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Exercise:
A single wave crest can be described by a Gaussian function. In its simplest form this can be written as fxe^-x^. Find a physically correct Gaussian shape function and erpret the parameters. Derive the corresponding wave function and plot this for several different times.

Solution:
In order to turn the Gauss function o a physically sensible function we need to include two additional parameters: fx Ae^-fracleftfracxsigmaright^ The effect of the amplitude A should be clear. The parameter sigma describes the width of the Gaussian wave crest. For xsigma the height is equal to fsigma A e^-frac fracAsqrteapprox .times A The figure below displays two such Gaussian wave crests with different amplitudes and widths. center includegraphicswidthcm#image_path:gaussian-shape# center To turn this expression o an expression describing a wave we need to substitute x by x-v t: yx t fx-v t Ae^-fracleftfracx-v tsigmaright^ The figure below displays snapshots of a Gaussian wave propagating in the positive direction vvO t_taO t_tbO t_tcO. center includegraphicswidthcm#image_path:gaussian-wave# center An animation of the wave propagation is linked to this exercise.
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Exercise:
A single wave crest can be described by a Gaussian function. In its simplest form this can be written as fxe^-x^. Find a physically correct Gaussian shape function and erpret the parameters. Derive the corresponding wave function and plot this for several different times.

Solution:
In order to turn the Gauss function o a physically sensible function we need to include two additional parameters: fx Ae^-fracleftfracxsigmaright^ The effect of the amplitude A should be clear. The parameter sigma describes the width of the Gaussian wave crest. For xsigma the height is equal to fsigma A e^-frac fracAsqrteapprox .times A The figure below displays two such Gaussian wave crests with different amplitudes and widths. center includegraphicswidthcm#image_path:gaussian-shape# center To turn this expression o an expression describing a wave we need to substitute x by x-v t: yx t fx-v t Ae^-fracleftfracx-v tsigmaright^ The figure below displays snapshots of a Gaussian wave propagating in the positive direction vvO t_taO t_tbO t_tcO. center includegraphicswidthcm#image_path:gaussian-wave# center An animation of the wave propagation is linked to this exercise.
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Mechanical Waves
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ENG (English)
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