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When two SHMs of the same frequency act along the same line, the resultant is also SHM at that frequency. Using phasor (vector) addition: $A_R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\delta}$ where $\delta$ is the phase difference. The resultant phase is $\tan\phi = (A_1\sin\phi_1 + A_2\sin\phi_2)/(A_1\cos\phi_1 + A_2\cos\phi_2)$.

Special cases: $\delta = 0$ (constructive, $A_R = A_1+A_2$), $\delta = \pi$ (destructive, $A_R = |A_1-A_2|$), $\delta = \pi/2$ ($A_R = \sqrt{A_1^2+A_2^2}$, Pythagorean). For equal amplitudes: $A_R = 2A\cos(\delta/2)$. Three equal-amplitude SHMs at 120 degrees apart cancel completely ($A_R = 0$). When frequencies differ slightly ($f_1 \neq f_2$), the superposition is NOT SHM but produces beats with beat frequency $|f_1-f_2|$. The amplitude modulates between $|A_1-A_2|$ and $A_1+A_2$. For perpendicular SHMs of the same frequency, the trajectory is generally an ellipse — a straight line when $\delta = 0$ or $\pi$, and a circle when $\delta = \pi/2$ with $A_1 = A_2$. These are Lissajous figures.