
\magnification=\magstep1


$$
y(t) \quad = \quad A \cos(\omega t + \phi)
$$

$$
y(x) \quad = \quad A \cos(k x + \phi)
$$


$$
v \quad = \quad {{\omega} \over {k}}
$$

$$
y(x, t) \quad = \quad A \cos(k x - \omega t + \phi)
$$


$$
\eqalign{
v_y(x, t) \quad &= \quad { { \partial y(x, t)} \over {\partial t} } \cr
}
$$


$$
v_y(x, t) \quad = \quad \omega A \sin(kx - \omega t + \phi) 
$$

$$
\eqalign{
a_y(x, t) \quad &= \quad { { \partial v_y(x, t)} \over {\partial t} } \cr
}
$$


$$
a_y(x, t) \quad = \quad - \omega^2 A \cos(kx - \omega t + \phi) 
$$

$$
{\rm KE\ } \quad = \quad {{1}\over{2}} M {v_y}^2
$$

$$
{\rm KE\ } \quad = \quad {{1}\over{2}} M \ 
           { \biggl ( - \omega A \sin(kx - \omega t + \phi) \biggr ) }^2
$$

$$
{\rm KE_{\rm avg}\ } \quad = \quad {{1}\over{2}} M \ 
           { \biggl ( \omega^2 A^2 \ {{1}\over{2}} \biggr ) }
$$

$$
\eqalign{
{\rm total\ E } \quad &= \qquad\quad  {\rm KE_{\rm avg}\qquad } + \qquad \quad {\rm \ PE_{\rm avg}} \cr
                       &= \quad {{1}\over{2}} M 
           { \biggl ( \omega^2 A^2 \ {{1}\over{2}} \biggr ) } 
                    +
                       \quad {{1}\over{2}} M 
           { \biggl ( \omega^2 A^2 \ {{1}\over{2}} \biggr ) }  \cr
}
$$

$$
{\rm total\ E } \quad = \quad {{1}\over{2}} M \omega^2 A^2   
$$


$$
\mu \quad \equiv \quad {{M}\over{d}} 
$$

$$
M \quad = \quad \mu v t
$$

$$
{\rm total\ E } \quad = \quad {{1}\over{2}} 
                            \thinspace ( \mu v t ) \thinspace
                        \omega^2 A^2   
$$

$$
{\rm power\ P\ } \quad = \quad {{1}\over{2}} 
                            \thinspace ( \mu v ) \thinspace
                        \omega^2 A^2   
$$


\bye
