Wave Mechanics: Sinusoidal OscillationWaveMechanicsSinusoidalOscillation2026-09-11 05:33:052026-09-11 05:59:59Topicwave mechanicssinusoidal oscillationcosineangular frequencyradiansphasephase constantperiodfrequencyamplitudeperiodic motion% this is the default PhysicsLibrary preamble. as your knowledge
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% define commands here\section*{Wave Mechanics: Sinusoidal Oscillation}
WM01 introduced oscillation at one point through the general time-dependent
quantity $u(t)$. It defined equilibrium, amplitude, period, and frequency
without assuming a particular shape for the motion. We now introduce the
most important smooth periodic shape in wave mechanics: the sinusoid.
The goal of this entry is to build, rather than merely state, the equation
\begin{equation}
\boxed{u(t)=A\cos(\omega t+\phi)}.
\end{equation}
Each symbol will be connected to a physical idea already developed in WM01.
This lesson still concerns only variation with time at one point. Wavelength,
wavenumber, spatial phase, and traveling waves are intentionally deferred to
later entries.
\section{Why sinusoidal motion matters}
A periodic function can have many shapes. It can be triangular, square,
pulsed, or irregular while still repeating after a period $T$. Sinusoidal
motion is special because it is smooth, mathematically simple, and appears
naturally in a large class of physical systems.
Examples include approximately small pendulum motions, ideal mass--spring
systems, acoustic pressure variations at a fixed point, alternating voltages,
and individual frequency components of more complicated signals.
At this stage we do not need to prove why a particular physical system becomes
sinusoidal. Our immediate task is to learn the mathematical language used to
describe a sinusoidal time history.
\section{The cosine function as a repeating cycle}
The cosine function repeats whenever its argument increases by $2\pi$:
\begin{equation}
\cos(\theta+2\pi)=\cos\theta.
\end{equation}
The variable $\theta$ is an angle. In wave mechanics and calculus, angles are
normally measured in \emph{radians}.
\subsection{What is a radian?}
For a circle of radius $r$, suppose an arc of length $s$ subtends an angle
$\theta$ at the center. The radian measure of the angle is defined by
\begin{equation}
\boxed{\theta=\frac{s}{r}}.
\end{equation}
\begin{center}
\includegraphics{WM02_fig01_radian_measure.png}
\vspace{0.45em}
\textbf{Figure.}
Radian measure compares arc length with radius. A full circle has arc length
$2\pi r$, so one complete revolution corresponds to $2\pi$ radians.
\end{center}
Because a full circumference has length $2\pi r$,
\begin{equation}
\theta_{\text{cycle}}
=\frac{2\pi r}{r}
=2\pi\ \text{rad}.
\end{equation}
Thus one complete cycle can be represented by an angular advance of
\begin{equation}
\boxed{2\pi\ \text{rad per cycle}.}
\end{equation}
Although the radian is dimensionless in the strict dimensional-analysis
sense, retaining the label ``rad'' is often useful because it reminds us that
the quantity represents angular or phase advance.
\section{From period to angular frequency}
WM01 defined the period $T$ as the time required for one complete cycle. In
that same time, the cosine argument must advance by $2\pi$ radians.
Therefore the angular advance per unit time is
\begin{equation}
\omega=\frac{2\pi\ \text{rad}}{T}.
\end{equation}
The symbol $\omega$ is the Greek letter omega. It is called the
\emph{angular frequency}. We therefore write
\begin{equation}
\boxed{\omega=\frac{2\pi}{T}}.
\end{equation}
Its commonly stated unit is radians per second:
\begin{equation}
[\omega]=\text{rad/s}.
\end{equation}
WM01 also established
\begin{equation}
f=\frac{1}{T}.
\end{equation}
Substituting this relation into the expression for $\omega$ gives
\begin{equation}
\boxed{\omega=2\pi f}.
\end{equation}
Equivalently,
\begin{equation}
\boxed{f=\frac{\omega}{2\pi}}.
\end{equation}
This distinction is fundamental:
\begin{itemize}
\item $f$ counts \emph{cycles per second};
\item $\omega$ counts \emph{radians of phase advance per second}.
\end{itemize}
Since one cycle contains $2\pi$ radians, the factor $2\pi$ is unavoidable.
\section{Building the first sinusoidal time history}
Consider the simplest case in which the oscillator is at its maximum positive
displacement at $t=0$. Let its amplitude be $A$.
A cosine naturally starts at its maximum because
\begin{equation}
\cos 0=1.
\end{equation}
The expression
\begin{equation}
u(t)=A\cos(\omega t)
\end{equation}
therefore satisfies
\begin{equation}
u(0)=A.
\end{equation}
After one period, $t=T$. Using $\omega T=2\pi$,
\begin{equation}
u(T)=A\cos(\omega T)
=A\cos(2\pi)
=A.
\end{equation}
The time history has returned to the same point in its cycle.
\begin{center}
\includegraphics{WM02_fig02_cosine_cycle.png}
\vspace{0.45em}
\textbf{Figure.}
A cosine time history with amplitude $A$ and period $T$. During one period the
cosine argument advances from $0$ to $2\pi$ radians.
\end{center}
At the quarter-period points,
\begin{align}
t=0 &: \qquad u=A,\\
t=T/4 &: \qquad u=0,\\
t=T/2 &: \qquad u=-A,\\
t=3T/4 &: \qquad u=0,\\
t=T &: \qquad u=A.
\end{align}
These five points provide a useful mental sketch of one cosine cycle.
\section{Frequency and angular frequency describe the same repetition rate}
Suppose one oscillator has
\begin{equation}
f=1\,\text{Hz}.
\end{equation}
Then
\begin{equation}
\omega=2\pi f=2\pi\,\text{rad/s}.
\end{equation}
A second oscillator with
\begin{equation}
f=2\,\text{Hz}
\end{equation}
has
\begin{equation}
\omega=4\pi\,\text{rad/s}.
\end{equation}
The second oscillator completes twice as many cycles per second, and its phase
angle advances twice as many radians per second.
\begin{center}
\includegraphics{WM02_fig03_angular_frequency.png}
\vspace{0.45em}
\textbf{Figure.}
Two sinusoidal oscillations with the same amplitude but different repetition
rates. Doubling $f$ doubles $\omega$ and halves the period.
\end{center}
The three quantities contain the same timing information:
\begin{equation}
\boxed{T=\frac{1}{f},\qquad
\omega=2\pi f,\qquad
T=\frac{2\pi}{\omega}.}
\end{equation}
The useful question is not which one is ``correct.'' The useful question is
which description is most convenient for the mathematics being performed.
\section{The phase angle}
The cosine does not fundamentally depend on time itself. It depends on its
\emph{argument}. We give that argument its own name:
\begin{equation}
\boxed{\theta(t)=\omega t+\phi.}
\end{equation}
The quantity $\theta(t)$ is the instantaneous phase angle. It tells us where
the oscillator is within its repeating cosine cycle.
The term $\omega t$ describes the phase accumulated as time passes. The
constant $\phi$ specifies the phase at the chosen time origin $t=0$:
\begin{equation}
\theta(0)=\phi.
\end{equation}
Substituting the phase angle into the cosine gives the general sinusoidal form
for this lesson:
\begin{equation}
\boxed{u(t)=A\cos(\omega t+\phi).}
\end{equation}
The constant $\phi$ is called the \emph{phase constant} or \emph{initial
phase}. It is normally expressed in radians.
\section{What the phase constant changes}
At $t=0$,
\begin{equation}
\boxed{u(0)=A\cos\phi.}
\end{equation}
Thus $\phi$ determines where the time history begins within the repeating
cycle.
For example,
\begin{align}
\phi=0 &: \qquad u(0)=A,\\
\phi=\frac{\pi}{2} &: \qquad u(0)=0,\\
\phi=\pi &: \qquad u(0)=-A.
\end{align}
\begin{center}
\includegraphics{WM02_fig04_phase_constant.png}
\vspace{0.45em}
\textbf{Figure.}
Changing $\phi$ changes the starting point in the sinusoidal cycle while
leaving the amplitude and angular frequency unchanged. WM03 will develop
phase and phase difference in greater detail.
\end{center}
A very important caution is that the displacement alone does not identify the
full point in the cycle. For example, the oscillator can pass through $u=0$
while moving in either direction. WM03 will use phase to distinguish such
situations systematically.
\section{Meaning of the four parameters}
The equation
\begin{equation}
u(t)=A\cos(\omega t+\phi)
\end{equation}
contains four essential quantities.
\begin{itemize}
\item $u(t)$ is the instantaneous value of the oscillating quantity.
\item $A$ is the amplitude, the maximum magnitude of displacement from equilibrium.
\item $\omega$ is the angular frequency, the rate at which phase advances with time.
\item $\phi$ is the phase constant, the phase angle at $t=0$.
\end{itemize}
Time $t$ is the independent variable.
Notice that $A$ controls the vertical scale of the graph, whereas $\omega$
controls the horizontal repetition rate. The phase constant shifts where the
cycle begins relative to the chosen time origin.
\section{Worked example 1: from frequency to a sinusoidal equation}
Suppose an oscillator has amplitude
\begin{equation}
A=4.0\,\text{cm}
\end{equation}
and frequency
\begin{equation}
f=2.0\,\text{Hz}.
\end{equation}
Assume that it begins at maximum positive displacement, so $\phi=0$.
First compute the angular frequency:
\begin{align}
\omega
&=2\pi f\\
&=2\pi(2.0\,\text{Hz})\\
&=4\pi\,\text{rad/s}.
\end{align}
The oscillation is therefore, with $t$ measured in seconds,
\begin{equation}
\boxed{u(t)=4.0\,\text{cm}\,\cos(4\pi t).}
\end{equation}
Its period is
\begin{equation}
T=\frac{1}{f}=0.50\,\text{s}.
\end{equation}
As a check,
\begin{equation}
\omega T=(4\pi\,\text{rad/s})(0.50\,\text{s})=2\pi\,\text{rad},
\end{equation}
which is exactly one full cycle of phase advance.
\section{Worked example 2: a nonzero phase constant}
Suppose
\begin{equation}
A=6.0\,\text{mm},\qquad
T=0.40\,\text{s},\qquad
\phi=\frac{\pi}{2}.
\end{equation}
The frequency is
\begin{equation}
f=\frac{1}{T}=2.5\,\text{Hz}.
\end{equation}
The angular frequency is
\begin{equation}
\omega=2\pi f=5\pi\,\text{rad/s}.
\end{equation}
Hence, with $t$ measured in seconds,
\begin{equation}
\boxed{u(t)=6.0\,\text{mm}\,
\cos\left(5\pi t+\frac{\pi}{2}\right).}
\end{equation}
At $t=0$,
\begin{equation}
u(0)=6.0\,\text{mm}\cos\left(\frac{\pi}{2}\right)=0.
\end{equation}
The oscillator therefore begins at equilibrium rather than at an extreme.
The phase constant encodes that different starting point.
\section{Cosine versus sine}
A sinusoid can be written using either cosine or sine. For example,
\begin{equation}
\cos\theta=\sin\left(\theta+\frac{\pi}{2}\right).
\end{equation}
Thus
\begin{equation}
A\cos(\omega t+\phi)
\end{equation}
and
\begin{equation}
A\sin(\omega t+\phi')
\end{equation}
can describe the same physical oscillation if the phase constants are chosen
appropriately.
Physics does not prefer cosine over sine. Cosine is used as the default in
this series because $\cos 0=1$, which makes the zero-phase case begin at
maximum positive displacement and gives a convenient reference convention.
\section{Common mistakes}
\subsection*{Confusing $f$ with $\omega$}
If
\begin{equation}
f=3\,\text{Hz},
\end{equation}
then
\begin{equation}
\omega=6\pi\,\text{rad/s},
\end{equation}
not $3\,\text{rad/s}$.
\subsection*{Putting hertz directly inside the cosine argument}
The cosine argument is an angle. Writing
\begin{equation}
\cos(ft)
\end{equation}
while $f$ is measured in cycles per second omits the conversion from cycles to
radians. The correct zero-phase form is
\begin{equation}
\cos(2\pi f t)=\cos(\omega t).
\end{equation}
\subsection*{Thinking phase changes amplitude}
Changing $\phi$ moves the starting point within the cycle, but it does not
change the maximum magnitude $A$.
\subsection*{Using degrees inside calculus formulas}
Degrees are useful geometrically, but the standard derivative identities for
sine and cosine take their simplest form when the argument is measured in
radians. Wave mechanics therefore uses radians by default.
\section{A compact derivation chain}
The central relationships of this lesson can be read as a sequence:
\begin{align}
\text{one cycle} &\longleftrightarrow 2\pi\ \text{rad},\\
f&=\frac{1}{T},\\
\omega&=\frac{2\pi}{T}=2\pi f,\\
\theta(t)&=\omega t+\phi,\\
u(t)&=A\cos\theta(t).
\end{align}
Combining the final two lines gives
\begin{equation}
\boxed{u(t)=A\cos(\omega t+\phi).}
\end{equation}
This is the basic sinusoidal time-history equation that will be reused
throughout the Wave Mechanics series.
\section{What comes next}
WM02 has introduced the phase angle and phase constant only far enough to
construct a sinusoidal oscillator. WM03 will focus specifically on phase:
what it means physically, why two oscillators can have the same amplitude and
frequency but different phases, how phase lead and lag are described, and why
adding $2\pi$ does not change the physical point in a cycle.
Spatial dependence still does not appear. The transition from temporal phase
to spatial phase begins later with $u(x)$, wavelength, and wavenumber.
\section{Summary}
A sinusoidal oscillation at one point can be written as
\begin{equation}
\boxed{u(t)=A\cos(\omega t+\phi).}
\end{equation}
The relations connecting period, frequency, and angular frequency are
\begin{equation}
\boxed{f=\frac{1}{T},\qquad
\omega=2\pi f=\frac{2\pi}{T}.}
\end{equation}
The phase angle is
\begin{equation}
\boxed{\theta(t)=\omega t+\phi.}
\end{equation}
Amplitude determines how large the oscillation is, angular frequency
determines how rapidly the phase advances, and the phase constant determines
where in the cycle the motion begins at the chosen time origin.