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\begin{document}

\frame{\frametitle{\b Waves II - 1} \bigskip

{\bf Wave Superposition (1):}
$$ A\,\sin(k\,x-\omega\,t) + A\,\sin(k\,x-\omega\,t+\phi) =
   2\,A\,\cos\left(\frac{\phi}{2}\right)\sin\left(
   k\,x-\omega\,t+\frac{\phi}{2} \right)$$ 
\smallskip

{\bf Guitar String (2):} 
$$ f_n = n\,\frac{v}{2L}\,~~n=1,~2,~3,~\dots\,,
   ~~~v=\sqrt{\frac{F_T}{\mu}}\,. $$
\smallskip

{\bf Harmonic Pipe (3):} 
$$ f_n = n\,\frac{v}{4L}\,~~n=1,~3,~5,~7,~\dots\,. $$
See previous equation for odd numbers.
\bigskip

{\bf Standing Wave in a Pipe (4):} See previous equation.
Then, $H_{\rm water}=L_0-L_1$.
\vskip 20pt }


\frame{\frametitle{\b Waves II - 2} \bigskip

{\bf Speed from Doppler Efffect (5):} 
$$ f_1 = f_0\,\left(1+\frac{u}{v}\right)\,.$$
Solve for $u$.
\bigskip

{\bf Doppler Effect for a passing Train (6):} 
$$ f_1\,\left(1-\frac{u}{v}\right)=f_0~~{\rm and}~~
   f_2\,\left(1+\frac{u}{v}\right)=f_0\,. $$
Solve for $u$.
\vskip 20pt }
\end{document}
