Waves

Waves:

• "waves are a disturbance that moves through a medium"
• 2 main types of waves:
• transverse waves:
• individual particles affected by the wave vibrate in directions perpendicular to the direction in which the wave travels
• requires that there be a shearing force in the medium & thus will only be propagated in those mediums that support a shearing stress
• eg. surface water waves
• velocity = distance travelled (ie. wavelength) / time elapsed => velocity = wavelength x frequency as frequency = 1/time elapsed
• a harmonic wave is one where the end is displaced with simple harmonic motion
• longitudinal waves:
• vibration of individual particles is parallel to the direction the wave travels
• do not require shearing stress & hence may be propagated through any elastic medium
• for a rod-like fluid or solid, velocity = sqrt(E/density), where E is Young's modulus for solids or bulk modulus for fluids
• eg. compression-decompression waves of sound, explosions
• mixed waves:
• waves may be a combination of transverse & longitudinal
• eg. surface waves of deep water, the individual particles have a circular motion & thus in addition to motion perpendicular to the wave, have a motion parallel to the waves
• superposition of waves:
• when two or more waves exist simultaneously in the same medium, each wave travels as though the other were not present, however, at any given point, the displacement of an individual particle is the vector sum of each wave affecting it
• thus, there will be points where the vector sum cancel each other with a result of zero thereby producing an interference pattern which is the hallmark of wave motion
• stationary waves:
• if two sinusoidal waves of same amplitude & frequency travel in opposite directions through a medium, the two waves will be superimposed in such a manner that stationary or standing waves will result (see also sound)
• stationary wave equation:
• = 2 * amplitude * cos(2pi x/wavelength) * sin(2pi * frequency * t), where t = time, x = position along the wave
• string instruments:
• when a pulse travelling along a string is reflected back from the end of the string by a boundary condition such as a violin string, in such case the string can vibrate with its lowest frequency - its fundamental frequency (=velocity/2xlength) & any integral multiple of that frequency which are called harmonic frequencies.
• if a finger is placed lightly on the string at the boundary, it will dampen the fundamental frequency, and the one can hear the harmonics which are of higher pitch.
• the string may be forced to vibrate with frequencies other than these, but in which case no stationary wave is set up & the amplitude is small and inconstant.
• wind instruments:
• an air column open at one end and closed at the other than vibrate in modes of the odd harmonics.
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• Huygen's principle:
• every point on a wave front may be considered as a new source of disturbance, sending wavelets in forward directions
• thus, as one passes from a medium of one density to a medium of another density there is a deviation in linear propagation due to the change in speed resulting in refraction
• for some types of waves, the speed is dependent on the frequency, resulting in dispersion
• if two waves of different frequency travel simultaneously, dispersion may cause separation of the waves due to refraction, and the resultant wave shape would change as the waves proceed, creating a pulse wave which is the resultant of many simple waves, but this pulse wave will not maintain the same shape in a dispersive medium.
• energy of a wave:
• in all travelling waves, energy travels through the medium in the direction in which the wave travels
• each particle of the medium has energy of vibration & passes energy on to succeeding particles
• in simple harmonic motion, where there is no damping, the energy of a vibrating particle changes from kinetic to potential energy and back, with the total energy constant.
• the energy of a wave = intensity = 2 pi2 x velocity x density x frequency2 x amplitude2
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