The difference \varphi(t) = \varphi_G(t) - \varphi_F(t) between the phases of two periodic signals F and G is called the
phase difference or
phase shift of G relative to F. When the phase difference \varphi(t) is a quarter of turn (a right angle, or ), sinusoidal signals are sometimes said to be in
quadrature, e.g.,
in-phase and quadrature components of a composite signal or even different signals (e.g., voltage and current). If the frequencies are different, the phase difference \varphi(t) increases linearly with the argument t. The periodic changes from reinforcement and opposition cause a phenomenon called
beating.
For shifted signals The phase difference is especially important when comparing a periodic signal F with a shifted and possibly scaled version G of it. That is, suppose that G(t) = \alpha\,F(t + \tau) for some constants \alpha,\tau and all t. Suppose also that the origin for computing the phase of G has been shifted too. In that case, the phase difference \varphi is a constant (independent of t), called the 'phase shift' or 'phase offset' of G relative to F. In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant. In this case, the phase shift is simply the argument shift \tau, expressed as a fraction of the common period T (in terms of the
modulo operation) of the two signals and then scaled to a full turn: \varphi = 2\pi \left[\!\!\left[ \frac{\tau}{T} \right]\!\!\right]. If F is a "canonical" representative for a class of signals, like \sin(t) is for all sinusoidal signals, then the phase shift \varphi called simply the
initial phase of G. Therefore, when two periodic signals have the same frequency, they are always in phase, or always out of phase. Physically, this situation commonly occurs, for many reasons. For example, the two signals may be a periodic soundwave recorded by two microphones at separate locations. Or, conversely, they may be periodic soundwaves created by two separate speakers from the same electrical signal, and recorded by a single microphone. They may be a
radio signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby. A well-known example of phase difference is the length of shadows seen at different points of Earth. To a first approximation, if F(t) is the length seen at time t at one spot, and G is the length seen at the same time at a
longitude 30° west of that point, then the phase difference between the two signals will be 30° (assuming that, in each signal, each period starts when the shadow is shortest).
For sinusoids with same frequency For sinusoidal signals (and a few other waveforms, like square or symmetric triangular), a phase shift of 180° is equivalent to a phase shift of 0° with negation of the amplitude. When two signals with these waveforms, same period, and opposite phases are added together, the sum F+G is either identically zero, or is a sinusoidal signal with the same period and phase, whose amplitude is the difference of the original amplitudes. The phase shift of the co-sine function relative to the sine function is +90°. It follows that, for two sinusoidal signals F and G with same frequency and amplitudes A and B, and G has phase shift +90° relative to F, the sum F+G is a sinusoidal signal with the same frequency, with amplitude C and phase shift -90^\circ from F, such that C = \sqrt{A^2 + B^2} \quad\quad \text{ and } \quad\quad \sin(\varphi) = B/C. File:Phase Comparison image two waves.gif|thumb|Representation of phase comparison. ==Phase comparison==