Calculating Energy of Vibrating String Problem

[e(ho0n3]

Problem. A string fixed at two ends (which are a length L apart) is pulled up at the center to a height of h. Assuming that the tension T remains constant, calculate the energy of the vibrations of the string when it is released. [Hint: What work does it take to strech the string up?]

The work to pull the string is

$$\int_0^h \frac{y}{c} \, T \, dy$$

where

$$c = \sqrt{y^2 + (L/2)^2}$$

right? And if I were to calculate the energy directly, I would need to know the frequency of vibration and the linear density of the string right?

 

[e(ho0n3]

Nevermind. One may find the frequency given the linear density using the fact that a standing wave is produced when the string is released so:

$$f = \frac{v}{\lambda} = \frac{v}{2L} = \frac{\sqrt{T/\mu}}{2L}$$

The energy is then given by:

$$E = \int_0^L 2 \pi^2 f^2 D(x)^2 \mu \, dx$$

where

$$D(x) = h \sin (\pi x / L)$$

right? I should, in theory, get the same answer using this method and the method in the first post.

 

[e(ho0n3]

I ask because the latter integral is much easier to calculate (at least for me) than the former one. For the latter one, I get $\pi T h / L$ as the answer.

 

[e(ho0n3]

It just dawned on me that

$$\frac{d}{dy} \sqrt{y^2 + (L/2)^2} = \frac{y}{\sqrt{y^2 + (L/2)^2}}$$

Duh! So the integral in the first post becomes $T(\sqrt{h^2 + L^2/4} - L/2)$. This doesn't agree with what I posted earlier. (After a quick dimensional analysis, I realize that the energy I calculated in post #3 is wrong.) Hmm...