\magnification=\magstep1

$$
{\rm dKE\ } \quad = \quad {{1}\over{2}} \thinspace dm \thinspace  v^2 \quad = \quad
                 {{1}\over{2}} \ ( \mu \ dx) \ ( v_y^2 ) 
$$

$$
y(x, t)
$$

$$
v_y \quad = \quad {{\partial y}\over{\partial t}} 
$$

$$
{\rm dKE\ } \quad = \quad {{1}\over{2}} \ ( \mu \ dx) \ 
                        \left( {{\partial y}\over{\partial t}} \right)^2
$$

$$
{\rm original\ length\ \ }  \quad =  \quad dx
$$

$$
{\rm stretched\ length\ \ } \quad = \quad ds \quad = \quad \sqrt{ dx^2 + dy^2} 
$$

$$
ds \ - \ dx \quad = \quad \sqrt{ dx^2 + dy^2} \ - \ dx 
$$

$$
\eqalign{
ds \ - \ dx \quad &= \quad \sqrt{ dx^2 \left[ 1 +  \left({{dy}\over{dx}}\right)^2  \right] } \ - \ dx  \cr
                  &= \quad dx \thinspace \sqrt{ 1 +  \left( {{dy}\over{dx}} \right)^2 } \ - \ dx  \cr
                  &= \quad dx \thinspace \left[ 1 +  \left( {{dy}\over{dx}} \right)^2 \right]^{1/2} \ - \ dx  \cr
                  &\simeq \quad dx \left[ 1 \ +  
                        {{1}\over{2}} \thinspace \left( {{dy}\over{dx}} \right)^2  \right]  \ - \ dx  \cr
                  &\simeq \quad {{1}\over{2}} \thinspace \left( {{dy}\over{dx}} \right)^2  \thinspace dx \cr
}
$$

$$
\quad \left( {{dy}\over{dx}} \right)^2  \thinspace dx 
$$

$$
\quad {{1}\over{2}} \thinspace \left( {{dy}\over{dx}} \right)^2  \thinspace dx 
$$

$$
\quad {{1}\over{2}} \thinspace \left( {{dy}\over{dx}} \right)  \thinspace dx 
$$

$$
\quad \left( {{dy}\over{dx}} \right)  \thinspace dx 
$$



$$
ds \ - \ dx \quad \simeq \quad {{1}\over{2}} \thinspace 
                   \left( {{\partial y}\over{\partial x}} \right)^2 
                    \thinspace dx
$$

$$
{\rm Work\ } \quad = \quad \int \vec{F} \cdot \vec{dx} 
$$

$$
{\rm dW \ } \quad = \quad \vec{F} \cdot \vec{dx} 
$$


$$
{\rm dW \ } \quad = \quad T \thinspace ( ds \ - \ dx ) \quad = \quad
                 T \thinspace {{1}\over{2}}  \left( {{\partial y}\over{\partial x}} \right)^2 \thinspace dx
$$

$$
{\rm dPE \ } \quad = \quad {{1}\over{2}} \thinspace T \thinspace \left( {{\partial y}\over{\partial x}} \right)^2 \thinspace dx
$$

$$
{\rm dE \ } \quad = \quad {\rm dKE} \ + \ {\rm dPE} \quad = \quad 
                  {{1}\over{2}} \ ( \mu \ dx) \ 
                        \left( {{\partial y}\over{\partial t}} \right)^2
                         \ + \   {{1}\over{2}} \thinspace T \thinspace \left( {{\partial y}\over{\partial x}} \right)^2 \thinspace dx
$$


$$
{\rm {{dE}\over{dx}} \ } \quad = \quad 
                  {{1}\over{2}} \thinspace \mu \thinspace
                        \left( {{\partial y}\over{\partial t}} \right)^2
                         \ + \   {{1}\over{2}} \thinspace T \thinspace \left( {{\partial y}\over{\partial x}} \right)^2 \thinspace 
$$

$$
\Delta x \quad = \quad v \thinspace \Delta t
$$

$$
({\rm energy\ of\ section}) \quad = \quad   {{ dE}\over{{dx}}} \thinspace {\Delta x} \quad = \quad
         \left( {{1}\over{2}} \thinspace \mu \omega^2 A^2 \right) \thinspace (v \thinspace \Delta t) 
$$


$$
{\rm Power \ } \quad = \quad  {{\rm energy\ of\ section}\over{\Delta t}} 
   \quad = \quad 
         {{ \left( {{1}\over{2}} \thinspace \mu \omega^2 A^2 \right) \thinspace (v \thinspace \Delta t) } \over{\Delta t}}
$$

$$
y(x, t) \quad = \quad A \sin(k x - \omega t)
$$

$$
{{\partial y}\over{\partial t}}  \quad = \quad 
          -\omega A \cos(kx - \omega t)
$$

$$
{{\partial y}\over{\partial x}}  \quad = \quad 
          k A \cos(kx - \omega t)
$$


$$
{{ dE}\over{{dx}}}  \quad = \quad
          {{1}\over{2}} \thinspace \mu \omega^2 A^2 \cos^2 (kx - \omega t)
    \ + \ 
          {{1}\over{2}} \thinspace T k^2 A^2 \cos^2 (kx - \omega t)
$$

$$
{\rm Power} \quad = \quad {{1}\over{2}} \thinspace \mu v \omega^2 A^2
$$

$$
{\rm Power} \quad = \quad {{1}\over{2}} \thinspace \left( \mu v \right) \ \left( \omega^2 A^2 \right) 
$$


$$
{\rm KE} \quad = \quad {{1}\over{2}} \thinspace m v^2
$$

$$
{\rm PE} \quad = \quad {{1}\over{2}} \thinspace k (\Delta x)^2 
   \quad = \quad 
    \int_0^{\Delta x} k \vec{x} \cdot \vec{dx}
   \quad = \quad 
    \int_0^{\Delta x} \vec{F}\cdot \vec{dx}
$$


\bye
