(a)Show that Eq. (16.30) can be written as Use the binomial theorem to show that if this is u « c is approximately equal to )

Solution 78P cv (a) The frequency measured by the receiver is given by f R = c+vf s c(1v/c) f R = c(1+v/c)f s 1v/c f R = 1+v/c f s (1v/c) f R = 1 f s (1+v/c)2 1 1 f R = f s1 v/c) (1 + v/c) 2…..(1) cv Therefore, the given equation f R = c+v f s can be simplified to obtain the form of equation 1. (b) Now, using binomial theorem, 1 (1 v/c) = 1 v/2c + ...... 2 (1 + v/c) = 1 v/2c + ...... Therefore, from equation (1) f R = f s1 v/2c + .....)(1 v/2c + .....) 2 2 f R = f s1 v/2c + v/2c + v /4c .....) f R = f s1 v/c + v /4c .....)…..(2) 2 Given that, v c, therefore any term containing v can be neglected. Therefore, the 2 2 term v /4c from equation (2) can be neglected. So, equation (2) can be rewritten as, f R = f s1 v/c)