Oscillations And Waves

SHM (Simple Harmonic Motion)

x (t) = A cos (ω t)

v (t) = - ω A sin (ω t)

a (t) = - ω^{2} A cos (ω t)

Where A is maximum displacement (AKA amplitude) and $ω$ is the angular frequency ( $\frac{2 π}{T}$ )

T = 2 π \frac{I}{m g d}

I = \frac{T ^{2} m g d}{4 π ^{2}}

Angular frequency:

ω = f 2 π

Wave speed is the product of frequency and wavelength

v = f λ

Beat frequency when two tones interfere:

f_{b e a t} = ∣ f_{1} - f_{2} ∣

Waves on a string, general:

y (x, t) = A sin (k x - ω t); k = \frac{2 π}{λ}

k is wave number?? number of waves per unit of space or some shit
Phase:

ϕ (x, t) = k x - ω t

λ = \frac{2 π}{k}

T = \frac{2 π}{ω}

v = \frac{ω}{k}

The transverse velocity:

y (x, t) v_{y} (x, t) = = = = A sin (k x - ω t + φ) \frac{\partial y ( x , t )}{\partial t} = \frac{\partial}{\partial t} (A sin (k x - ω t + φ)) - A ω cos (k x - ω t + φ) - v_{y max} cos (k x - ω t + φ) .

The maximum transverse speed of an antinode:

v_{ma x} = A ω

The transverse acceleration:

a_{y} (x, t) = \frac{\partial v _{y}}{\partial t} = \frac{\partial}{\partial t} (- A ω cos (k x - ω t + φ)) = - A ω^{2} sin (k x - ω t + φ) = - a_{y max} sin (k x - ω t + φ) .

The maximum transverse acceleration of an antinode (when the argument to the function results in an output of 1):

a_{ma x} = A ω^{2}

Doppler shift:

f^{'} = \frac{v _{w a v e} - v _{o b ser v er}}{v _{w a v e} - v _{so u rce}} f

In a String (F is tension, $μ$ is mass/length):

v = \frac{F}{μ}

Speed of Sound in a Fluid ( $β$ is bulk modulus, $ρ$ is density):

v = \frac{β}{ρ}

Speed of Sound in a Solid (Y is youngs modulus, $ρ$ is density):

v = \frac{Y}{ρ}

λ O_{n} = \frac{λ}{n + 1}

f O_{n} = (n + 1) f

λ H_{n} = \frac{λ}{n}

f H_{n} = n f