Vibrations and Waves

• Vibration - The cyclical motion of an object about an equilibrium point

• Mechanical Wave - The transfer of energy through a material due to vibration

  • Medium - The material that permits the transmission of energy through vibrations

  • The nature of the medium dictates the actual speed (particle spacing, forces between particles)

• Waves - Disturbances that carry energy and cause particles to vibrate

  • The particle will eventually return to its original position

Waves Properties

• Waves can be transverse or longitudinal

  • Transverse Wave - A wave in which particles vibrate perpendicular to the direction of the flow of energy

    • ex. light, electromagnetic waves

  • Longitudinal Wave - A wave in which particles vibrate parallel to the direction of the flow of energy

    • ex. sound

    • Can’t travel through liquids

    • Compression - The region in a longitudinal wave in which the medium’s particles are closer together

    • Rarefaction - The region in a longitudinal wave in which the medium’s particles are farther apart

• Amplitude - The maximum height of the wave from equilibrium position

• Wavelength - The distance between a point on one wave and the same point of the next wave

  • Represented by λ

• Period - The time for a vibrating particle to complete one cycle

  • Represented by T

• Frequency - The number of complete cycles that occur in a unit of time

  • Measured in hertz, or Hz (one cycle per second)

  • Represented by f

  • f = 1/T

• Phase - The x-coordinate of a unique point of the wave in a continuous wave

  • The relative displacement of two waves at the same time

  • In Phase - The state of two identical waves that have the same phase shift

  • Out of Phase - The state of two identical waves that have different phase shifts

Wave Speed

• Wave Speed - The rate at which a wave in travelling through a medium; also a measure of how fast the energy in the wave is moving

  • Represented by v, and measured in distance/time

  • v = λf

• Factors affecting wave speed

  • v = √(Tension/Linear Density)

  • Temperature - Increase in temperature causes an increase in motion and a faster wave

  • Density - Decrease in density increases wave speed, as they are easier to move

  • Tension - Increase in tension causes an increase in speed, as it transmits energy faster

  • Linear Density - The mass per unit distance of string

    • The mass divided by length

    • An increase in linear density will decrease the speed

Sound Waves

• Human hearing ranges from 20 Hz to 20,000 Hz

  • Sound waves with a frequency higher than 20,000 Hz are ultrasonic

  • Sound waves with a frequency lower than 20 Hz are infrasonic

• The speed of sound in air is 330m/s

  • Contingent on altitude (sea level) and temperature (20 degrees)

  • To calculate varying temperatures:

    • 331.4 + 0.606T, where T is temperature

• Mach Number - The ratio of the airspeed of an object to the local speed of sound

  • Mach = (speed of object)/(local speed of sound)

• Sound Intensity - The amount of sound energy being transferred per unit area

  • Related to the air pressure

  • Pressure = Force/Area

    • Measured in N/m², or Pascals (Pa)

  • Loudness is also related to the amount of energy carried by the wave

    • Loudness is measured in Watts/m² or on the logarithmic decibel (dB) scale

Interference

• Interference - The process of generating a new wave when two or more waves meet

  • When waves interact

  • Interference doesn’t affect the properties of the waves

• Superposition - At any point the amplitude of two interfering waves is the sum of the amplitudes of the individual waves

  • Constructive Interference - The process of forming a wave with a larger amplitude when two or more waves combine

  • Destructive Interference - The process of forming a wave with a smaller amplitude when two or more waves combine

  • The waves must be in phase

Waves at Media Boundaries

• In a free-end, the waves are reflected in phase

• In a fixed end, the waves are reflected out of phase

• At boundaries that are neither fixed nor free ends:

  • When moving from a slower medium to a faster medium, the transmitted and reflected wave will be in phase

  • When moving from a faster medium to a slower medium, the transmitted wave will be in-phase, and the reflected wave will be inverted

Standing Waves

• Standing Wave - A wave created by interference of waves that have the same amplitude and wavelength that appears to be stationary

  • The wave from the source interacts with the wave from the boundary

  • A pattern of nodes (no vibrations) and antinodes (maximum vibrations) form

• Fundamental Frequency - The lowest frequency that can produce a standing wave in a given medium

• Harmonics - Whole-number multiples of the fundamental frequency

• If both ends are fixed, the ends act as nodes, and there will always be one less antinode than nodes

  • For the first harmonic, there are two nodes and one single wave

  • This occurs at the natural frequency, or f₀

  • L = (nλ)/2

• If one end is fixed, the end is a node

  • This takes place in an air column

  • The first harmonic is a quarter of the wavelength, and every harmonic after is an additional half

  • L = [(2n-1)/4](λ)

Beats

• Beat - Periodic change in sound intensity caused by the interference between two nearly identical sound waves

  • This creates a tone whose volume changes

• The frequency at which the volume changes is equal to f_b = f₁ - f₂

• As the frequency difference increases, the beat frequency increases

Resonance

• Damping - A reduction in the amplitude of a wave as a result of energy absorption or destructive interference

• Resonant Frequency - The frequency at which a medium vibrates most easily

  • If a wave is constant at the resonant frequency, it will reach the maximum amplitude

• Resonance - The condition in which the frequency of a wave equals the resonant frequency of the wave’s medium

Doppler Effect

• Doppler Effect - When a source of waves approaches an observer, the observed frequency of the waves increases; when the source moves away from an observer, the observed frequency of the wave decreases

  • In sound - moving towards is a higher pitch, and moving away is a lower pitch

  • In light - moving towards is blue/violet, and moving away is red

• λ decreases as it moves closer

  • f₂ = (f₁v_s)/(v_s - v₀)

• λ increases as it moves away

  • f₂ = (f₁v_s)/(v_s + v₀)

• In equations:

  • f₁ - frequency of source

  • f₂ - frequency detected

  • v_s - speed of sound

  • v₀ - speed of object

• The effect is also observed when the source of the waves is stationary and the observer is moving