Serway & Jewett — Wave Motion: Problem 52

A train whistle (\(f = 400 \text{ Hz}\)) sounds higher or lower in frequency depending on whether it approaches or recedes. (a) Prove that the difference in frequency between the approaching and receding train whistle is \[ \Delta f = \frac{2u/v}{1 - u^2/v^2} f \] where \(u\) is the speed of the train and \(v\) is the speed of sound. (b) Calculate this difference for a train moving at a speed of \(130 \text{ km/h}\). Take the speed of sound in air to be \(340 \text{ m/s}\).