Calculate Doppler Shift: A Train Horn

To find the observed frequency, $f_{obs}=f_s\left(\frac{v_w}{v_w\pm v_s}\right)$ must be used because the source is moving. The minus sign is used for the approaching train, and the plus sign for the receding train.

(1) Enter known values into $f_{obs}=f_s\left(\frac{v_w}{v_w-v_s}\right)$ to calculate the frequency observed by a stationary person as the train approaches:

$$f_{obs}=f_s\left(\frac{v_w}{v_w-v_s}\right)=\left(150\text{ Hz}\right)\left(\frac{340\text{ m/s}}{340\text{ m/s}-35\text{ m/s}}\right)$$ $$=167\text{ Hz}\approx \mathrm{170 }\text{Hz (2 sig. figs.)}$$

(2) Use the same equation but with the plus sign to find the frequency heard by a stationary person as the train recedes.

$$f_{obs}=f_s\left(\frac{v_w}{v_w+v_s}\right)=\left(150\text{ Hz}\right)\left(\frac{340\text{ m/s}}{340\text{ m/s}+35\text{ m/s}}\right)$$ $$=136\text{ Hz}\approx \text{140 Hz (2 sig. figs.)}$$