The displacement of an elastic wave is given by the function y = 3 sin omega t + 4 cos omega t.where y is in cm and t is in second. Calculate the resultant amplitude.

The displacement of an elastic wave is given by the function y = 3 sin

1.	The displacement of an elastic wave is given by the function y = 3 sin omega t + 4 cos omega t.where y is in cm and t is in second. Calculate the resultant amplitude.
Answer» Solution :Here `y =3 sin (OMEGA t ) + 4 cos (omega t) ""…(1)` But for a harmonic wave, `y = sin (omega t + phi)` `therefore y = a [sin (omega t) cos phi+ cos (omega t ) sin phi]` `therefore y = (a cos phi ) sin (omega t ) + (a sin phi) cos (omega t ) ...(2)` Comparing equations (1) and (2), we get, `a cos phi =3""...(3)` `a sin phi =4""...(4)` Squaringand adding `a ^(2) cos ^(2) phi + a ^(2) sin ^(2) phi = 9 + 16 ` `therefore a ^(2) (sin ^(2) phi + cos ^(2) phi ) = 25` `thereofre a ^(2) (1) = 25` `therefore a ^(2) = 25` `therefore a = 5 cm (because a gt 0)` Taking ratio of EQUATION (4) to equation (3), `(a sin phi)/( a cos phi) = 4/3` `therefore tan phi = (4)/(3) implies phi = tan^(-1) ((4)/(3))`