The superposition of two simple harmonic motions (SHM) with the same angular frequency $\omega$ results in a new simple harmonic motion. The resultant displacement $y$ is the sum of the individual displacements: $y = y_1 + y_2 = a \sin(\omega t) + b \cos(\omega t)$ Using the trigonometric identity $\cos(\omega t) = \sin(\omega t + \frac{\pi}{2})$, we see that the two waves have a phase difference of $\phi = \frac{\pi}{2}$. The resultant amplitude $R$ for the superposition of two waves with amplitudes $a$ and $b$ and phase difference $\phi$ is given by: $R = \sqrt{a^2 + b^2 + 2ab \cos\phi}$ Substituting $\phi = \frac{\pi}{2}$ (since $\cos \frac{\pi}{2} = 0$): $R = \sqrt{a^2 + b^2}$ Thus, the resulting motion is simple harmonic with an amplitude of $\sqrt{a^2 + b^2}$. This concept is consistent with the principles of phasor addition used in analyzing AC circuits and oscillations .