Wave Mechanics: Oscillation at One Point

Before studying a wave that varies in both space and time, it is useful to understand the simpler idea of oscillation at a single point. This article introduces a one-dimensional time-dependent displacement u(t) and develops the ideas of equilibrium, displacement, amplitude, cycle, period, and frequency. These quantities form the temporal foundation for later wave mechanics.

Spatial periodicity, wavelength, wavenumber, angular frequency, phase, and traveling-wave equations are intentionally deferred to later lessons.

1 The simplest possible wave-mechanics starting point

Wave mechanics will eventually study fields such as

(1)

which can vary from place to place and from one instant to another. WM01 temporarily removes the spatial coordinate. We study only

(2)

This is not yet a traveling wave. It is one quantity changing with time at one location or for one mechanical degree of freedom.

A mass attached to a spring is a useful mental model. The mass can move to either side of an equilibrium position. Its displacement from equilibrium is described by the single number u(t).

The important abstraction is not the spring itself. Many systems can be described by a single time-dependent variable: a pendulum angle, the vertical displacement of a floating object, the voltage across an oscillating circuit element, or the pressure measured by a microphone at one fixed location. The physical meaning changes, but the idea of a quantity varying with time remains the same.

2 Equilibrium and displacement

An oscillator needs a reference position. We call this the equilibrium position. In this series the displacement coordinate is normally chosen so that

(3)

at equilibrium.

The sign of u identifies which side of equilibrium the system occupies. For example,

u(t)	> 0		means displacement in the chosen positive direction,	(4)
u(t)	= 0		means the oscillator is at equilibrium,	(5)
u(t)	< 0		means displacement in the opposite direction.	(6)

The choice of positive direction is arbitrary, but it must be used consistently. Reversing the axis changes the sign of u but does not change the physical motion.

2.1 Displacement is not distance traveled

Suppose the mass begins at u = 0, moves to u = +2 cm, then returns to u = 0. Its final displacement is zero, but it has traveled a total distance of

(7)

Displacement describes location relative to equilibrium. Distance traveled describes the length of the path taken during the motion. These are different quantities.

3 Amplitude

For an oscillation centered on equilibrium, the amplitude A is the largest magnitude of the displacement:

(8)

Amplitude is therefore nonnegative:

(9)

If the motion reaches +3 mm on one side and −3 mm on the other, then

(10)

not 6 mm. The full distance from one extreme to the other is 2A.

Amplitude answers the question

How large is the oscillation relative to equilibrium?

It does not tell us how quickly the oscillator moves or how many times per second the motion repeats.

4 A time history

A graph of u versus t is called a time history. The horizontal axis represents time. The vertical axis represents the value of the oscillating quantity.

A time-history graph should not be confused with a picture of the physical path in space. In the figure above, the curve does not mean that the oscillator travels along a wavy road. The curve records the value of u(t) as time passes.

This distinction becomes even more important later, when a wave is described by a function of both position and time.

5 Cycles and periodic motion

An oscillation is periodic when its motion repeats after a fixed amount of time.

Mathematically, a function u(t) is periodic if there exists a positive number T such that

(11)

for every time t for which the motion is defined.

A complete repetition of the motion is called a cycle. The time required for one cycle is called the period.

For a basic periodic oscillator, we use the symbol

(12)

The SI unit of period is the second:

(13)

5.1 The smallest positive repeat time

If a motion repeats after T, it also repeats after 2T, 3T, and so on. When we speak of the period, we normally mean the smallest positive time for which the complete pattern repeats.

For example, if

(14)

for the complete repeating motion, then the period is

(15)

The motion also repeats after 0.50 s and 0.75 s, but these are multiples of the fundamental period rather than new fundamental periods.

6 Returning to the same displacement is not necessarily one cycle

During one oscillation the system can pass through the same displacement more than once.

At the two marked times in the figure above,

(16)

but one crossing occurs while u is decreasing and the other while u is increasing.

For mechanical motion, a complete state includes more than position alone. velocity also matters. When the full periodic motion repeats after one period, both the displacement and its direction/rate of change repeat. This observation will later help motivate the concept of phase.

7 Frequency

The period tells us how long one cycle takes. Frequency tells us how many cycles occur per unit time.

If one cycle takes T seconds, then the number of cycles completed in one second is

(17)

The SI unit of frequency is the hertz:

(18)

Thus period and frequency contain the same timing information in reciprocal forms:

(19)

7.1 Example: period to frequency

Suppose an oscillator completes one cycle every

(20)

Then

(21)

So the oscillator completes four cycles each second.

7.2 Example: frequency to period

Suppose an oscillator has frequency

(22)

Then

(23)

Each cycle lasts 0.05 s.

8 Reading oscillation information from a graph

Given a time-history graph, use the following procedure.

1. Identify the equilibrium value, normally u = 0.
2. Measure the largest magnitude of the displacement to obtain the amplitude A.
3. Choose a recognizable point in the cycle, such as a maximum.
4. Find the next occurrence of the same point with the same direction of motion.
5. Measure the time separation; this is the period T.
6. Compute the frequency from f = 1∕T.

The phrase “same point with the same direction of motion” prevents the common mistake illustrated earlier.

9 Units and dimensional checks

The basic quantities introduced in WM01 are summarized below.

The relation

(24)

is dimensionally consistent because the reciprocal of seconds is inverse seconds. This kind of unit check will become increasingly useful as more wave quantities are introduced.

10 What WM01 is deliberately not doing yet

Several familiar wave and oscillation quantities are intentionally absent from this lesson.

• No sinusoidal equation such as A cos(ωt + ϕ) is required yet.
• No angular frequency ω is required yet.
• No phase ϕ is required yet.
• No spatial coordinate x is being used.
• No wavelength λ or wavenumber k exists in this one-point description.
• No wave speed is being discussed because nothing is propagating through space yet.

This separation is intentional. Period and frequency are temporal concepts. Wavelength and wavenumber will later be introduced as spatial concepts. Only after both sides are understood separately will they be combined into a traveling wave.

11 Common misconceptions

11.1 Amplitude is not peak-to-peak motion

If the oscillator reaches +A and −A, its peak-to-peak range is

(25)

The amplitude remains A.

11.2 Frequency is not speed

A high frequency means many cycles occur per second. It does not by itself specify how quickly a wave travels through space. Propagation speed is a different concept introduced later.

11.3 Period is not wavelength

Period T measures a time interval. Wavelength λ measures a spatial interval. They have different physical dimensions and should not be interchanged.

11.4 Crossing equilibrium is not the same as completing a cycle

An oscillator commonly crosses equilibrium twice during one complete cycle. Counting equilibrium crossings without accounting for direction can therefore produce a factor-of-two error.

12 A preview of the next step

WM01 has described periodic motion without specifying a particular mathematical shape. In WM02 we will study the most important smooth periodic motion in physics: the sinusoid.

The generic time-dependent variable

(26)

will become a specific function involving amplitude, a temporal rate of cycling, and an initial position within the cycle. That will introduce angular frequency and phase while preserving the period and frequency concepts developed here.

Summary

The essential results of WM01 are:

• An oscillation at one point is described by a time-dependent quantity u(t).
• Equilibrium is the reference position, normally chosen as u = 0.
• Displacement is signed; distance traveled is not the same thing as displacement.
• The amplitude A measures the maximum magnitude of the oscillation about equilibrium.
• A periodic motion repeats after a period T:

• Frequency is the number of cycles per second and is the reciprocal of period:

• Returning to the same displacement does not necessarily mean a complete cycle has occurred; direction of motion matters.
• No spatial wave quantities are needed yet. WM02 next introduces sinusoidal oscillation, angular frequency, and phase.

Further reading

For complementary treatments of introductory oscillations, see A. P. French, Vibrations and Waves; Frank S. Crawford, Waves; and standard introductory university physics texts covering oscillatory motion. The later articles in this series will develop the mathematical structure needed for traveling waves, energy transport, boundary phenomena, Fourier methods, Helmholtz problems, and quantum wave mechanics.