Harmonic Waves

# Harmonic Waves - given by: = k is sometimes called the wave...

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Harmonic Waves One set of extremely important solutions to the differential wave equation are sinusoidal functions. These are called the harmonic waves. One of the reasons they are so important is that it turns out that any wave can be constructed from a sum of harmonic waves--this is the subject of Fourier analysis. The solution in its most general form is given by: ψ ( x , t ) = A sin[ k ( x - vt )] (we could, of course, equally well choose a cosine since the two functions only differ by a phase of Π /2 ). The argument of the sine is called the phase. A is called the amplitude of the wave and corresponds to the maximum displacement the particles of the medium can experience. The wavelength of a wave (the distance between similar points (eg. peaks) on adjacent cycles) is
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Unformatted text preview: given by: = k is sometimes called the wave number. The period of the wave (the amount of time taken for a complete cycle to pass a fixed point) is given by T = = As usual, the frequency, , is just the inverse of this, = 1/ T = v / . If a complete cycle comprises 2 radians, then the number of radians of a cycle that pass a fixed point per interval of time is given by the angular frequency, = 2 / T = 2 . Thus the harmonic wave may also be expressed as: ( x , t ) = A sin( kx- t ) . A fixed point on the wave, such as a particular peak, moves along with the wave at the phase velocity v = / k ....
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