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

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