The displacement (from the mean / central position) of a particle, executing a SHM, is expressed as:
x = a sin(ωt +ϕ)
The instantaneous displacement, of the particle, also helps us to find its instantaneous velocity and its instantaneous acceleration. It is worth noting here that it is the instantaneous value of the quantity (ωt +ϕ), (the arguement of the sine function in the expression for displacement), that enables us to find all these quantities, related to the instantaneous ‘state of motion’ of the particle. We call the term (ωt +ϕ) as the (instantaneous) phase of the particle executing SHM
Thus the ‘phase’, of the oscillating particle, at any instant (any position), is a ‘term’ / ‘quantity’ that determines the parameters (displacement, velocity, acceleration) that define / determine the instantaneous ‘state of motion’ of the particle.
Thus, for a particle, executing a SHM,
Phase, at an instant, t = δ = (ωt +ϕ)
It follows that the ‘phase difference’, for two time instants, t, and (t + Δt), is
where T is the ‘time period’ of the oscillating particle. It follows that if Δt = nT , we would have
Δδ = n.2π
The value of sin (2nπ + θ) (as well as other trigonometric functions) being the same as that of sin θ , it follows that the particle would be back in the same ‘state of motion’, after time intervals that are an integral multiple of its ‘time period’. In terms of ‘phase’, we express this by saying that at time intervals, that are nT apart, the partile is in the ‘same phase’. Thus a phase difference
Δδ = 2πn (n = 0,1,2,3,....)
implies the same, or identical, ‘phase states’ of the particles.
Again, since
The particle would be having ‘equal in magnitude but opposite in direction’ displacement at time intervals that are half integral multiples of its time period. We express this by saying that at time intervals, that are (2n +1)T/2 apart, the particle is in ‘opposite in phase’ states of its motion.
Thus a phase difference
implies the ‘opposite in phase’ states of motion of the particle.
‘Epoch’
Another simple point, about the ‘phase term’ is worth noting. The phase term
δ = (ωt + ϕ)
has the value δ0 = ϕ, at t = 0. The term, ϕ , thus represents the value of the phase at the start of the motion of the oscillating particle. We call this initial value of the phase (i.e. phase at t = 0) as the ‘epoch’ of the oscillating particle.
It is conventional, and convenient, to (usually) take ϕ = 0 for an oscillating particle. However, when the oscillations of two, or more, oscillating particles are to be described by taking a common instant as t = 0, (for all of them), the value of the epoch may be ‘zero’ for one of them but may, or may not, be zero for the others.
