Find the wavelength of a train whistle heard by a fixed observer as the train moves toward the observer with a velocity of 40.0 m/s. A wind blows...

Tuesday, July 9, 2013

Find the wavelength of a train whistle heard by a fixed observer as the train moves toward the observer with a velocity of 40.0 m/s. A wind blows...

Hello!

The observed frequency of a train whistle may be different from its natural frequency when an observer moves relative to a train. This is called the Doppler effect. The cause of the Doppler effect is that each next maximum or minimum of a wave is emitted closer or farther from the observer then the previous. This way, wavelength shortens when an observer and a wave source are moving towards each other, and elongates otherwise.

The formula for the observed frequency if $\displaystyle{f}_{{o}}=\frac{{{v}+{v}_{{o}}}}{{{v}-{v}_{{s}}}}{f}_{{n}},$ where $\displaystyle{v}\approx{344}\frac{{m}}{{s}}$ is the speed of sound in air, $\displaystyle{v}_{{o}}$ is the speed of an observer towards a source and $\displaystyle{v}_{{s}}$ is the speed of a source towards an observer. This formula assumes no wind.

To take wind speed $\displaystyle{v}_{{w}}$ into account, consider a frame of reference associated with air, it is an inertial one. There is no wind in this system and the speeds $\displaystyle{v}_{{o}}$ and $\displaystyle{v}_{{s}}$ become $\displaystyle{v}_{{s}}-{v}_{{w}}$ and $\displaystyle{v}_{{s}}-{v}_{{w}}$ (directions must be considered).

In our case $\displaystyle{v}_{{o}}={0}$ and the formula becomes $\displaystyle{f}_{{o}}=\frac{{{v}-{v}_{{w}}}}{{{v}-{\left({v}_{{s}}-{v}_{{w}}\right)}}}{f}_{{n}},$ in numbers it is about $\displaystyle\frac{{{344}-{7}}}{{{344}-{33}}}\cdot{500}\approx{541.8}{\left({H}{z}\right)}.$

We are asked about the wavelength, it is of course $\displaystyle\frac{{v}}{{f}_{{o}}}\approx{0.63}{m}.$

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Tuesday, July 9, 2013