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.

Explanation

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)`
Squaring and 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))`