Oscillations/Waves/Sounds

Kinematic equations.

Momentum review.

• p = mv
  • Formula for an object’s momentum.
• (m1)(v1i) + (m2)(v2i) = (m1)(v1f) + (m2)(v2f)
  • Two objects collide and stick together.
• (m1)(v1i) + (m2)(v2i) = vf(m1 + v1)
  • Two objects collide and stick together.

Wave characteristics.

• A wave is a disturbance propagating through space that usually transfers energy.
  • Acts in a ripple effect.
• Types of waves:
  • A transverse wave is a wave with oscillations perpendicular to the direction of propagation.
    • 
  • A longitudinal wave is a wave with oscillations parallel to the direction of propagation.
    • 
• A periodic wave follows the pattern of a sinusoidal function and moves with constant speed.
  • The wave’s peak is its highest point.
  • The wave’s trough is its lowest point.
  • A wave’s amplitude is the distance from its resting position to its peak.
  • A wave’s period (T) is the time taken for each cycle.
  • A cycle is resetting to the same position & direction.
  • A wave’s frequency (f) is the number of cycles per second.
    • f = 1/T, the period and frequency are reciprocals of each other.
    • Measured in hertz (Hz), aka. cycles/second.
  • A wavelength (λ) is how far the wave has traveled after 1 period.
  • A cycle can be measured from peak to peak, or trough to trough.
  • The wave’s velocity is v = λ/T = λf
    • Wavelength in the time of a period.

Wave interference.

• To figure out what happens when two wave pulses collide, you can add the waves visually.
  • This applies for partial overlap as well.
  • The wave interference is only during the overlap, but after they pass through each other, they return to their initial states.
• This process of "adding up" waves is called superposition.
• Constructive interference is when overlapping waves produce a wave with an amplitude that is the sum of the individual waves.
  • This is when both pulses have the same direction of displacement, the heights are the sum of each individual heights.
• Destructive interference is when overlapping waves produce a wave with an amplitude that is less than the sum of the individual waves.
  • This is have both pulses are in opposing directions.
  • One of the waves is deemed “negative” and the addition becomes a subtraction of the “positive” pulse - the pulse in the opposing direction.

Waves traveling mediums.

Standing waves.

• In confined mediums, waves will reflect and bounce off of a medium’s boundaries.
• This reflection will make the wave overlap with itself.
• This is called standing waves.
  • When standing waves happen, they select preferred wavelengths and frequencies.
• Nodes are points on a standing wave where the wave stays in a fixed position over time because of destructive interference.
• Anti-nodes are points on a standing wave where the wave vibrates with maximum amplitude.
• Wavelengths & frequencies make standing waves when there is a node at each end.
  • Special wavelengths:
    • 
    • This wavelengths make standing waves because the nodes are evenly spaced across the medium.
    • Formula for special wavelengths:
      • λn = (2L)/n
        • L is the length of the medium.
        • n is the number of the harmonic, the positive integer multiple of the fundamental frequency.

Simple harmonic motion.

• Oscillator goes back & forth to return to equilibrium position.
  • The restoring force returns it to initial position.
    • Overshoots the restore, the mass moves too far in the restoring direction, which makes for the oscillation. (Assume no friction).
• Simple Harmonic Oscillators have a restoring force that’s proportional to the displacement of the mass.
  • Restoring force changes with the amount of the change in mass’s position.
  • Described by sin and cos functions, as those are oscillating functions.
• Important points in the oscillation:
  • At the points of maximum compression or maximum extension, v = 0, no speed.
    • The restoring force has stopped the mass and this is right before the mass reverses direction.
  • Parts of the system:
    • Amplitude (A) is the maximum magnitude of displacement.
    • Period (T) is the time required for an “entire cycle”.
      • A cycle is the time from one end to another and back.
      • Change in amplitude doesn’t change the period.
    • Frequency (f) is the inverse of the period.
      • f = 1/T
  • Graphing the system:
    • Sinusoidal function: position (x) vs. time (t).
    • 
    • Equations:
      • Start equilibrium: x(t) = Asin(ωt)
      • Start maximum: x(t) = Acos(ωt)
      • Start minimum: x(t) = -Acos(ωt)
      • ω = 2π/T
        • Remember T is the period.

Spring oscillators.

• The spring itself provides the restoring force.
• Hooke’s Law: Fs = -kx.
  • In the vertical direction, mg = -kx.
  • k is the spring constant [N/m].
    • More stiff means k is higher, and vice versa.
  • x is the displacement.
  • The - sign makes it a restoring force, it counters the motion direction.
• The equation for Hooke’s law shows that the force is proportional to the displacement.
  • Masses on springs are Simple Harmonic Oscillators.
• Period formula: _T = 2π _ √(m/k)*
  • Increasing m, the mass increases the period.
  • Increasing k, spring constant, decreases period.
  • Works in the vertical direction as well.
  • Doesn’t depend on gravity.

Simple pendulums.

• A mass, m, attached to a string of length l, that can swing back & forth. It is a simple harmonic oscillator, which means it can be represented by a sinusoidal function. Gravity is the restoring force. Adapted motion equations: Start equilibrium: θ(t) = θMAXsin(ωt) Start maximum: θ(t) = θMAXcos(ωt) Start minimum: θ(t) = -θMAXcos(ωt) ω = 2π/T Remember T is the period. θMAX is the amplitude The maximum angle the pendulum can make. Period formula: T = 2π * √(L/g) L is length of string. Increase in L increases period. g is the ACCELERATION due to gravity. Increase in g decreases period. Changing the mass doesn’t affect the period.

Energy in oscillators.

• As mass oscillates, PE will turn to KE, and vice versa, repeatedly.
• Formulas:
  • KE = (1/2)mv^2
  • PE = (1/2)kx^2
• ETOTAL = PE + KE = (1/2)kx^2 + (1/2)mv^2
• When PE is maximum, KE is zero, v (velocity) is zero.
• When PE is zero, KE is maximum, v is maximum.

Sound characteristics.

• Sound displaces air molecules and oscillates them.
  • The molecules get pushed, and return to their initial state, repeatedly.
  • The displacement of the air cause other air molecules to displace as well, and these displace more molecules, etc.
  • This ripple effect reaches your ear to create sound.
  • The air molecules oscillate in a sinusoidal pattern, aka. a periodic wave.
• A higher volume makes the air oscillations larger, and the sound loudens.
• Connection to periodic waves.
  • The amplitude is the maximum displacement of the air molecule before it starts to return to its initial state.
    • This is not the length of the entire displacement.
    • It's the maximum displacement measured from equilibrium point.
  • The period (T) is the time an air molecule takes to move a cycle, or fully back & forth to the equilibrium position.
    • Decreasing the period decreases the time the air takes to oscillate and results in a higher perceived pitch, and vice versa.
  • The frequency (f) is the number cycles an air molecules displaces in a second.
    • Describes as f = 1/T
    • Lower frequency = lower pitch of the sound.
    • Higher frequency = higher pitch of the sound.
  • The wavelength (λ) is the distance between two compressed regions of air.
    • Can't be calculated normally from the dt graph because that graph represents an individual air molecule, so the distance between peaks on that graph would be the period.
  • Sound as a whole is a longitudinal wave.

Standing sound waves.

• Tube with both ends open:
  • Molecules on both ends can oscillate.
  • However, the air in the middle of the tube is unable to oscillate.
    • 
  • This is a standing wave because the compressed regions aren’t explicitly moving, rather the particles appear to bounce back and forth the middle of the tube, where the air particles aren’t oscillating.
    • 
  • The particles further from the center are displacing by increasing amounts.
  • However, as you get close to the center, the displacements get closer to zero.
    • 
  • The middle doesn’t displace, so it’s a node.
  • The ends move the most, so they’re anti-nodes.
  • The length of this wave in this tube is half the wavelength.
    • If L is the tube’s length, L = 0.5λ or λ = 2L
  • Applying this to different harmonics:
    • λN = (2L)/N
      • This would have N nodes between the two end anti-nodes.
• Tube with one end open, one end closed:
  • Air oscillates on the open side of the tube.
  • However, the air near the closed end doesn’t have space to oscillate.
    • 
    • Those molecules try to oscillate, but bump into the closed end and lose their energy.
  • This is a standing wave because the compressed regions aren’t explicitly moving, rather the particles appear to bounce back and forth from the closed side.
    • 
  • On the open end, there is an anti-node, because there is a lot of movement and oscillation.
  • On the closed end, there is a node, as it stays in place and doesn’t oscillate.
  • The length of this wave in this tube is ½ the wavelength.
  • If L is the tube’s length, L = 0.25λ or λ = 4L
  • Applying this to different harmonics:
    • λN = (4L)/N, where N is an odd whole number.
      • This would have N nodes between the two end anti-nodes.
• Important note: the term resonance is synonymous with standing sound waves.
• Pressure/Displacement (Anti-)Nodes:
  • Pressure Nodes are where the pressure stays constant.
    • At the open ends of closed tubes because the pressure at the ends equalizes with the atmosphere & stays constant throughout.
    • In the middle of open tubes because the pressure cancel out.
  • Displacement Nodes are where the particles don't change position or displace.
  • Displacement Anti-Nodes are where the molecules displace the most.
    • At the open ends of tubes because the particles at the end displace (by the amplitude) more than at anywhere else.
  • 

Sound wave interference.

• Same frequencies:
  • If two waves with same frequencies are exactly overlapped, the sound would get louder due to constructive interference.
  • If they are are overlapped inversely, meaning wave A’s peaks are concurrent with wave B’s troughs and vice versa, destructive interference should cancel the whole thing out.
• Different frequencies:
  • This overlap starts off constructively, but over time, the differences in frequency would make an increasing discrepancy between the waves.
    • 
  • The sound will cycle between being loud and soft, because the period discrepancy results in a constant back & forth between constructive and destructive interference.
  • Resultant wave:
    • 
• This is called beat frequency.
• Wobbles in volume when waves of different frequencies overlap.
  • fB = |f1 - f2|
  • The beat frequency is the difference between the two frequencies.
  • The ## of times the superposed wave goes from constructive interference to destructive interference and back per second.

Doppler effect.

• The Doppler Effect is a phenomenon where if a moving body is emitting a wave to a stationary object, the perceived frequency will be different that the wave’s true frequency.
• Stationary object emitting a 1 Hz wave:
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• Object moving at 5 [m/s] emitting a 1 Hz wave:
  •