- #1

DavideGenoa

- 155

- 5

##\Delta E=\Big[ \frac{1}{2}\mu\Big(\frac{\partial y}{\partial t}\Big)^2+\frac{1}{2}F\Big(\frac{\partial y}{\partial x}\Big)^2 \Big]\Delta x##

where ##\mu## is the linear density of the string and ##F## is its tension. Opportune approximations are made to get this result.

By using an explicit notation for the variables, I would say that the formula means

##\Delta E=\Big[ \frac{1}{2}\mu\Big(\frac{\partial y(x_0,t_0)}{\partial t}\Big)^2+\frac{1}{2}F\Big(\frac{\partial y(x_0,t_0)}{\partial x}\Big)^2 \Big](x-x_0)##

Everything clear to me until here.Then, from the formula, my book infers that "the energy propagates along the string with velocity ##v=\Delta x/\Delta t##" and "the power of th wave is ##P=(\Delta E/\Delta x)(\Delta x/\Delta t)##" i.e.

##P=v\Big[ \frac{1}{2}\mu\Big(\frac{\partial y}{\partial t}\Big)^2+\frac{1}{2}F\Big(\frac{\partial y}{\partial x}\Big)^2 \Big]##

but I do not understand this step, because I do not understand what ##\Delta x/\Delta t## really is... I mean: the ##x## in the expression of ##\Delta E## is **not a function of time**and ##\Delta E## is defined for any choice of ##x##, ##x_0## and ##t_0## in ##\mathbb{R}##, and ##y## is defined on all ##\mathbb{R}^2##, and not only for ##x=vt##, therefore I do not see how we can define ##\Delta x/\Delta t##, which I explicitly write as ##(x-x_0)/(t-t_0)##, as a well defined velocity, since we cannot consider it as ##(x(t)-x(t_0))/(t-t_0)##: ##x## and ##t## can be arbitrarily chosen and ##x## is not a function of ##t##...

Could anybody explain that step to me? I ##\infty##-ly thank you!