\magnification=\magstep1

$$
y(x, t)
$$

$$
y(x) \quad = \quad e^{-x^2}
$$


$$
y(x) \quad = \quad e^{-(x - 3)^2}
$$


$$
y(x) \quad = \quad e^{-(x - 6)^2}
$$

$$
y(x, t) \quad = \quad e^{-(x \ - \ 3 {\rm {{m}\over{s}}} t )^2}
$$

$$
y(x, t) \quad = \quad e^{-(x \ - \ v t )^2}
$$

$$
y(x, t) \quad = \quad A f(x) \ g(t) \quad = \quad 
           A \sin \left( {{n \pi x}\over{L}} \right) \ 
             \cos ( \omega t ) 
$$

$$
k \quad = \quad {{n \pi}\over{L}}  
$$

$$
k \quad = \quad {{2 \pi}\over{\lambda}}  
$$


$$
y(x, t) \quad = \quad A f(x) \ g(t) \quad = \quad 
           A \sin \left( k x \right) \ 
             \cos ( \omega t ) 
$$

$$
y(x, t) \quad = \quad A \sin \left( kx \ - \ \omega t \right)
$$

$$
{{\partial^2 y}\over{\partial x^2}} \quad = \quad {{1}\over{v^2}} \ 
         {{\partial^2 y}\over{\partial t^2}}
$$

$$
{{\partial^2 y}\over{\partial x^2}} \quad = \quad - k^2 \  A \sin(kx - \omega t)
$$

$$
{{\partial^2 y}\over{\partial t^2}} \quad = \quad - \omega^2 \  A \sin(kx - \omega t)
$$


$$
{{\partial^2 y}\over{\partial x^2}} \quad = \quad {{k^2}\over{\omega^2}} \ 
         {{\partial^2 y}\over{\partial t^2}}
    \quad = \quad  
         {{1}\over{v^2}} \ {{\partial^2 y}\over{\partial t^2}}
$$

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

$$
\sin(\alpha) \cos(\beta) \quad = \quad {{1}\over{2}} 
    \biggl[ \sin(\alpha + \beta) \ + \ \sin(\alpha - \beta) \biggr]
$$

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

$$
y(x, t) \quad = \quad {{A}\over{2}} \ 
        \biggl[ \sin(kx + \omega t) \ + \ \sin(kx - \omega t) \biggr]
$$

$$
\omega_n \quad = \quad {{n \pi}\over{L}} \ \sqrt{ {{T}\over{\mu}} }
$$

$$
y(x, t) \quad = \quad {{A}\over{2}} \ 
        \biggl[ \sin \left( {{n \pi}\over{L}} x  +   
                   {{n \pi}\over{L}} \sqrt{{{T}\over{\mu}}} t \right) \ + \ 
                 \sin \left({{n \pi}\over{L}} x  -   
                   {{n \pi}\over{L}} \sqrt{{{T}\over{\mu}}} t \right) \biggr]
$$

$$
y(x, t) \quad = \quad {{A}\over{2}} \ 
        \biggl[ \sin \left\{ {{n \pi}\over{L}} ( x  + 
                     \sqrt{{{T}\over{\mu}}} t ) \right\} \ + \ 
                \sin \left\{ {{n \pi}\over{L}} (x  -   
                     \sqrt{{{T}\over{\mu}}} t ) \right\} \biggr]
$$

$$
y(x, t) \quad = \quad 
           y \thinspace (x \ \pm \ vt \thinspace )
    \quad = \quad 
       y\left( x \ \pm \ \sqrt{ {{T}\over{\mu}}} t \right) 
$$

$$
v \quad = \quad \sqrt{ {{T}\over{\mu}} }
$$

$$
y(x, t) \quad = \quad A \ \sin \left( {{n \pi x}\over{L}} \right) 
                        \ \cos(\omega_n t)  
$$

$$
y(x, t) \quad = \quad {{A}\over{2}} \ \biggl[ 
           \sin \left( {{n \pi x}\over{L}} \ + 
                        \ \omega_n t \right)
              \ + \ 
           \sin \left( {{n \pi x}\over{L}} \ - 
                        \ \omega_n t \right)  \biggr] 
$$


\bye
