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  `f_o=(v+v_o)/(v-v_s) f_n,` where  `v approx 344 m/s` is the speed of sound in air, `v_o` is the speed of an observer towards a source and `v_s` is the speed of a source towards an observer. This formula assumes no wind.

To take wind speed `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 `v_o` and `v_s` become `v_s-v_w` and `v_s-v_w` (directions must be considered).

In our case `v_o=0` and the formula becomes `f_o=(v-v_w)/(v-(v_s-v_w)) f_n,` in numbers it is about `(344-7)/(344-33)*500 approx 541.8 (Hz).`

We are asked about the wavelength, it is of course `v/f_o approx 0.63 m.`