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Power of a wave in a string

  1. Aug 19, 2015 #1
    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!
     
  2. jcsd
  3. Aug 19, 2015 #2

    olivermsun

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    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...
     
  4. Aug 20, 2015 #3
    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##...
     
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