Amplitude and Energy: A Simple Explanation

[DaydreamNation]

I tried to come up with a simple calculus-free explanation for why the energy in a sound wave is proportional to the square of the wave's amplitude for my musical acoustics class. I think this makes sense, and seems to just be an elaboration of what Donald Hall writes, but I haven't seen it explained this way elsewhere so please let me know if there are some problems here.

If we use the SHM model and imagine a ball on a spring, A (amplitude) is the maximum displacement. To start the vibrating system, the ball must be displaced by A.

Then, how much work is done when starting the vibration?
W=Fd and d=A
F=ma
a=v/t
v=d/t
a=d/t²
F=md/t²
W=md²/t² or mA²/t²
Energy transferred = work done
.˙. E is proportional to A²

 

[DaydreamNation]

It might work better if I explain it with Hooke's Law but they haven't learned this yet and don't really need it for most of the course.

 

[Andy Resnick]

I tried to come up with a simple calculus-free explanation for why the energy in a sound wave is proportional to the square of the wave's amplitude for my musical acoustics class. <snip>
a=v/t
v=d/t
a=d/t²
<snip>

Argh! Don't do this! first, a≠v/t, a=Δv/Δt= (v_f-v_i)/(t_f-t_i), and they are not the same, even if you set v_i = 0 and t_i = 0. I realize you are trying to provide some foundations behind the formulas, but this abuse of notation has significant consequences.

A better explanation could be that since the pressure amplitude can have negative values with respect to a zero-point but the energy carried by sound cannot be negative, the energy is proportional to the amplitude^2, which is always positive.

 

[DaydreamNation]

Yes, I see the error. Thanks for that.

 

[DaydreamNation]

Throwing this out here for anyone: the problem is I already gave the class that explanation of E ∝A² yesterday; I was perhaps feeling hubristic and rushed into it before I got replies on the thread; will not do that sort of thing again. (I did say it was crude and simplified, and that they wouldn't solve problems with those intermediate equations.) Any suggestions on how to recover and avoid future problems with this, without losing too much face? (Hall's explanation in the textbook is that if you pull a mass on a spring twice as far, you also have to pull it twice as hard; I was really just trying to elaborate on why his explanation makes sense. Does it??) I'm not sure I totally get your explanation: why would the need for a positive value mean that E ∝A² as opposed to e.g. E∝|A|?

:(

 

[DaydreamNation]

Btw, the textbook is Hall, Donald. Musical Acoustics.

Maybe it's just a matter of emphasizing the final formula E ∝A² going forward, and reiterating that the equations in the other explanation can't all be used, that it was a way to try to give an explanation for something?

:(