(a) -83.6 Hz
Due to the Doppler effect, the frequency of the sound of the train horn appears shifted to the observer at rest, according to the formula:
![f' = (\frac{v}{v\pm v_s})f](https://tex.z-dn.net/?f=f%27%20%3D%20%28%5Cfrac%7Bv%7D%7Bv%5Cpm%20v_s%7D%29f)
where
f' is the apparent frequency
v = 343 m/s is the speed of sound
is the velocity of the source of the sound (positive if the source is moving away from the observer, negative if it is moving towards the observer)
f is the original frequency of the sound
Here we have
f = 350 Hz
When the train is approaching, we have
![v_s = -40.4 m/s](https://tex.z-dn.net/?f=v_s%20%3D%20-40.4%20m%2Fs)
So the frequency heard by the observer on the platform is
![f' = (\frac{343 m/s}{343 m/s - 40.4 m/s})(350 Hz)=396.7 Hz](https://tex.z-dn.net/?f=f%27%20%3D%20%28%5Cfrac%7B343%20m%2Fs%7D%7B343%20m%2Fs%20-%2040.4%20m%2Fs%7D%29%28350%20Hz%29%3D396.7%20Hz)
When the train has passed the platform, we have
![v_s = +40.4 m/s](https://tex.z-dn.net/?f=v_s%20%3D%20%2B40.4%20m%2Fs)
So the frequency heard by the observer on the platform is
![f' = (\frac{343 m/s}{343 m/s + 40.4 m/s})(350 Hz)=313.1 Hz](https://tex.z-dn.net/?f=f%27%20%3D%20%28%5Cfrac%7B343%20m%2Fs%7D%7B343%20m%2Fs%20%2B%2040.4%20m%2Fs%7D%29%28350%20Hz%29%3D313.1%20Hz)
Therefore the overall shift in frequency is
![\Delta f = 313.1 Hz - 396.7 Hz = -83.6 Hz](https://tex.z-dn.net/?f=%5CDelta%20f%20%3D%20313.1%20Hz%20-%20396.7%20Hz%20%3D%20-83.6%20Hz)
And the negative sign means the frequency has decreased.
(b) 0.865 m
The wavelength and the frequency of a wave are related by the equation
![v=\lambda f](https://tex.z-dn.net/?f=v%3D%5Clambda%20f)
where
v is the speed of the wave
is the wavelength
f is the frequency
When the train is approaching the platform, we have
v = 343 m/s (speed of sound)
f = f' = 396.7 Hz (apparent frequency)
Therefore the wavelength detected by a person on the platform is
![\lambda' = \frac{v}{f'}=\frac{343 m/s}{396.7 Hz}=0.865m](https://tex.z-dn.net/?f=%5Clambda%27%20%3D%20%5Cfrac%7Bv%7D%7Bf%27%7D%3D%5Cfrac%7B343%20m%2Fs%7D%7B396.7%20Hz%7D%3D0.865m)