
\magnification=\magstep1


$$
\mu \quad \equiv \quad { {\rm mass }\over{\rm length} } 
$$


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

$$
{\rm velocity\ } v \quad = \quad {{\omega}\over{k}} 
           \qquad = \quad f \lambda
$$

$$
y_1(x, t) \quad = \quad A \sin(k x - \omega t) 
$$

$$
y_2(x, t) \quad = \quad A \sin(k x + \omega t) 
$$

$$
y_1 + y_2  \quad = \quad 2 A \sin(k x) \thinspace \cos(\omega t)
$$

$$
 2 A \sin(k x) \quad = \quad 0
$$

$$
{\rm nodes\ at\ \ } x \quad = \quad 0, \thinspace\thinspace {{\pi}\over{k}} , 
                                       \thinspace\thinspace
                                       {{2 \pi}\over{k}} , 
                                       \thinspace\thinspace
                                       {{3 \pi}\over{k}} , 
                                       \thinspace\thinspace
                                     \ldots ,
                                       \thinspace\thinspace
                                       {{N \pi}\over{k}} 
$$
            
$$
{\rm nodes\ at\ \ } x \quad = \quad 0, \thinspace\thinspace 
                                     {{1}\over{2}} \lambda , 
                                       \thinspace\thinspace
                                                   \lambda , 
                                       \thinspace\thinspace
                                     {{3}\over{2}} \lambda , 
                                       \thinspace\thinspace
                                     \ldots ,
                                       \thinspace\thinspace
                                       {{N}\over{2}} \lambda 
$$
            
$$
 2 A \sin(k x) \quad = \quad \pm \thinspace 2 A 
$$


$$
{\rm antinodes\ at\ \ } x \quad = \quad {{\pi}\over{2 k}} , 
                                       \thinspace\thinspace
                                       {{3 \pi}\over{2 k}} , 
                                       \thinspace\thinspace
                                       {{5 \pi}\over{2 k}} , 
                                       \thinspace\thinspace
                                     \ldots ,
                                       \thinspace\thinspace
                                       {{ (2N + 1) \pi}\over{2 k}} 
$$
            

            
$$
{\rm antinodes\ at\ \ } x \quad = \quad 
                                     {{1}\over{4}} \lambda , 
                                       \thinspace\thinspace
                                     {{3}\over{4}} \lambda , 
                                       \thinspace\thinspace
                                     {{5}\over{4}} \lambda , 
                                       \thinspace\thinspace
                                     \ldots ,
                                       \thinspace\thinspace
                                       {{(2N + 1)}\over{4}} \lambda 
$$
            

$$
\mu_s \quad = \quad {\_\_\_\_\_\_\_} \pm {\_\_\_\_\_\_}  \quad {\rm kg/m}
$$

$$
v \quad = \quad {\_\_\_\_\_\_\_} \pm {\_\_\_\_\_\_}  \quad {\rm m/s}
$$


\bye


