What is the power of a wave in a string?

[DavideGenoa]

Hi, friendsi! My text of physics, Gettys', shows how the energy, both kynetic and potential, of a small element $\Delta x$ of a string, through which a wave (whose wave function is $y:\mathbb{R}^2\to\mathbb{R}$, $(x,t)\mapsto y(x,t)$) runs, is:

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

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!

 

[olivermsun]

In the notation $v = \Delta x/\Delta t$, $x = x(t)$ is usually the position of a point of fixed phase in the traveling wave, e.g., a wave crest. That might help you to interpret the rest of the notation...

 

[DavideGenoa]

Thank you very much, oliversum! The problem is that $y(x,t)$, a wave function, is defined on all $\mathbb{R}^2$, not only for some $x=x(t)$: the $x$ in its argument can be any real value independently from $t$...