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

(N/A) Consider two progressive harmonic waves traveling along a stretched string with the same angular frequency $(\omega)$,angular wave number $(k)$,and amplitude $(a)$.
Let the two waves be represented as:
$y_{1}(x, t) = a \sin(kx - \omega t)$
$y_{2}(x, t) = a \sin(kx - \omega t + \phi)$
where $\phi$ is the constant phase difference between the two waves.
According to the principle of superposition,the resultant displacement $y(x, t)$ is the algebraic sum of the individual displacements:
$y(x, t) = y_{1}(x, t) + y_{2}(x, t)$
$y(x, t) = a \sin(kx - \omega t) + a \sin(kx - \omega t + \phi)$
Using the trigonometric identity $\sin C + \sin D = 2 \sin \left( \frac{C+D}{2} \right) \cos \left( \frac{C-D}{2} \right)$,where $C = kx - \omega t$ and $D = kx - \omega t + \phi$:
$y(x, t) = 2a \sin \left( \frac{kx - \omega t + kx - \omega t + \phi}{2} \right) \cos \left( \frac{kx - \omega t - (kx - \omega t + \phi)}{2} \right)$
$y(x, t) = 2a \sin \left( kx - \omega t + \frac{\phi}{2} \right) \cos \left( -\frac{\phi}{2} \right)$
Since $\cos(-\theta) = \cos(\theta)$:
$y(x, t) = [2a \cos(\frac{\phi}{2})] \sin(kx - \omega t + \frac{\phi}{2})$
This is the equation of the resultant progressive wave with an amplitude of $2a \cos(\frac{\phi}{2})$ and an initial phase of $\frac{\phi}{2}$.