\magnification=\magstep1


$$
F_y \quad = \quad -T \sin(\theta) \ + \ T \sin(\theta + \Delta \theta) 
$$


$$
F_y \quad = \quad -T (\theta) \ + \ T (\theta + \Delta \theta) 
   \quad = \quad T \ \Delta \theta
$$


$$
\mu \quad = \quad {{M}\over{L}}
$$

$$
F_y \quad = \quad (\mu \thinspace \Delta x) \ {{\partial^2y}\over{\partial t^2}} \quad = \quad T \ \Delta \theta
$$

$$
\theta \quad = \quad {\rm \ slope\ of\ line\ \ } \quad = \quad 
        {{\partial y}\over{\partial x}}
$$

$$
\Delta \theta \quad = \quad {{\partial \theta}\over{\partial x}} \ \Delta x
$$

$$
{{\partial \theta}\over{\partial x}} \quad = \quad 
            {{\partial^2 y}\over{\partial x^2}}
$$

$$
(\mu \thinspace \Delta x) \ {{\partial^2y}\over{\partial t^2}} \quad = \quad T \ {{\partial^2 y}\over{\partial x^2}} \ \Delta x
$$

$$
 {{\partial^2 y}\over{\partial x^2}} \quad = \quad 
      {{\mu}\over{T}}   \ {{\partial^2y}\over{\partial t^2}} 
$$

$$
{\rm speed\ of\ disturbance\ along\ string\ } \ v \quad = \quad 
      \sqrt{{T}\over{\mu}}
$$

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

$$
y(x, t) \quad = \quad f(x) \ \ \cos(\omega t)
$$

$$
{{\partial^2 y}\over{\partial x^2}} \quad = \quad 
   {{\partial^2 f}\over{\partial x^2}} \ \cos(\omega t)
$$

$$
{{\partial^2 y}\over{\partial t^2}} \quad = \quad 
   -\omega^2 \ f(x) \ \  \cos(\omega t)
$$

$$
   {{\partial^2 f}\over{\partial x^2}} \ \cos(\omega t)
 \quad = \quad
   \left( {{1}\over{v^2}} \right) \ -\omega^2 \ f(x) \ \  \cos(\omega t)
$$

$$
   {{\partial^2 f}\over{\partial x^2}} \ \quad = \quad 
   - \left( {{\omega^2}\over{v^2}} \right) \ f(x)
$$
      
$$
f(x) \quad = \quad A \cos({{\omega}\over{v}} x) \ + \ 
                   B \sin({{\omega}\over{v}} x)
$$

$$
\sin( {{\omega}\over{v}} L )  \quad = \quad 0 \qquad \quad
        \Rightarrow \quad {{\omega}\over{v}} L \quad = \quad n \pi
$$

$$
{{\omega}\over{v}} \quad = \quad {{n \pi}\over{L}}
$$

$$
f(x) \quad = \quad B \sin \left( {{n \pi x}\over{L}} \right) 
$$


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


$$
\omega_1 \quad = \quad {{\pi}\over{L}} \ \sqrt{{{T}\over{\mu}}} 
$$

$$
\omega_n \quad = \quad n \thinspace \omega_1
$$

$$
y(x, t) \quad = \quad B \sin{ \biggl( {{n \pi x}\over{L}} \biggr) }
            \ \cos(\omega_n t)
$$

$$
{\rm slope} \quad = \quad {{\partial y}\over{\partial x}} 
     \quad = \quad \theta
     \qquad \Rightarrow \qquad
     {{\partial \theta}\over{\partial x}} 
            \quad = \quad
     {{\partial}\over{\partial x}} \left( {{\partial y}\over{\partial x}} \right) 
              \quad = \quad
     {{\partial^2 y}\over{\partial x^2}} 
$$


\bye
