Question

An engine whistling at a constant frequency and moving with a constant velocity goes past a stationary observer. As the engine crosses him, the frequency of the sound heard by him changes by a factor f. The actual difference in the frequencies of sound heard by him before and after the engine crosses him is

Answer 1

The sound of frequency n heard by a stationary observer from a source of sound moving with velocity Vs towards the observer is given by,         ν'=VnV−Vs−−−(1)and when source of sound moves away from the observer then,       ν''=VnV+Vs−−−(2) But we are given,       ν'−v''=f=Vn(1V−Vs−1V+Vs)=−2V n Vs(V2−Vs2)or                          f = −2n ×VVs[(VVs)2−1]−−−−(3)again      ν'=VnV−Vs= fVnV+Vsor               f(V−Vs)=V+Vsor               V(f−1)=Vs(1+f)or             VVs=1+ff−1−−−(4)Now putting the value of VVs from eqn (4) in eqn (3), we get,     ν'=−2n×1+ff−1[(1+ff−1)2−1]=2n×(1−f2)4f=n×(1−f2)2f. proved.