Oscillations & Waves — Wave Motion Applied (Medium)

Quick Recap — Wave Motion

• Basics: $v=f\lambda$; a progressive wave $y=A\sin(\omega t-kx)$ has speed $v=\dfrac{\omega}{k}$; phase difference $=\dfrac{2\pi}{\lambda}\times$ path difference; intensity $\propto A^2$.
• Wave speeds: on a string $v=\sqrt{\dfrac{T}{\mu}}$; sound in a gas $v=\sqrt{\dfrac{\gamma P}{\rho}}\propto\sqrt{T}$.
• Standing waves: string fixed both ends $f_n=\dfrac{nv}{2L}$ (all harmonics); open pipe $f_n=\dfrac{nv}{2L}$ (all harmonics); closed pipe $f_n=\dfrac{(2n-1)v}{4L}$ (odd harmonics only, fundamental half the open-pipe value).
• Beats: $f_{beat}=|f_1-f_2|$. Frequency is set by the source and does not change when a wave enters a new medium (the wavelength does).

Worked mini-example. A string of linear density $0.01$ kg/m under $100$ N tension has $v=\sqrt{\dfrac{T}{\mu}}=\sqrt{\dfrac{100}{0.01}}=100$ m/s; fixed over $0.5$ m its fundamental is $f_1=\dfrac{v}{2L}=100$ Hz.