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

• A
• Chump
In summary, the energy of a wave is related to a Taylor series, where the amplitude is squared. The energy doesn't depend on phase, and only even terms will occur.

#### 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 = ao + 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?

Chump said:
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 = ao + 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.

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.

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.

Chump said:
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.

Chump said:
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.

Dale said:
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?

CWatters said:
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.

## 1. What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms. It is used to approximate a function at a given point by using the values of its derivatives at that point.

## 2. How is the energy of a wave represented as a Taylor series?

The energy of a wave can be represented as a Taylor series by using the wave equation, which relates the energy of a wave to its amplitude and frequency. The Taylor series allows us to approximate the energy at a given point by using the values of its derivatives.

## 3. What is the significance of using a Taylor series to represent the energy of a wave?

Using a Taylor series to represent the energy of a wave allows us to better understand the behavior of the wave and make predictions about its energy at different points. It also allows us to analyze the wave in a more precise and mathematical manner.

## 4. Are there any limitations to using a Taylor series to represent the energy of a wave?

Yes, there are limitations to using a Taylor series to represent the energy of a wave. One limitation is that the series is only accurate for a finite number of terms, so the approximation becomes less accurate as we move further away from the given point. Additionally, the series may not always converge for certain functions.

## 5. How can we use a Taylor series to analyze the energy of a wave?

We can use a Taylor series to analyze the energy of a wave by evaluating the series at different points and comparing the results to the actual energy values. This can help us understand the behavior of the wave and make predictions about its energy at different points. Additionally, we can use the derivatives of the series to determine the rate of change of the energy at a given point.