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

(1)

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.

1 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.

2 The cosine function as a repeating cycle

The cosine function repeats whenever its argument increases by 2π:

(2)

The variable 𝜃 is an angle. In wave mechanics and calculus, angles are normally measured in radians.

2.1 What is a radian?

For a circle of radius r, suppose an arc of length s subtends an angle 𝜃 at the center. The radian measure of the angle is defined by

(3)

Because a full circumference has length 2πr,

(4)

Thus one complete cycle can be represented by an angular advance of

(5)

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.

3 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 2π radians.

Therefore the angular advance per unit time is

(6)

The symbol ω is the Greek letter omega. It is called the angular frequency. We therefore write

(7)

Its commonly stated unit is radians per second:

(8)

WM01 also established

(9)

Substituting this relation into the expression for ω gives

(10)

Equivalently,

(11)

This distinction is fundamental:

• f counts cycles per second;
• ω counts radians of phase advance per second.

Since one cycle contains 2π radians, the factor 2π is unavoidable.

4 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

(12)

The expression

(13)

therefore satisfies

(14)

After one period, t = T. Using ωT = 2π,

(15)

The time history has returned to the same point in its cycle.

At the quarter-period points,

t = 0	: u = A,	(16)
t = T∕4	: u = 0,	(17)
t = T∕2	: u = −A,	(18)
t = 3T∕4	: u = 0,	(19)
t = T	: u = A.	(20)

These five points provide a useful mental sketch of one cosine cycle.

5 Frequency and angular frequency describe the same repetition rate

Suppose one oscillator has

(21)

Then

(22)

A second oscillator with

(23)

has

(24)

The second oscillator completes twice as many cycles per second, and its phase angle advances twice as many radians per second.

The three quantities contain the same timing information:

(25)

The useful question is not which one is “correct.” The useful question is which description is most convenient for the mathematics being performed.

6 The phase angle

The cosine does not fundamentally depend on time itself. It depends on its argument. We give that argument its own name:

(26)

The quantity 𝜃(t) is the instantaneous phase angle. It tells us where the oscillator is within its repeating cosine cycle.

The term ωt describes the phase accumulated as time passes. The constant ϕ specifies the phase at the chosen time origin t = 0:

(27)

Substituting the phase angle into the cosine gives the general sinusoidal form for this lesson:

(28)

The constant ϕ is called the phase constant or initial phase. It is normally expressed in radians.

7 What the phase constant changes

At t = 0,

(29)

Thus ϕ determines where the time history begins within the repeating cycle.

For example,

ϕ = 0	: u(0) = A,	(30)
ϕ =	: u(0) = 0,	(31)
ϕ = π	: u(0) = −A.	(32)

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.

8 Meaning of the four parameters

The equation

(33)

contains four essential quantities.

• u(t) is the instantaneous value of the oscillating quantity.
• A is the amplitude, the maximum magnitude of displacement from equilibrium.
• ω is the angular frequency, the rate at which phase advances with time.
• ϕ is the phase constant, the phase angle at t = 0.

Time t is the independent variable.

Notice that A controls the vertical scale of the graph, whereas ω controls the horizontal repetition rate. The phase constant shifts where the cycle begins relative to the chosen time origin.

9 Worked example 1: from frequency to a sinusoidal equation

Suppose an oscillator has amplitude

(34)

and frequency

(35)

Assume that it begins at maximum positive displacement, so ϕ = 0.

First compute the angular frequency:

ω	= 2πf	(36)
	= 2π(2.0 Hz)	(37)
	= 4π rad/s.	(38)

The oscillation is therefore, with t measured in seconds,

(39)

Its period is

(40)

As a check,

(41)

which is exactly one full cycle of phase advance.

10 Worked example 2: a nonzero phase constant

Suppose

(42)

The frequency is

(43)

The angular frequency is

(44)

Hence, with t measured in seconds,

(45)

At t = 0,

(46)

The oscillator therefore begins at equilibrium rather than at an extreme. The phase constant encodes that different starting point.

11 Cosine versus sine

A sinusoid can be written using either cosine or sine. For example,

(47)

Thus

(48)

and

(49)

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.

12 Common mistakes

Confusing f with ω

If

(50)

then

(51)

not 3 rad/s.

Putting hertz directly inside the cosine argument

The cosine argument is an angle. Writing

(52)

while f is measured in cycles per second omits the conversion from cycles to radians. The correct zero-phase form is

(53)

Thinking phase changes amplitude

Changing ϕ moves the starting point within the cycle, but it does not change the maximum magnitude A.

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.

13 A compact derivation chain

The central relationships of this lesson can be read as a sequence:

one cycle	↔2π rad,	(54)
f	= ,	(55)
ω	= = 2πf,	(56)
𝜃(t)	= ωt + ϕ,	(57)
u(t)	= A cos 𝜃(t).	(58)

Combining the final two lines gives

(59)

This is the basic sinusoidal time-history equation that will be reused throughout the Wave Mechanics series.

14 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 2π 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.

15 Summary

A sinusoidal oscillation at one point can be written as

(60)

The relations connecting period, frequency, and angular frequency are

(61)

The phase angle is

(62)

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.