to what's commonly taught, isn't strictly related to the square of the amplitude. It can be seen to be related to a Taylor series, where E = a_o + a₁ A + a₂A² ... Also, that the energy doesn't depend on phase, so only even terms will occur and the Taylor series gets truncated to only be proportional to the amplitude squared.

My questions are:

• Is there a derivation/more in-depth explanation of how the Taylor series came about for relating energy to amplitude?
• Why doesn't energy depend on phase? (My guess is because it's based on a simple harmonic oscillator model)
• Since phase isn't important, why will the odd terms in the series not be important? How are those two things related?

 

[ZapperZ]

I've read that, in general, the energy of a wave, as opposed to what's commonly taught, isn't strictly related to the square of the amplitude. It can be seen to be related to a Taylor series, where E = a_o + a₁ A + a₂A²

Wait... what? How exactly is the energy of wave is due to "a Taylor series" and how does the Taylor series have anything to do with a description of a wave?

Where exactly did you get this idea?

Zz.

 

[Chump]

I didn't know if I was allowed to post references to another site, but it's in reference to Ben Crowell's answer to this question: https://physics.stackexchange.com/q...y-in-a-wave-proportional-to-amplitude-squared. The question is the first post on the page, there's an answer below it, and then Ben's answer is second from the top.

 

[ZapperZ]

Then shouldn't you be asking this THERE? I don't understand why we at PF are the ones who have to clean up other people's mess.

Zz.

 

[Chump]

OK. First, I've asked this there already. It was pretty much the first thing I did. I could not get in touch with Ben Cromwell on the site. Also, no one else from that site gave an answer. Further, I could not get in touch with Ben via his outside site. I believe I've taken all of the proper channels, and I'd just like a little bit of insight from other avenues, if possible. You don't need to "clean up any mess." If you don't have any helpful insight, please move along because I didn't approach this in a disrespectful way. I'd appreciate it if no disrespect came my way. Thank you.

 

[Dale]

Since phase isn't important, why will the odd terms in the series not be important?

I am puzzled by that one too. @bcrowell knows his stuff, so I am pretty confident it is right, but it isn’t obvious to me either.

 

[vela]

Is there a derivation/more in-depth explanation of how the Taylor series came about for relating energy to amplitude?

Ben is just saying that the energy E of the wave can be thought of as a function of its amplitude A, so we can expand this function E(A) in a Taylor series about A=0.

 

[CWatters]

I am puzzled by that one too. @bcrowell knows his stuff, so I am pretty confident it is right, but it isn’t obvious to me either.

Nearest explanation I can come up with is...

Taylor series for a sin wave has odd powers...

Sin(X) = X¹ - X³/3! + X⁵/5! -...

and a cos has even powers.

Cos(X) = 1 - X²/2! + X⁴/4! -...

Difference between cos and sin is phase.

But I'm not sure I believe that deleting one set or the other from a general series of both makes it ignore phase.

If your function is a sin wave and you delete the odd terms to remove phase what are you left with?

 

[Dale]

Difference between cos and sin is phase.

That makes sense, but then why pick the cos instead of the sin. That seems like cos waves carry energy and sin waves don’t.

Edit: oh, that is even and odd powers of X not of A.