(i) Two sinusoidal wave are given as `y_(1)=a_(1) sin (omegat+kx+delta)` and `y_(2)=a_(2) sin (omegat-kx)`. They superimpose.
(a) Calculate the resultant amplitude of oscillation at a position x. Is amplitude time dependent?
(b) Calculate the ratio of maximum and minimum amplitudes observed.
(ii) A speaker (producing a sound of a single wavelength l) and a microphone are placed as shown in the figure. The microphone detects the sound and converts it into electrical signal. This way we can obtain the waveform of the sound. Assume that there is no attenuation of the sound. The waveform detected by the microphone is sinusoidal with amplitude a. In one experiment 6 microphones are placed in front of the speaker with distance between two neighbouring microphones being `d=(5 lambda)/(6)`
(a) he output from all the 6 microphones is superimposed. What is amplitude of the resultant?
(b) If large number of microphone are kept with separation L between two consecutive ones, how will the combined output change with L? Given that `L ne n lambda (n=1, 2, 3...........)`

Answer

(i) (a) `sqrt(a_(1)^(2)+a_(2)^(2)+2a_(1)a_(2) cos (2kx+delta))`
(b) `A_("max")/A_("min")=(a_(1)+a_(2))/(|a_(1)-a_(2)|)`
(ii) (a) zero
(b) zero