On aCartesianco-ordinatesystem.2identicalsmallspeakers,at(X =-3m, Y=8m), (X=3m, Y=-8m).Frequency85Hz.flywalked alongx-axisfrom x=-100m-x=100m.Howmany local max?

Question

Spider laid her large plane web on a Cartesian co-ordinate system.She located 2 identical small speakers,at (X =-3m, Y=8m), (X=3m, Y=-8m).She drove both speakers with the same electrical signal,they sent out identical sound waves of frequency 85Hz.fly walked slowly along x-axis from x=-100m to x=100m,listening to the resulting sound.How many local maxima did he detect?Use 340 m/s for the speed of sound in air if need it.

The answers are:

A) 1

B) 2

C) 3

D) 4

E) 5

 

ps. I need to know how to one of this answer given by my teacher, thx!

Answer 1

My analysis indicates that the number of local maxima will be 101, which is not one of the options.

 

The traveling wave equation (340 =  λ f)   gives  λ =  4 m.     

The acoustic amplitude due to the speaker on the left is   proportional to   cos( (2 pi/4)) (x +100) ,

 The acoustic amplitude due to the speaker on the right is proportional to  cos ( (2 pi/4) (x - 100).

        ( I have neglected the 1/r^2 factors - these are not important here)

  Adding these amplitudes and simplifying  using some trigonometric identities results in

 

       A  = 2 cos ( (2 pi/4) x )   .   The intensity, I, is the square of this so

   

   I  is proportional to   cos²( (2 pi/4) x)  =   1/2 + 1/2 cos( pi x)  

 

     To get a maximum,    we need   pi x =  n 2 pi   for any integer n.

        So   x =  2n,    that is: x must be an even integer.    

 

     There are 101 even integers between -100 and +100.