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 is the length seen at time at one spot, and 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 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 and with same frequency and amplitudes and , and has phase shift +90° relative to , the sum is a sinusoidal signal with the same frequency, with amplitude and phase shift from , such that and . A real-world example of a sonic phase difference occurs in the warble of a Native American flute. The amplitude of different harmonic components of same long-held note on the flute come into dominance at different points in the phase cycle. The phase difference between the different harmonics can be observed on a spectrogram of the sound of a warbling flute. Phase comparison Phase comparison is a comparison of the phase of two waveforms, usually of the same nominal frequency. In time and frequency, the purpose of a phase comparison is generally to determine the frequency offset (difference between signal cycles) with