Key Equations

Key Equations

Wave speed	$v=\frac{\lambda }{T}=\lambda f$
Linear mass density	$\mu =\frac{\text{mass of the string}}{\text{length of the string}}$
Speed of a wave or pulse on a string under tension	$\left|v\right|=\sqrt{\frac{F_T}{\mu }}$
Speed of a compression wave in a fluid	$v=\sqrt{\frac{{\rm B}}{\rho }}$
Resultant wave from superposition of two sinusoidal waves that are identical except for a phase shift	$y_R\left(x,t\right)=\left[2A\text{cos}\left(\frac{\varphi }{2}\right)\right]\text{sin}\left(kx-\omega t+\frac{\varphi }{2}\right)$
Wave number	$k\equiv \frac{2\pi }{\lambda }$
Wave speed	$v=\frac{\omega }{k}$
A periodic wave	$y\left(x,t\right)=A\text{sin}\left(kx\mp \omega t+\varphi \right)$
Phase of a wave	$kx\mp \omega t+\varphi$
The linear wave equation	$\frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}=\frac{1}{v_w^2}\frac{{\partial }^{2}y\left(x,t\right)}{\partial t^2}$
Power averaged over a wavelength	$P_{\text{ave}}=\frac{E_{\lambda }}{T}=\frac{1}{2}\mu A^2{\omega }^2\frac{\lambda }{T}=\frac{1}{2}\mu A^2{\omega }^{2}v$
Intensity	$I=\frac{P}{A}$
Intensity for a spherical wave	$I=\frac{P}{4\pi r^2}$
Equation of a standing wave	$y\left(x,t\right)=\left[2A\text{sin}\left(kx\right)\right]\text{cos}\left(\omega t\right)$
Wavelength for symmetric boundary conditions	${\lambda }_n=\frac{2}{n}L,n=1,2,3,4,5\text{...}$
Frequency for symmetric boundary conditions	$f_n=n\frac{v}{2L}=n{f}_1,n=1,2,3,4,5\text{...}$