\[ y(x,t) = y_m sin(kx-\omega t) \]
k = angular wave number w = angular frequency
Traveling waves are generally expressed as:
\[ y(x,t) = h(kx\pm \omega t) \]
Attributes
Wavelength to k
\[ k = \frac{2\pi}{\lambda} \]
Period to frequency
\[ \frac{\omega}{2\pi} = f = T^{-1} \]
Wave speed
\[ v =\frac{\omega}{k} =\frac{\lambda}{T} = \lambda f \]
Wave speed on stretched string
\[ v=\sqrt{\frac{\tau}{\mu}} \]
\(\tau\) = tension \(\mu\) = linear density
Average Power of a wave on a stretched string
\[ P_{avg} = \frac{1}{2}\mu v \omega^2 y_m^2 \]
Interactions
Interference
\[ y'(x,t) = [2y_m \cos \frac{1}{2}\phi] \sin(kx-\omega t +\frac{1}{2}\phi) \]
Interference of two waves with same direction, amplitude, frequency (thus wavelength) but different phase constant \(\phi\)
Power
\[ P_{avg} = \frac{1}{2}\mu v \omega^2 y_m^2 \]
Wave equation
\[ \pdv[2]{y}{x} = \frac{1}{v^2}\pdv[2]{y}{t} \]
Standing waves
\[ y'(x,t) =[2y_m \sin kx] \cos \omega t \]
nodes: \[ x=n {\lambda \over 2} \]
antinodes: \[ x=(n+{1\over2}){\lambda \over 2} \]
harmonic series:
\[ \lambda = \frac{2L}{n} \]
\[ f=n\frac{v}{2L} \]