Phase in Simple Harmonic Motion

The quantity $(\omega t + \phi)$ appearing inside the cosine (or sine) function of an S.H.M. is called the PHASE of the oscillation at time $t$. It is the single number that completely determines the instantaneous state of the motion at that instant -- both WHERE the oscillating particle currently is (its displacement) and WHICH WAY it is currently moving (the sign of its velocity).\n\nThe constant $\phi$ -- the value of the phase at the instant $t=0$ -- is called the PHASE CONSTANT, or the initial phase. It is fixed entirely by the particle's own displacement and velocity at the moment the stopwatch is started; a different choice of WHEN to start timing an otherwise physically identical oscillation simply corresponds to a different value of $\phi$, without changing either the amplitude $A$ or the angular frequency $\omega$ in the slightest -- those two are fixed purely by the physical system itself and by how far it was originally displaced.\n\nThe PHASE DIFFERENCE between two particles executing S.H.M. of the same frequency -- written $\Delta\phi = \phi_2 - \phi_1$ -- describes how far apart the two motions are running within their respective cycles:\n\n- If $\Delta\phi = 0$, the two particles are said to be IN PHASE: they reach their extreme positions together, pass through the mean position together, and move in the same direction at every instant.\n- If $\Delta\phi = \pi$ (exactly half a cycle apart), the two particles are exactly OUT OF PHASE: whenever one is at its positive extreme, the other is simultaneously at its negative extreme, and their velocities are always opposite to each other.\n- If $\Delta\phi = \pi/2$ (exactly a quarter-cycle apart, sometimes described as being IN QUADRATURE), the two motions are offset in a special way: one of the two particles is at an extreme position (momentarily at rest) at exactly the instant the other is passing through its own mean position at maximum speed. This is precisely the relationship that exists between the displacement and the velocity of a SINGLE S.H.M. particle -- as Section 13.6 shows next, $v(t)$ is always a quarter-cycle ahead of $x(t)$ for one and the same oscillating particle.\n\nThe idea of phase can also be pictured geometrically through the REFERENCE-CIRCLE construction: S.H.M. along a straight line is mathematically identical to the shadow (the pr …