\magnification=\magstep1

$$
\eqalign{
{\rm x} \quad &= \quad {\rm location\ on\ the\ rod\ } \cr
{\rm z} \quad &= \quad {\rm displacement\ from\ original\ location\ } \cr
}
$$

$$
dm \quad = \quad \rho \thinspace V \quad = \quad \rho \thinspace A \thinspace dx
$$



$$
{\rm stress\ } \quad = \quad {{\rm force}\over{\rm area}} \quad = \quad
     {{F_2 - F_1 }\over{A}}
$$

$$
{\rm strain\ } \quad = \quad {{\rm change\ in\ length}\over{\rm original\ length}} \quad = \quad
     {{\Delta z }\over{\Delta x}}
$$

$$
{\rm stress\ } \quad = \quad Y \ {\rm strain}
$$

$$
\left( {{F_2 - F_1}\over{A}} \right)  \quad = \quad 
          Y \ \left( {{\Delta z}\over{\Delta x}} \right) 
$$

$$
F_2 - F_1 \quad = \quad \left( \rho \ A \ \Delta x \right) 
                  \ {{\partial^2 z}\over{\partial t^2}}
$$

$$
{{F_2 - F_1}\over{A}} \quad = \quad 
                        \left( \rho \ \Delta x \right) 
                  \ {{\partial^2 z}\over{\partial t^2}}
$$


$$
\left( {{F_2 - F_1}\over{A}} \right)  \quad = \quad 
          Y \ \left( {{\Delta z}\over{\Delta x}} \right) 
$$

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

$$
{{\Delta z}\over{\Delta x}} \quad \simeq 
   \quad {{\partial}\over{\partial x}} ( \Delta z ) \quad = \quad
   \quad {{\partial}\over{\partial x}} \left( {{\partial z}\over{\partial x}} \ \Delta x \right)
   \quad = \quad 
   {{\partial^2 z}\over{\partial x^2}} \ \Delta x
$$

$$
\left( {{F_2 - F_1}\over{A}} \right)  \quad = \quad 
          Y \ \left( {{\Delta z}\over{\Delta x}} \right) 
  \quad \quad \Rightarrow \quad \quad 
       \left( \rho \ \ \Delta x \right) 
           {{\partial^2 z}\over{\partial t^2}} 
      \quad = \quad 
         Y \ {{\partial^2 z}\over{\partial x^2}} \ \Delta x
$$

$$
 {{\partial^2 z}\over{\partial x^2}} 
   \quad = \quad 
       \left( {{\rho}\over{Y}} \right) 
           {{\partial^2 z}\over{\partial t^2}} 
$$

$$
{{1}\over{v^2}} \quad = \quad {{\rho}\over{Y}}
  \quad \quad \quad \Rightarrow 
  \quad v \quad = \sqrt{ {{Y}\over{\rho}} }
$$

$$
\lambda_n \quad = \quad {{4 L}\over{2 n - 1}}
$$

$$
f_n \quad = \quad {{v}\over{\lambda_n}} \quad = \quad
            \sqrt{{Y}\over{\rho}} \ \left( {{2 n - 1}\over{4 L}} \right) 
$$




\bye
