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For the damped string($u_{tt}-c^2u_{xx}+\gamma u_t=0,c=\sqrt{\frac{T}{p}}$), show that the energy decreases.

$E = \dfrac{\int_{-\infty}^{\infty}p{u_t}^2+T{u_x}^2 dx}{2}$

$\dfrac{dE}{dt}=\dfrac{\int_{-\infty}^{\infty}2pu_tu_{tt}+2T{u_x}u_{xt} dx}{2}$ $=\dfrac{\int_{-\infty}^{\infty}2pu_t(c^2u_{xx}-\gamma u_t)+2T{u_x}u_{xt} dx}{2} = \dfrac{\int_{-\infty}^{\infty}2T(u_tu_x)_x-2p\gamma {u_t}^2 dx}{2}= {Tu_tu_x|_{-\infty}^\infty}-p\gamma \int_{-\infty}^\infty{u_t}^2dx$

Then , how can I get further?

Peter Phipps
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dlfjsemf
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1 Answers1

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First of all a string is usually modelled as a fixed interval $[a,b]$, not $(-\infty,\infty)$. Secondly you don't mention any boundary conditions. Standard boundary conditions are to fix the end-points of the string (so $u_t = 0$ at $x=a$ and $x=b$). This would make the boundary term you end up with zero and we are left with

$$\frac{dE}{dt} = - p\gamma \int_a^b u_t^2{\rm d}x$$

Finally if $p\gamma > 0$ then the right hand side is negative (the integral of a positive function over a positive directed interval is positive) so $\frac{dE}{dt} < 0$.

Winther
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  • I can't find the boundary condition in the problem..then is it fine to set the boundary condition? – dlfjsemf Mar 31 '18 at 13:14
  • @dlfjsemf The boundary conditions is a crucial part of defining a PDE so you need to know them. See e.g. page 6 https://www.math.hmc.edu/~ajb/PCMI/lecture7.pdf for some standard boundary conditions for this kind of problem. These would imply that the boundary term vanish – Winther Mar 31 '18 at 13:21
  • Thank you for your help! – dlfjsemf Mar 31 '18 at 13:29