Waves and Sounds

General Wave Characteristics

• Transverse Wave: direction of particle oscillation is perpendicular to the propagation of the wave. Think above moving a string with a fixed point by moving the hand up and down.
  • Includes electromagnetic waves like visible light, microwaves, and x-rays.
  • In any waveform, energy is delivered in the direction of wave travel
• Longitudinal Waves: particles of wave oscillate parallel to the direction of transfer
  • Sound waves, causes air particles to oscillate through cycles of compression and rarefaction (decompression) along the direction of the wave.

Describing Waves

• Wavelength: Distance from one maximum (Crest) to another crest or valley to valley.
• Frequency: Number of wavelengths passing a fixed point per second [Hz].
• Propagation Speed: v = fλ
• Period: The inverse of frequency and is defined as the number of seconds per cycle.
• Angular Frequency: w = 2πf = (2π)/T
• Equilibrium Position: Waves oscillate about this center point.
  • Displacement is how far a particular point on the wave is from the equilibrium position. Maximum magnitude is called the Amplitude.

Phase

• Can calculate a phase difference to compare two waves passing through the same phase.
• If completely out of phase, expressed as a difference of λ/2 or 180 degrees.

Principle of Superposition

• When waves interact with each other, the displacement of the resultant wave at any point is the sum of the displacements of the two interacting waves.
  • Constructive Interference: If waves are in-phase, the displacements always add together. Resultant amplitude will be 2x the original amplitudes.
  • Destructive Interference: Two out of phase waves cancelling each other out completely.

Travelling and Standing Waves

• Travelling Waves: are when the waves are moving and can be modelled as a string with one free end and one fixed end.
• Standing Waves: Only apparent movement of string is fluctuation of amplitude at fixed points along the length of the string. Can be modelled by a string with two fixed end or an open ended pipe
  • Node: Where amplitude is constantly zero
  • Antinode: At midway point between nodes, fluctuate at maximum amplitudes.

Resonance

• Every object has a natural (resonant) frequency when struck and allowed to vibrate freely.
• Timbre: The quality of sound is determined by the natural frequency.
• Objects that produce multiple non-harmonious or unrelated frequencies produce noise.
• Musical sounds are usually objects that vibrate with multiple natural frequencies which are whole-number multiples of the fundamental frequency (fundamental pitch & overtones).
  • Can usually hear about 20-20000 Hz for healthy adult, high frequency hearing declines with age.
• If frequency of forced oscillated is close to or equal to natural frequency, amplitude of wave becomes exponentially higher (Resonating).
  • Amplitude would go to infinity without the presence of damping, which is caused by frictional forces.
  • Damping: The decrease in amplitude of a wave caused by a non-conservative force.

Sound

A longitudinal wave transmitted by the oscillation of particles in a deformable medium. Cannot travel through a vacuum. Speed of sound is given by:

• v = √(B/ρ) 
• B is the bulk modulus (increases from gas to solid)

Production of Sound

• The mechanical distribution of particles in a material along the sound wave’s direction of propagation. Particles vibrate along an equilibrium position which causes area of compression and decompression.
  • Particles themselves do not move, but alternating areas allows for propagation of wave.

Frequency and Pitch

• Pitch is the same thing as frequency. A lower pitch indicates a lower frequency. We just perceive pitch while frequency is more mathematics.
• Soundwaves below 20 Hz are called infrasonic & soundwaves above 20,000 Hz are called ultrasonic waves.

Doppler Effect

• Describes the differences between the actual frequency of a sound and the perceived frequency of a sound.
  • If the source and the detector are moving towards each other, then the frequency is perceived to be higher. Vice Versa for if they’re moving away from each other.
    • f’ = f (v ± v_D)/(v ± v_s )
    • v is the speed of sound in the medium, v_D is the speed of the detector, and v_s is the speed of the source.
    • Upper sign used when the source moving towards, lower signs used when the source is moving away.
• Can be visualized as sound waves in front of a moving object as being compressed, while the sound waves behind the object are stretched out

Shock Waves

• If an object is producing sound and moving at a speed above the speed of sound, then wave fronts begin to build upon one another at the front of the object.
• Shock Wave: The highly condensed wave which creates a high pressure, followed by a low pressure (sonic boom).

Intensity and Loudness of Sound

• Loudness is the way humans perceive the intensity of a sound.
• Intensity: the average rate of energy transfer per area across a surface that Is perpendicular to the wave. (Power transported per unit area)
  • I = P/A, P is the power & A is the Area
• Intensity is proportional to the square of amplitude.
• Intensity is inversely proportional to the square of the distance from the source.
• Sound Level is measured in decibels (dB): β = 10 log (I/Io)
  • I₀ is the threshold of hearing which is 1 x 10^-12 W/m²
  • If intensity is changed by some factor, can use: βf = βi + 10 log (If/Ii)

Attenuation

• Real world measurements of sound will be lower than those expected from calculations. Which is a direct result of damping/attenuation.
• Since sound is simply oscillations in simple linear motion, they are subject to non-conservative forces which cause a decrease in amplitude as the oscillation progresses.
• Pitch does not change because of damping.

Beat Frequency

• When two sounds of slightly different frequencies are produced in proximity: 
• fbeat = f1 – f2

Standing Waves

• Occur when two waves of the same frequency are travelling in different directions and interfere with each other.
• As waves move in opposite directions, they interfere to produce new wave patterns characterized by alternating points of maximum amplitude (Antinode) & points of no displacement (node).
• Objects that support standing waves have boundaries at both ends
  • Closed Boundaries: do not allow oscillation and correspond to nodes.
  • Open Boundaries: Allow maximum oscillation and correspond to antinodes.

Strings

• λ = (2L)/n
  • n is an integer number called the harmonic, which corresponds to how many half-wavelengths are supported by the string. Equal to number of antinodes.
• f = (nv)/(2L)
  • The fundamental frequency is the lowest frequency of a standing wave. At n=2, the frequency is known as the first overtone or the second harmonic.
    • First overtone has half the wavelength and twice the frequency of the first harmonic.
• All possible frequencies that can be supported are known as the harmonic series.

Open Pipes

• Open pipes are open at both ends, while closed pipes have one end closed.
• These are basically the exact opposite of strings. The number of nodes, between the antinodes on each end, corresponds to what harmonic frequency the pipe is in.
• Same equations as above are used.

Closed Pipes

• A node on the closed end and an antinode on the open end. First harmonic is when there are only these two initial points. Corresponds to one quarter of a wavelength.
  • Each harmonic is equal to the number of quarter-wavelengths supported by the pipe. (half-wavelengths in open pipes and strings)
• There can only be odd numbered harmonics since an even number would require synonymous pipe ends.
  • λ = (4L)/n 
  • f = (nv)/(4L)
  • n can only be odd number interger.

Ultrasound

• Use high frequency sound waves outside the range of human hearing to compare the relative densities of tissues in the body.
• The transmitter produces a pressure gradient and acts as a receiver which processes the reflected sound.
• Ultimately relies on the reflection of the wave.
• Doppler Ultrasound: used to determine flow of blood within body by measuring the frequency shift that is associated with movement towards or away from the receiver.