Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• Consider a source of light of frequency fsce moving obliquely to an observer Q as in Fig. 1.24(a). (a) Prove that Q receives the light with frequency fobs given by the general Doppler formula

(b) Check that this formula reduces to our previous result (1.51) when the source is approaching Q head-on.

The analysis in part (a) is quite similar to that leading to (1.51) but the geometry is more complicated. Consider two successive wave crests emitted at points A and B as in Fig. 1.(b). Since A and B are in practice very close together, the rays AQ and BQ are effectively parallel. Show that the difference between the lengths AQ and BQ is approximately v Δt cos θ and hence that the distance between successive crests as they approach Q is (c − v cos θ) Δt. This is the appropriate generalization of (1.47), and from here the discussion is closely parallel.

FIGURE 1

(a) Light from the moving source to the observer Q makes an angle θ with the velocity v. (b) If two successive wave crests are emitted at A and B, a time Δt apart, then AB is v Δt.