# Oscillations

Term | Units | Description |
---|---|---|

Wave Number (k) | radians m^{-1} |
k = ω / v k = 2 π ƒ / v |

Wavelength (λ) | meters | The distance between two equally displaced points on a wave. λ = v / ƒ |

Angular Frequency (ω) | rad s^{-1} |
ω^{2} = k / m
ω = 2 π ƒ = 2 π / T |

Frequency (ƒ) | seconds^{-1} |
ƒ = 1 / T = ½ ω / π |

Period (T) | seconds | Time between wave crests. Time for the object to undergo one full motion: from its leftmost, to its rightmost, and back to its leftmost position. T = 2 π / ω |

Velocity (v) | m s^{-1} |
v(t) = - ω A sin ( ω t + Φ ) v = ω / k v = 2 π ƒ / k v = ± ω ( A v = ƒ λ v_{max} = ω A |

Amplitude (A) | meters | A = ( x_{0} + v_{0}^{2} / ω^{2} )^{½} |

Kinetic Energy (K) | Joules | K = ½ m v _{max} = ½ mv_{max}^{2} = ½ k A^{2} |

Potential Energy (U) | Joules | U = ½ k x^{2} = ½ k A^{2} cos ^{2} ( ω t + Φ ) |

Total Energy = E | E = K + U = ½ k A^{2} | |

Dampening Constant (b) | ||

Critically Damped (b_{c}) |
The system does not oscillate at and beyond this level of dampening; it is said to be overdamped. | |

Node | Point of minimum vibration for a wave. The distance between two nodes is equal t the wavelength. | |

Antinode (n) | Point of maximum vibration for a wave. The distance between two antinodes is equal to the wavelength. | |

Resonance Harmonic | A description of the antinodes in a standing wave. For example, the first resonance/harmonic will be when there is one antinode. |

## Simple Harmonic Motion

x = A cos ( ω t + Φ )

a = - ω^{2} A cos ( ω t + Φ )

a_{max} = ω^{2} A

2 π = [ ω ( t + T ) + Φ ] - ( ω t + Φ )

tan Φ = - v_{0} / ( ω x_{0} )

m = k T^{2} / ( 4 π^{2} )

## Damped Oscillations

x = ( A e^{-&frac;bm/t ) cos ( ω t + Φ )}

ω^{2} = ( k / m ) - ( ½ b / m )^{2} = ω_{0}^{2} - ( ½ b / m )^{2}

b_{c} = 2 m ω_{0}

## Forced Oscillations

x = A cos ( ω t + Φ )

A = F_{0} / [ m^{2} ( ω_{d}^{2} - ω^{2} )^{2} + ( b ω_{d} )^{2} ]^{½}

ω_{d} represents the angular frequency of the oscillator, which is usually different from ω, the natural angular frequency of the spring, aka the resonance frequency.

## Pendulum

T = 2 π ( L / g )^{½}

## Standing Waves

y = A sin [ 2 π λ^{-1} ( x - v t ) ] = A sin ( k x - ω t )

v = λ / T = λ ƒ

k = 2 π / λ

℘ = Power transmitted by a sinusoidal wave on a stretched string = ½ µ ω^{2} A^{2} v

v = ( Tension / µ )^{½} = ( Tension · L / mass )^{½}

ƒ_{n} = [ n / 2 L ] [ F / µ ]^{½}

For octaves, frequency is 2^{octave} ƒ_{0} and tension is 2^{2 · octave} ƒ_{0}

## Superposition

y_{1} = A sin ( k x - ω t )

y_{2} = A sin ( k x - ω t + Φ )

y_{resultant} = [ 2 A cos ( ½ Φ ) ] [ sin ( k x - ω t + ½ Φ ) ]

Note that if Φ is 0 then the amplitude of the resultant wave is twice the amplitude of the two individual waves. However, if Φ is any odd multiple of π (ie, π, 3π, 5π) then the individual waves completely cancel out each other.

## Boundary Conditions

Imagine sound emitted into a pipe of air. An open end will always be a displacement antinode and pressure node (a maximum of height displacement, creating the most pressure); and if there is a closed end, it will be a displacement node and pressure antinode (a minimum of height displacement, creating the least pressure).

Pipe open at both ends: λ = 2 L / n and ƒ = v / λ = n v / 2 L

You will see antinodes in quantities of 1, 2, 3 etc

Open at just one end: λ = 4 L / n and ƒ = v / λ = n v / 4 L

You will see antinodes in quantities of 1, 3, 5 etc

## Doppler Effect for Sound

ƒ' = ƒ ( v + v_{O} ) / ( v - v_{S} )

v_{O} and v_{S} are positive when the observer and source are getting nearer, and negative when getting further. Two items can go from nearing each other to moving further away. Consider two cars headed in the same direction: the one in back is moving faster than the one ahead, so they are moving *toward* one another; yet when it finally bypasses the one originally ahead, then they are thenceforth moving *away* from one another.