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 ( $\frac{2 π}{T}$ )

Mechanical Pendulum

$T = 2 π \frac{I}{m g d}$ $I = \frac{T ^{2} m g d}{4 π ^{2}}$

Mechanical Waves

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

Wave speed equations

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): $v = \frac{Y}{ρ}$

Overtones and Shit Vocab

Overtones

$λ O_{n} = \frac{λ}{n + 1}$ $f O_{n} = (n + 1) f$

Harmonics

$λ H_{n} = \frac{λ}{n}$ $f H_{n} = n f$

Luca's Notes

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