I have trouble understanding why the energy is constant in a part of the string. In my calculations, when I integrated the energy density over some interval, I obtained:
$$ \int_{0}^{a} u(x,t) \, dx ------(1) $$ where $$ u(x, t) = \frac{\partial^2 y}{\partial t^2}\frac{1}{2}p+\frac{\partial^2 y}{\partial x^2}\frac{1}{2}T $$ I guess that if the energy is constant on the interval, then: $$\frac{\partial}{\partial t}\int_{0}^{a} u(x,t) \, dx =0 $$ But can the above expression be written as:
$$ \int_{0}^{a} \frac{\partial u(x,t)}{\partial t} \, dx =0 $$ This seems to never happen since y(x,t) changes like a sine function, so energy is not conserved.
On the other hand, if we divide the string into small parts, then we can approximate all things like a point mass (or can we?), and we have simple harmonic motion (SHM). So, energy is conserved. Where is the mistake?
Also, I am interested in understanding why, for a small part of the string, where the initial length was dx, and after the wave propagation, the length is dl, the potential energy density is: $$u_pdx=T(dl-dx)$$ I mean, if this is energy per unit length, why do we simply say: $$u_pdl=T(dl-dx)$$ Side Question: Do you know of any problem books that contain problems requiring the use of continuity equations, flux, and general formulas for waves, available for free access?
