Phase (waves)
Imagine a wave moving through time. At any moment, the phase tells you exactly how far along that cycle has progressed. This angle-like quantity represents the fraction of the cycle covered up to a specific point in time. In physics and mathematics, scientists use the symbol phi or lowercase phi to denote this value. It varies by one full turn as the variable goes through each period. You might measure it in degrees or radians. When the variable completes a full period, the phase increases by 360 degrees or 2 pi radians. Whole turns are often ignored when expressing the phase. So if two points differ by a whole number of periods, they are said to be at the same phase.
Let the signal f be a periodic function of one real variable t. Let T be its period, which is the smallest positive real number such that f(t + T) equals f(t) for all t. Then the phase of f at any argument t is defined using the fractional part function. This function discards the integer part of a real number. An arbitrary origin value t0 serves as the beginning of a cycle. To get the phase as an angle between 0 and 2 pi, you apply the formula involving the modulo operation. If you want the phase as an angle between 0 degrees and 360 degrees, replace 2 pi with 360 degrees in the calculation. The numeric value depends on your choice of where the period starts. For a sinusoid, a convenient choice is any time where the function changes from zero to positive values.
Picture a clock with a hand turning at constant speed. It makes a full turn every T seconds. At time t0, the hand points straight up. The phase phi is then the angle from the 12:00 position to the current position of the hand. You measure this clockwise direction. This concept becomes most useful when the origin t0 is chosen based on features of the signal itself. For example, for a sinusoid, pick any point where the function's value changes from zero to positive. In the clock analogy, each signal is represented by a hand or pointer of the same clock. Both hands turn at constant but possibly different speeds. The phase difference is simply the angle between these two hands measured clockwise.
The difference delta_phi between the phases of two periodic signals f(t) and g(t) is called the phase shift or phase difference of g relative to f. When this difference equals zero, the two signals are in phase. Otherwise, they are out of phase with each other. If the phase difference is 180 degrees or pi radians, the phases are opposite. Then the signals have opposite signs and destructive interference occurs. A phase reversal implies a 180-degree phase shift. When the phase difference is a quarter of a turn, or a right angle of pi over 2 radians, sinusoidal signals are sometimes said to be in quadrature. If frequencies differ, the phase difference increases linearly with the argument t. These periodic changes from reinforcement and opposition cause a phenomenon known as beating.
Consider two periodic sound waves emitted by two sources and recorded together by a microphone. This usually happens in linear systems when the superposition principle holds. For arguments when the phase difference is zero, the two signals will have the same sign and reinforce each other. One says that constructive interference is occurring. At arguments where the phases differ, the value of the sum depends on the waveform. A well-known example involves the length of shadows seen at different points on Earth. To a first approximation, if L1 is the length seen at time t at one spot, and L2 is the length seen at the same time at a longitude 30 degrees west, then the phase difference between the two signals will be 30 degrees. Another real-world example occurs in the warble of a Native American flute. The amplitude of different harmonic components comes into dominance at different points in the phase cycle.
Connect two signals to a two-channel oscilloscope to perform a phase comparison. The oscilloscope displays two sine signals side by side. If the two frequencies were exactly the same, their phase relationship would not change and both would appear stationary on the display. Since the frequencies are not exactly the same, the reference appears stationary while the test signal moves. By measuring the rate of motion of the test signal, you determine the offset between frequencies. Vertical lines mark the points where each sine signal passes through zero. Bars below show width representing the phase difference. In this case, the increasing phase difference indicates that the test signal is lower in frequency than the reference. This method allows engineers to determine frequency offsets with respect to a standard reference.
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Common questions
What does the phase of a periodic function represent in physics and mathematics?
The phase represents the fraction of the cycle covered up to a specific point in time. It is an angle-like quantity that varies by one full turn as the variable goes through each period.
How do scientists denote the phase value for a periodic signal f at argument t?
Scientists use the symbol phi or lowercase phi to denote this value. The numeric value depends on the choice of where the period starts, often using 0 to 2 pi radians or 0 degrees to 360 degrees.
When are two signals considered to be in phase versus out of phase?
Two signals are in phase when their phase difference equals zero, meaning they have the same sign and reinforce each other. They are out of phase if the difference is not zero, with opposite signs occurring at a 180-degree shift.
What phenomenon occurs when the phase difference between two frequencies increases linearly with time?
Beating occurs when the phase difference increases linearly with the argument t due to differing frequencies. This causes periodic changes from reinforcement and opposition in the combined signal.
How can engineers determine frequency offsets using a two-channel oscilloscope?
Engineers measure the rate of motion of the test signal relative to a stationary reference to find the offset. Vertical lines mark points where sine signals pass through zero while bars below show width representing the phase difference.