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4. a) A jet of water coming out of a tap always breaks up into a series of drops. It has been
   observed experimentally that these drops vibrate. Use dimensional analysis to predict
   how the frequency of vibration of the water drops will depend on their density, radius
   and surface tension. f6g
   b) An ideal polarising filter is oriented such that incident horizontally polarised light
   is transmitted 100% and vertically polarised light is all absorbed. (These directions
   relate to the electric field).
   (i) When light of amplitude E0 = 200 Vm􀀀1 and linearly polarised at 45% to the
   vertical is incident, what polarisation and what amplitude is transmitted? f4g
   (ii) When circularly polarised light of the same amplitude is incident, what polarisation
   and what fraction of the incident amplitude is transmitted? f3g
   c) An outdoor sound system comprises two identical loudspeakers distance d = 5 m
   apart as sketched in the diagram below. As part of a test they are both fed the same
   pure note, of same amplitude A0 and phase, corresponding to a sound wavelength
   l = 0:6 m (and same frequency w).
   L
   y
   d
   
   P
   y = 0
   S1
   S2
   (i) The speakers S1 and S2 emit waves
   y1 = A0 cos (kx1􀀀wt) ;
   y2 = A0 cos (kx2􀀀wt) ;
   respectively, where k is the wavenumber and x1 and x2 are distances measured
   from S1 and S2. Show that the total wave at P is given by
   y (x; t) = 2A0 cos (kX 􀀀Wt)cos(kD) ;
   and define X;W and D. f4g
   (ii) Show that for y << L the sound amplitude at point, P, in the diagram is expected to be proportional to cos pdy lL f4g: (iii) An observer stands L = 20 m away, at y = 0. They move along the positive y-direction. At what value of y(> 0) will they first hear a maximum amplitude.
   f4g