Useful Relations for the Final Exam in 123, WI 1999.

		$\displaystyle v~=~\nu\lambda \qquad \qquad , \qquad \qquad \nu~=~{\omega\over 2... ...\qquad k~=~{2\pi\over\lambda} \qquad \qquad , \qquad \qquad v~=~{\omega\over k}$

		$\displaystyle \nu_{\rm obs.}~=~\nu_0 \ \left( { c_s \pm v_{\rm obs.} \over c_s ... ...ight) \qquad \qquad \qquad , \qquad \qquad \qquad \nu_b~=~\vert\nu_1-\nu_2\vert$

		$\displaystyle I~=~{P\over A} \qquad \qquad , \qquad \qquad c_{\rm air}~=~v_{\rm... ...40\ {\rm m/s} \qquad \qquad , \qquad \qquad c~=~2.99792458\times 10^8~{\rm m/s}$

		$\displaystyle {\rm SL}~=~10\ \log_{10}\left({I\over I_0}\right) \qquad \qquad ,... ...quad I_0~=~10^{-12}\ {\rm W/m^2} \qquad \qquad , \qquad \qquad P~=~{dE\over dt}$

		$\displaystyle \nu_n = n\ {v\over 2 L} \qquad \qquad , \qquad \qquad \nu_m = (2 m+1) \ {v\over 4 L} \qquad \qquad , \qquad \qquad A~=~4\pi r^2$

		$\displaystyle v = \sqrt{T\over\mu} \ =\ \sqrt{B\over\rho} \qquad \qquad , \qquad \qquad y(x\ ,\ t) = A\ \sin\left( k x - \omega t + \phi\right)$

		$\displaystyle {\bf S}~=~{1\over\mu_0}{\bf E}\times {\bf B} \qquad , \qquad I~=~... ...{\bf E}_{\rm max}\vert^2 \qquad , \qquad \mu_0~=~4\pi\times 10^{-7}~{\rm T m/A}$

		$\displaystyle n_1\sin\theta_1~=~n_2\sin\theta_2 \qquad \qquad , \qquad \qquad \... ...p}\vert~=~{E\over c} \qquad \qquad , \qquad \qquad {\bf F}~=~{d{\bf p}\over dt}$

		$\displaystyle {1\over O}+{1\over I}~=~{1\over f} \qquad \qquad , \qquad \qquad ... ...quad \qquad {1\over O}+{1\over I}~=~(n-1)\left({1\over R_1}-{1\over R_2}\right)$

		$\displaystyle I(\theta)~=~I_0 \cos^2\beta \left({\sin\alpha\over\alpha}\right)^... ...r\lambda} \qquad \qquad , \qquad \qquad \alpha~=~{\pi a \sin\theta\over\lambda}$

		$\displaystyle a\sin\theta~=~m\lambda \qquad \qquad , \qquad \qquad d\sin\theta~=~m\lambda \qquad \qquad , \qquad \qquad d\sin\theta~=~(m+{1\over 2})\lambda$

		$\displaystyle r~=~\sqrt{m\lambda R} \qquad \qquad , \qquad 2 n d~=~m\lambda \qquad \qquad , \qquad 2 n d~=~(m+{1\over 2})\lambda$

		$\displaystyle P~=~{{\bf F}\cdot {\bf n}\over A} \qquad \qquad , \qquad \qquad a... ...quad \qquad , \qquad \qquad \phi~=~{\bf k}\cdot {\bf x}~=~2\pi {x\over \lambda}$

		$\displaystyle I~=~I_0\cos^2\theta \qquad \qquad , \qquad \qquad I~=~{1\over 2} I_0\cos^2\theta \qquad \qquad , \qquad \qquad \tan\theta_p~=~{n_2\over n_1}$

		$\displaystyle \sin\theta_c~=~{n_2\over n_1} \qquad \qquad , \qquad \qquad m~=~-... ...u_{\rm obs.}~=~\nu_0 \ \ {\sqrt{1-{v^2\over c^2}}\over 1-{v\over c}\cos\theta }$