WAVES

wave = disturbance that propagates
``disturbance''
=, e.g., displacement of medium element from its equilibrium position;
propagation can be in medium or in space (disturbance of a ``field'');

mechanical waves:

• When matter is disturbed, energy emanates from the disturbance, is propagated by interaction between neighboring particles;
  this propagation of energy is called wave motion;
• a traveling mechanical wave is a self-sustaining disturbance of a medium that propagates from one region to another, carrying energy and momentum.
• examples:
  - waves on strings,
  - surface waves on liquids,
  - sound waves in a gas (e.g. in air),
  - compression waves in solids and liquids;
• it is the disturbance that advances, not the material medium

• transverse wave
  displacements perpendicular to direction of propagation;
• longitudinal wave
  sustaining medium displaced parallel to direction of propagation
  (e.g. sound waves, some seismic waves, compression waves in a bell);

• periodic wave motion:
  particles oscillate back and forth, same cycle of displacement repeated again and again;
  (unless stated otherwise, we only discuss periodic waves)
• terms describing waves:
  • crest of the wave = position of maximum displacement
    (``highest point of the wave'')
  • amplitude = amount of maximum displacement
    (height of crest above undisturbed position)
  • wave velocity = velocity of propagation of wave crest
  • wavelength = distance between successive same-side crests
  • frequency = number of same-side crests passing by a fixed point per second
  • period = time for one complete wave oscillation
    

    

• unit of frequency: 1 hertz = 1Hz = 1/second

• speed of waves depends on properties of the carrying medium;
  in general:
  speed of mechanical waves in solids greater than in liquids,
  and greater in liquids than in gases.

• relation between speed, wavelength and frequency:

  

  i.e. speed = frequency wavelength
  

• intensity of a wave
  is a measure of how much power is transported to a point by the wave
  intensity = energy flow per unit time, per unit area,
  power per unit area,
  (where area = area perpendicular to propagation direction)
  
• energy flow carried by wave: is proportional to the
  square of the amplitude and the square of the frequency;
• ``inverse square law of wave intensity'':
  the intensity of a wave is inversely proportional to the square of the distance from the source of the wave

  

  (source = object emitting the wave)
  (I = intensity, P = total power emitted by source,
  R = distance from source)
  (strictly speaking, only for point-like or spherically symmetric sources,
  or if size of the source)
  

SOUND

• Sound waves propagate in any medium that can respond elastically and thereby transmit vibrational energy.
• sound waves in gases and liquids are longitudinal
  (alternating compression and rarefaction);
  in solids, both longitudinal and transversal;
• speed of sound is independent of frequency;
• speed of sound in air
  increases with temperature,
  in water;
• three frequency ranges of sound waves:
  • below 20 Hz: infrasonic
  • 20 Hz to 20 kHz: audible, i.e. sound proper
  • above 20 kHz: ultrasonic, ``ultrasound''
• pitch is given by frequency
  e.g. ``standard a'' corresponds to 440 Hz
• intervals between tones given by ratio of frequencies
  (e.g. doubling of frequency - one octave)
• male voice range 80 Hz to 240 Hz for speech, up to 700 Hz for song;
• female voice range 140 Hz to 500 Hz for speech,
  up to 1100 Hz for song.

SUPERPOSITION OF WAVES

• Superposition principle:
  two or more waves moving through the same region of space will superimpose and produce a well-defined combined effect. the resultant of two or more waves of the same kind overlapping is the algebraic sum of the individual contributions at each point
  i.e. the (signed) displacements (elongations) add
• Huygens' principle
  every point on a wavefront can be considered as a source, emitting a wave; the superposition of all these waves results in the observed wave.

• consequences: interference, diffraction
• interference:
  superposition of two waves of same frequency can lead to reinforcement (constructive interference) or partial or complete cancellation (destructive interference
  
  • constructive interference:
    two waves ``in phase'', i.e. crests of two waves coincide in time reinforce each other, resultant amplitude bigger than that of individual waves
  • destructive interference:
    two waves ``completely out of phase'' i.e. out of phase by 1/2 period, so that crests of one wave coincide with troughs of the other cancellation;
    complete cancellation (extinction) if both waves have same amplitude.

• phase differences can be caused by differences in pathlength;
  given a pathlength difference, the phase difference depends on the wavelength
  
• examples:
  colors of thin films (oil on water, soap bubbles)
  dead spots in auditorium
• diffraction grating:
  • many narrow parallel slits spaced closely together;
  • every slit forms source for wave;
  • differences in pathlength from different slits to some point in space
    phase difference
    wavelength dependent interference pattern;
  • can be used to measure wavelength.