Obtain the equation of resultant displacement of two progressive harmonic waves on a stretched string.

Explanation

Suppose two progressive harmonic waves are moving on a stretched string of same angular frequency `(omega),` angular wave number (k) and wavelength `(lamda).`
Both waves are of same speed but there will be some initial phase difference.
Suppose, these two waves have same amplitude and both are moving in +X-axis.
Suppose, `y _(1) (x,t)` has initial phase zero and `y _(2) (x,t)` has initial phase `phi.`
From equation of progressive harmonic wave,
`y(x,t) =a sin (kx - omega t + phi)`
`y _(1) (x,t) = a sin (kx - omega t + phi) ...(1) and `
`y _(2) (x,t) = a sin (kx - omega t + phi) ...(2)`
From the principle of superposition,
`y (x,t) = y _(1) (x,t) + y _(2) (x,t)`
`=a sin (kx - omega t ) + a sin (kx - omega t + phi)`
`=a [ 2 sin { ((kx - omega t ) + (kx - omega t + phi ))/(2 )} cos "" (phi)/(2) ]" "...(3)`
`[because sin C + sin D = 2 sin ((C+D)/(2)) cos ((C -D)/(2)) ]`
`therefore y (x,t) =2 a cos "" (phi)/( 2) sin (kx - omega t + (phi)/(2))" "...(4)`
Here, equation (4) is of progresive harmonic wave moving in + x direction. Its frequency and wavelength are same as of original wave but the initial phase is `(phi)/(2).`