Question

Derive an analytical expression for a simple harmonic plane progressive wave.

Solution

Let a plane wave originate from O and progress in all the directions.
All the particles of media will execute simple harmonic motion of same amplitude and time period about mean position when wave propagates in media.
Let A be the amplitude, co be the frequency and λ be the wavelength of the wave.

The displacement of particle O at any instant is given by y(0,t) = A sin ωt
The disturbance is handed over from one particle to next and will reach point P at a distance x from O, a bit later.
Therefore, phase of particle at P lags behind the phase of particle at O by,
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where, $fraction numerator 2 straight pi over denominator straight lambda end fraction equals straight k$
Therefore the displacement of particle at P is,
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$equals straight A space sin left parenthesis ωt minus kx right parenthesis$

Therefore simple harmonic progressive wave propagating towards positive direction of x-axis is,
y(x,t) = A sin (ωt – kx).
If wave propagates towards left direction then replacing x by -x, we get
y(x,t) = A sin (ωt + kx).

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Mechanical Properties Of Fluids

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