\magnification=\magstep1

$$
\eqalign{
y_1(x, t) \quad &= \quad A_1 \cos(k_1 x \ - \ \omega_1 t \ + \ \phi_1) \cr
y_2(x, t) \quad &= \quad A_2 \cos(k_2 x \ - \ \omega_2 t \ + \ \phi_2) \cr
}
$$

$$
y_1(x, t) + y_2(x, y) \quad = \quad ... {\rm \ messy\ } ...
$$

$$
\eqalign{
y_1(x, t) \quad &= \quad A \cos(k x \ - \ \omega t \ + \ \phi_1) \cr
y_2(x, t) \quad &= \quad A \cos(k x \ - \ \omega t \ + \ \phi_2) \cr
}
$$

$$
\Phi \quad \equiv \quad \phi_1 - \phi_2
$$

$$
\eqalign{
y_1(x, t) \quad &= \quad A \cos(k x \ - \ \omega t ) \cr
y_2(x, t) \quad &= \quad A \cos(k x \ - \ \omega t \ + \ \Phi) \cr
}
$$

$$
\cos(\alpha) \ +\  \cos(\beta) \quad = \quad 
               2 \cos\left( {{\alpha + \beta}\over{2}} \right) \thinspace
                 \cos\left( {{\alpha - \beta}\over{2}} \right) 
$$

$$
y_1 \ + \ y_2 \quad = \quad 2 \thinspace
                 \cos \left( {{\Phi}\over{2}} \right) \thinspace
                 A \thinspace \cos \left( k x \ - \ \omega t \ + \ {{\Phi}\over{2}} \right)
$$


$$
\eqalign{
y_1(x, t) \quad &= \quad A_1 \cos(k_1 x \ - \ \omega_1 t) \cr
y_2(x, t) \quad &= \quad A_2 \cos(k_2 x \ - \ \omega_2 t) \cr
}
$$

$$
\omega_1 \quad \simeq \quad \omega_2
$$


$$
y_1 \ + \ y_2 \quad = \quad A \thinspace
                 \cos \left( {{\omega_1 - \omega_2}\over{2}} \thinspace t \right) \thinspace
                 \cos \left( k_{\rm avg} x \ - \ \omega_{\rm avg} t \right)
$$


$$
\eqalign{
y_1(x, t) \quad &= \quad A \cos(k_1 x \ - \ \omega_1 t \ + \ \phi_1) \cr
y_2(x, t) \quad &= \quad A \cos(k_2 x \ - \ \omega_2 t \ + \ \phi_2) \cr
}
$$


$$
\eqalign{
y_1(x, t) \quad &= \quad A \cos(\omega_1 t) \cr
y_2(x, t) \quad &= \quad A \cos(\omega_2 t) \cr
}
$$

$$
y_1 \ + \ y_2 \quad = \quad 2A \thinspace
                 \cos \left( {{\omega_1 - \omega_2}\over{2}} \thinspace t \right) \thinspace
                 \cos \left( {{\omega_1 + \omega_2}\over{2}} \thinspace t \right)
$$


$$
y_1 \ + \ y_2 \quad = \quad 2 A \thinspace
                 \cos \left( {{\omega_1 - \omega_2}\over{2}} \thinspace t \right) \thinspace
                 \cos \left(\omega_{\rm avg} t \right)
$$

$$
\omega \quad = \quad 2 \pi \thinspace f
$$

$$
\eqalign{
y_1(x, t) \quad &= \quad A \cos(2 \pi f_1 \thinspace t) \cr
y_2(x, t) \quad &= \quad A \cos(2 \pi f_2 \thinspace t) \cr
}
$$

$$
y_1 \ + \ y_2 \quad = \quad 2 A \thinspace
                 \cos \left( 2 \pi (1 {\rm \ Hz})  \thinspace t \right) \thinspace
                 \cos \left( 2 \pi (499 {\rm \ Hz})  \thinspace t \right)
$$

$$
f_{\rm beat} \quad = \quad 2 \ \left( {{f_1 - f_2}\over{2}} \right) \quad 
          = \quad (f_1 \thinspace - \thinspace f_2) 
$$


\bye

