waves

Transverse wave

Vibrations perpendicular to direction of travel

Longitudinal

Vibrations parallel to direction of travel

Wave speed

v_0=f\lambda

Phase angle

\phi=\frac{2\pi x}{\lambda}

wavenumber

k=\frac{2\pi}{\lambda} \left(k\ne\right. spring constant)

displacement in y and x

y=A\sin(\omega t-\frac{2\pi}{\lambda}x)

(can sub in k)

Doppler effect: moving source

\lambda^{\prime}=\lambda_0\left(1-\frac{v_{s}}{v_0}\right)

\left(v_{s}\right. is velocity of source and v_0 is speed of propagation eg speed of light)

f^{\prime}=\frac{f_0}{\left(1-\frac{v_{s}}{v_0}\right)}

Doppler effect: moving observer

v^{\prime}=v_0+v_{obs}

f^{\prime}=\left(1+\frac{v_{obs}}{v_0}\right)f_0 (on eq sheet)

Nyquist frequency (on eq sheet)

the highest detectable frequency in the data:

N=\frac{1}{2\Delta t}

where \Delta t is the cadence (how often measurements are taken)

Frequency resolution

smallest detectable change in frequency

\delta f=\frac{1}{t}

where t is the observing time

superposition

A^{\prime}=2A\cos(-\frac{\phi}{2})

y=A^{\prime}\sin(kx-\omega t+\frac{\phi}{2})

beat frequency

f_{beat}=\frac12(f_1-f_2)

beat wavelength

\frac{1}{\lambda_{beat}}=\frac12(\frac{1}{\lambda_1}-\frac{1}{\lambda_2})