Total energy of a wave on a string

[Brian_D]

Homework Statement

Consider a wave train propagating as a sine wave along a string. Obtain an expression for the total energy in this wave train, in terms of the string tension F, the wave amplitude A and the wavelength lambda.

Relevant Equations

$\mathit{total} \mathit{energy} \mathit{of} a \mathit{wave}= 0.5 \mu A^{2} \omega^{2} \lambda$

$\mu =\frac{F}{V^{2}}$

$\omega =\frac{2 \pi\mathrm{V}}{\lambda}$

P.S. I tried to clean up this post, but the program does not seem to be working correctly. When I click on "preview" I get one thing, but when I go back to my original text, it does not match the preview.
I plugged the expressions for mu and omega into the equation for total energy and simplified. I got $\textit{total\_}\mathit{energy}=\frac{2 \pi^{2} A^{2} F}{\lambda}$. However, the book answer key says, $\textit{total\_}\mathit{energy}=\frac{4 \pi^{2} A^{2} F}{\lambda}$. (The book gives 4 as the coefficient, not 2). Is the answer key wrong? If not, where was my mistake.I talenergyofawave=0.5μA2ω2
ω=2πVλ

 

[TSny]

Relevant Equations: total energy of a wave $= 0.5 \mu A^{2} \omega^{2} \lambda$

This expression does not give the total energy of a wavetrain. It provides the energy of a specific length of the wavetrain. See your textbook to determine this length.

The question asks for the total energy in the wave train. This will depend on the overall length of the wavetrain, but this length is not given in the problem statement. Check to make sure that you have provided the complete problem statement.

 

[Brian_D]

Thank you, TSny, you have clarified the whole thing. The problem statement indicated two cycles and included a diagram showing two cycles; I modified the problem statement because I didn't want to have to reproduce the diagram. I didn't think the number of cycles mattered, and that was my mistake.