Wave Mechanics: Phase and Phase Difference WaveMechanicsPhaseAndPhaseDifference 2026-09-11 05:59:39 2026-09-11 05:59:39 Topic wave mechanics phase phase angle phase difference phase lead phase lag time shift angular frequency sinusoidal oscillation periodic motion equivalent phase modulo 2 pi % this is the default PhysicsLibrary preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics). % If you use \includegraphics{figure} with no file extension, PhysicsLibrary % will prefer figure.png before falling back to figure.eps. \usepackage{graphicx} \DeclareGraphicsExtensions{.png,.jpg,.jpeg,.pdf,.eps} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \section*{Wave Mechanics: Phase and Phase Difference} WM02 introduced the sinusoidal oscillator \begin{equation} u(t)=A\cos(\omega t+\phi) \end{equation} and defined the phase angle \begin{equation} \theta(t)=\omega t+\phi. \end{equation} The purpose of this entry is to make the idea of phase physically useful. Phase tells us where an oscillator is within its repeating cycle. Phase difference tells us how far one oscillator is advanced or delayed relative to another. This lesson still considers oscillation at one point. Spatial phase, the term $kx$, wavelength, wavenumber, and traveling waves are introduced later in the series. \section{Phase as a coordinate within a cycle} For the cosine function, one full cycle corresponds to an increase of $2\pi$ radians in the phase angle. Thus the values \begin{equation} 0,\qquad \frac{\pi}{2},\qquad \pi,\qquad \frac{3\pi}{2},\qquad 2\pi \end{equation} mark familiar locations within one cycle. \begin{center} \includegraphics{WM03_fig01_phase_cycle.png} \vspace{0.45em} \textbf{Figure.} Phase may be pictured as an angular coordinate around a cycle. Advancing the phase by $2\pi$ returns to the same location in that cycle. \end{center} For \begin{equation} u(t)=A\cos\theta(t), \end{equation} the quarter-cycle points are \begin{align} \theta=0 &: \qquad u=+A,\\ \theta=\frac{\pi}{2} &: \qquad u=0,\\ \theta=\pi &: \qquad u=-A,\\ \theta=\frac{3\pi}{2} &: \qquad u=0,\\ \theta=2\pi &: \qquad u=+A. \end{align} The two zero-displacement points are especially instructive. They have the same displacement, but they correspond to different phases and opposite directions of motion. Phase therefore carries more information about the state of an oscillation than displacement alone. \section{The phase advances uniformly for a sinusoid} WM02 defined \begin{equation} \theta(t)=\omega t+\phi. \end{equation} If $\omega$ is constant, then the phase increases uniformly with time. Over a time interval $\Delta t$, the phase advance is \begin{equation} \Delta\theta=\omega\Delta t. \end{equation} Using \begin{equation} \omega=\frac{2\pi}{T}, \end{equation} we can also write \begin{equation} \boxed{\Delta\theta=2\pi\frac{\Delta t}{T}}. \end{equation} This equation gives a useful conversion between elapsed time and position within the cycle. For example, after one quarter of a period, \begin{equation} \Delta t=\frac{T}{4}, \end{equation} so \begin{equation} \Delta\theta =2\pi\frac{T/4}{T} =\frac{\pi}{2}. \end{equation} A quarter period in time corresponds to a quarter cycle, or $\pi/2$ radians, in phase. \section{Equivalent phases} Because cosine is periodic, \begin{equation} \cos(\theta+2\pi)=\cos\theta. \end{equation} More generally, \begin{equation} \boxed{\cos(\theta+2\pi n)=\cos\theta,\qquad n=0,\pm1,\pm2,\ldots} \end{equation} Therefore phase angles separated by an integer number of complete cycles are equivalent for a periodic sinusoid. \begin{center} \includegraphics{WM03_fig03_equivalent_phase.png} \vspace{0.45em} \textbf{Figure.} The phases $\phi$ and $\phi+2\pi$ differ numerically, but they identify the same location within the repeating cycle. Phase is naturally interpreted modulo $2\pi$. \end{center} For example, \begin{equation} \frac{5\pi}{2} =\frac{\pi}{2}+2\pi, \end{equation} so $5\pi/2$ and $\pi/2$ represent the same phase location. Similarly, \begin{equation} -\frac{\pi}{2} =\frac{3\pi}{2}-2\pi, \end{equation} so $-\pi/2$ and $3\pi/2$ are also equivalent modulo one cycle. This does not mean that accumulated phase is always unimportant. In some problems we care about how many complete cycles have occurred. But when the question is only where the oscillator lies within its present cycle, phase can be reduced modulo $2\pi$. \section{Comparing two sinusoidal oscillators} Consider two oscillators with the same angular frequency $\omega$: \begin{align} u_1(t)&=A_1\cos(\omega t+\phi_1),\\ u_2(t)&=A_2\cos(\omega t+\phi_2). \end{align} Their instantaneous phase angles are \begin{align} \theta_1(t)&=\omega t+\phi_1,\\ \theta_2(t)&=\omega t+\phi_2. \end{align} Subtracting gives \begin{equation} \theta_2(t)-\theta_1(t) =\phi_2-\phi_1. \end{equation} Because the $\omega t$ terms cancel, the difference is constant in time. We define \begin{equation} \boxed{\Delta\phi=\phi_2-\phi_1.} \end{equation} For same-frequency sinusoids, $\Delta\phi$ is their constant phase difference. The amplitudes $A_1$ and $A_2$ do not enter this expression. Two oscillators can have different amplitudes and still have a well-defined phase difference. \section{What phase lead and phase lag mean} Using the convention \begin{equation} u(t)=A\cos(\omega t+\phi), \end{equation} a larger phase constant means that the oscillator is further advanced through its cycle at the same clock time. Thus, if \begin{equation} \Delta\phi=\phi_2-\phi_1>0, \end{equation} then oscillator 2 \emph{leads} oscillator 1 by $\Delta\phi$ under this sign convention. If \begin{equation} \Delta\phi<0, \end{equation} then oscillator 2 \emph{lags} oscillator 1 by the magnitude of that phase difference. \begin{center} \includegraphics{WM03_fig02_phase_difference.png} \vspace{0.45em} \textbf{Figure.} Two equal-frequency sinusoids separated by a constant phase difference. In the convention $\cos(\omega t+\phi)$, the curve with the larger phase constant is advanced in time and therefore leads. \end{center} The words ``lead'' and ``lag'' are meaningful only after the sign convention has been stated. Some disciplines write harmonic motion with a minus sign in the phase argument. The safest practice is therefore to inspect the actual equation rather than memorize a verbal sign rule without context. \section{Converting phase difference into a time shift} A constant phase difference can be expressed as an equivalent time shift. Suppose \begin{equation} u_1(t)=A\cos(\omega t+\phi_1) \end{equation} and \begin{equation} u_2(t)=A\cos(\omega t+\phi_2). \end{equation} Write \begin{equation} \phi_2=\phi_1+\Delta\phi. \end{equation} Then \begin{align} u_2(t) &=A\cos(\omega t+\phi_1+\Delta\phi)\\ &=A\cos\left[\omega\left(t+\frac{\Delta\phi}{\omega}\right)+\phi_1\right]. \end{align} Therefore the corresponding time shift is \begin{equation} \boxed{\Delta t=\frac{\Delta\phi}{\omega}.} \end{equation} Using $\omega=2\pi/T$ gives another useful form: \begin{equation} \boxed{\Delta t=\frac{\Delta\phi}{2\pi}T.} \end{equation} This relation says that the fraction of a period represented by a phase difference is the same as the fraction of a full $2\pi$ phase cycle: \begin{equation} \boxed{\frac{\Delta t}{T}=\frac{\Delta\phi}{2\pi}.} \end{equation} For example, a phase difference of $\pi/2$ is one quarter of a complete phase cycle, so it corresponds to one quarter of a period: \begin{equation} \Delta t=\frac{\pi/2}{2\pi}T=\frac{T}{4}. \end{equation} \section{Important special phase differences} Several phase differences occur repeatedly in physics and engineering. \begin{itemize} \item $\Delta\phi=0$: the oscillators are \emph{in phase}. \item $|\Delta\phi|=\pi/2$: the oscillators are separated by one quarter cycle. \item $|\Delta\phi|=\pi$: the oscillators are separated by one half cycle and are often called \emph{opposite in phase} or \emph{antiphase}. \item $|\Delta\phi|=2\pi$: the numerical phase difference is one full cycle, so the oscillators are again equivalent in phase. \end{itemize} Because phase is cyclic, a difference such as \begin{equation} \frac{3\pi}{2} \end{equation} can also be represented as \begin{equation} -\frac{\pi}{2}. \end{equation} The first description says ``lead by three quarters of a cycle.'' The second says ``lag by one quarter of a cycle.'' They describe the same relative phase modulo $2\pi$. When comparing phase differences, it is often convenient to choose a principal interval such as \begin{equation} -\pi<\Delta\phi\leq\pi. \end{equation} This convention selects the smaller signed angular separation, but other interval conventions are possible. The convention should be stated whenever ambiguity matters. \section{Worked example: phase lead and equivalent time shift} Consider \begin{align} u_1(t)&=3\cos(8\pi t),\\ u_2(t)&=5\cos\left(8\pi t+\frac{\pi}{3}\right). \end{align} The amplitudes are different, but both oscillators have \begin{equation} \omega=8\pi\ \text{rad/s}. \end{equation} Their phase constants are \begin{equation} \phi_1=0, \qquad \phi_2=\frac{\pi}{3}. \end{equation} Thus \begin{equation} \Delta\phi =\phi_2-\phi_1 =\boxed{\frac{\pi}{3}}. \end{equation} Under the $+\phi$ cosine convention, oscillator 2 leads oscillator 1. The corresponding time lead is \begin{align} \Delta t &=\frac{\Delta\phi}{\omega}\\ &=\frac{\pi/3}{8\pi}\\ &=\boxed{\frac{1}{24}\ \text{s}}. \end{align} The period is \begin{equation} T=\frac{2\pi}{8\pi}=\frac14\ \text{s}, \end{equation} and indeed \begin{equation} \frac{T}{6}=\frac{1}{24}\ \text{s}. \end{equation} A phase lead of $\pi/3$ is one sixth of a complete $2\pi$ cycle, so the time lead is one sixth of a period. \section{Worked example: reducing a phase difference} Suppose \begin{equation} \Delta\phi=\frac{7\pi}{4}. \end{equation} Subtract one full cycle: \begin{align} \Delta\phi_{\text{equiv}} &=\frac{7\pi}{4}-2\pi\\ &=\frac{7\pi}{4}-\frac{8\pi}{4}\\ &=-\frac{\pi}{4}. \end{align} Thus \begin{equation} \boxed{\frac{7\pi}{4}\equiv-\frac{\pi}{4}\pmod{2\pi}.} \end{equation} The relative phase can be described either as a lead of $7\pi/4$ or, more compactly, as a lag of $\pi/4$. \section{What if the frequencies are different?} A constant phase difference requires equal angular frequencies. Consider \begin{align} \theta_1(t)&=\omega_1 t+\phi_1,\\ \theta_2(t)&=\omega_2 t+\phi_2. \end{align} Their phase difference is \begin{align} \Delta\theta(t) &=\theta_2(t)-\theta_1(t)\\ &=(\omega_2-\omega_1)t+(\phi_2-\phi_1). \end{align} Therefore \begin{equation} \boxed{\Delta\theta(t)=(\omega_2-\omega_1)t+\Delta\phi_0.} \end{equation} If $\omega_1\neq\omega_2$, the relative phase changes continuously with time. One oscillator gradually gains phase on the other. \begin{center} \includegraphics{WM03_fig04_phase_drift.png} \vspace{0.45em} \textbf{Figure.} When two angular frequencies differ, their relative phase drifts with time. Only equal-frequency sinusoids maintain a constant phase difference. \end{center} This distinction becomes important later in interference, beats, Fourier analysis, and wave propagation. \section{Phase is not amplitude} Amplitude and phase describe different features of a sinusoid. For \begin{equation} u(t)=A\cos(\omega t+\phi), \end{equation} \begin{itemize} \item $A$ tells how large the oscillation is; \item $\omega$ tells how rapidly phase advances; \item $\phi$ tells where the cycle is positioned relative to the chosen time origin. \end{itemize} Changing $A$ stretches the graph vertically. Changing $\phi$ shifts the cycle horizontally in time without changing its amplitude or period. This separation of roles will become increasingly important when several waves are superposed. \section{Common mistakes} \begin{itemize} \item \textbf{Mistake:} treating equal displacement as equal phase. A sinusoid normally passes through most displacement values twice per cycle, once in each direction. \item \textbf{Mistake:} forgetting that phase is cyclic. Angles differing by $2\pi n$ represent the same location within a cycle. \item \textbf{Mistake:} comparing phase constants when the frequencies are different and calling the result a permanent phase lead. If $\omega_1\neq\omega_2$, the relative phase changes with time. \item \textbf{Mistake:} converting phase directly to seconds without using $\omega$. A phase angle becomes a time shift through $\Delta t=\Delta\phi/\omega$. \item \textbf{Mistake:} stating ``positive phase means lead'' without specifying the sinusoidal sign convention. In this series the present result follows from $\cos(\omega t+\phi)$. \end{itemize} \section{Summary} For a sinusoidal oscillator, \begin{equation} \theta(t)=\omega t+\phi \end{equation} identifies position within the repeating phase cycle. One complete cycle is $2\pi$ radians, so equivalent phases satisfy \begin{equation} \theta\equiv\theta+2\pi n. \end{equation} For two oscillators with the same $\omega$, \begin{equation} \boxed{\Delta\phi=\phi_2-\phi_1} \end{equation} is constant, and the corresponding time shift is \begin{equation} \boxed{\Delta t=\frac{\Delta\phi}{\omega} =\frac{\Delta\phi}{2\pi}T.} \end{equation} If the angular frequencies differ, the phase difference instead evolves as \begin{equation} \Delta\theta(t)=(\omega_2-\omega_1)t+\Delta\phi_0. \end{equation} WM04 will move from oscillation in time to variation in space. That step will introduce wavelength and prepare the separate idea of spatial phase.