Case I : When two SHMs are of same angular frequency
Figure shows two particles P´ and Q´ in SHM with same angular frequency ω. P and Q are the corresponding particles in circular motion for SHM of P´ and Q´. Let P and Q both starts their circular motion at the same time at t = 0 then at the same instant P´ and Q´starts their SHM in upward direction as shown. As frequency of both are equal, both will reach their extreme position (topmost point) at the same time and will again reach their mean position simultaneously at time t = T/2. [T = Time period of SHM = 2π/ω] and move in downward direction together or we can state that the oscillations of P´ and Q´ are exactly parallel and at every instant the phase of both P´ and Q´ are equal, thus phase difference in these two SHMs is zero. These SHMs are called same phase SHMs.
Now consider figure, where we assume if P´starts its SHM at t = 0 but Q´ will start at time t1. In this duration from t = 0 to t = t1, P´ will move ahead in phase by ωt1 radians while Q´ was at rest. Now Q´starts at time t1 and move with same angular frequency ω. It can never catch P´ as both are oscillating at same angular frequency. Thus here Q´ will always lag in phase by ωt1 then P´ or we say P´ is leading in phase by ωt1 then Q´ ans as ω of both are constant their phase difference will also remains constant so keep it in mind that in two SHMs of same angular frequency, if they have some phase difference, it always remains constant.
Now we consider a spacial case when time lag between the starting of two SHMs is T/2 i.e. half of oscillation period. Consider figure, here if we assume, particle P´ starts at t = 0 and Q´ at t = T/2 when P´ completes its half oscillation. Here we can see that the phase difference in the two SHM is π by which Q´ is lagging. Here when Q´starts its oscillation in upward direction P´ moves in downward direction.As angular velocity of P and Q are same, both complete their quarter revolution in same time. Thus when P´ reaches its bottom extreme position, Q´ will reach its upper extreme position and then after Q´starts moving downward, P´starts moving upward and both of these will reach their mean position simultaneously but in opposite directions, P´ has completed its one oscillation where as Q´ is at half of its oscillation due to a phase lag of π.
Thus if we observe both oscillations simultaneously we can see that oscillations of the two particles P´ and Q´ are exactly anti parallel i.e. when P´ goes up, Q´ comes down and at all instants of time their displacements from mean position are equal but in opposite direction if there amplitudes are equal. Such SHMs are called opposite phase SHMs.
Case II : When two SHMs are of different angular frequency
We’ve discussed that when two SHMs are of same angular frequency, their phase difference does not change with time.Consider two particles in SHM as shown in figure. Their corresponding particles for circular motion are A and B respectively as shown. If both A´ and B´starts their SHM from mean position at t =0 with angular frequencies ω1 and ω2 , then we say at t = 0 their phase difference is zero but after time t, their respective phase are ω1t and ω2t. Thus after time t, the phase difference in the two SHMs is ω = (ω1 – ω2)t thus equation above shows that if angular frequencies of the two SHMs are different their phase difference continuously changes with time.
