Fourier Analysis, Wave Packets, and Dispersion

Fourier Analysis, Wave Packets, and Dispersion

A single sinusoidal wave is mathematically simple, but real disturbances are usually localized in space and time.

A pluck, pulse, acoustic click, or lattice disturbance cannot be represented by one wave number alone. It requires a superposition of many sinusoidal components.

Fourier analysis provides the natural language for that superposition.

For a continuous one-dimensional field, one useful transform convention is

(1)

with inverse

(2)

A traveling wave packet then has the form

(3)

The function

(4)

is the wave-number spectrum.

The dispersion relation

(5)

controls how each Fourier component evolves.

Two velocities then become important:

(6)

for the phase velocity, and

(7)

for the group velocity.

When

(8)

is nonlinear, different spectral components accumulate phase at different rates and a packet can broaden or distort. That is dispersion.

1 Why Fourier analysis appears in linear wave physics

The wave equations considered in M06-06 through M06-09 are linear. If

(9)

and

(10)

are solutions, then

(11)

is also a solution for constants

(12)

This superposition principle makes sinusoidal modes especially useful.

In translationally invariant systems, complex exponentials

(13)

are spatial eigenfunctions. The dynamics of a complicated waveform can therefore be reduced to the independent evolution of its Fourier components.

2 Fourier series on a finite interval

Suppose

(14)

is periodic with period

(15)

Define

(16)

The real Fourier series is

f(x)

= + ∑ n=1∞ .

(17)

The coefficients are

(18)

and

(19)

The constant coefficient is

(20)

These formulas are direct consequences of sine and cosine orthogonality.

3 Complex Fourier series

The same periodic function can be written more compactly as

(21)

The coefficients are

(22)

For a real function,

(23)

Thus negative wave numbers do not provide independent information for a real field, but they remain extremely useful in the complex representation.

4 From a Fourier series to a Fourier transform

As the interval becomes very large, the spacing between allowed wave numbers becomes small:

(24)

The discrete sum over

(25)

approaches an integral.

This leads to the Fourier-transform pair

(26)

and

(27)

A finite-domain normal-mode expansion and an infinite-domain Fourier transform are therefore closely related versions of the same spectral idea.

5 Wave-number spectrum

The function

(28)

contains the amplitude and phase of each wave-number component.

A broad waveform in position space often has a narrow spectrum in

(29)

while a sharply localized waveform requires a broad range of wave numbers.

This is a general Fourier-localization principle.

6 Gaussian example

Consider

(30)

Its Fourier transform is proportional to

(31)

Therefore the characteristic spectral width is

(32)

for this amplitude-width convention.

A narrow Gaussian in space has a broad Gaussian spectrum, and vice versa.

7 Standard-deviation localization inequality

If a square-integrable waveform is normalized and standard deviations are defined from

(33)

and

(34)

then Fourier analysis gives

(35)

A Gaussian attains equality.

This is a purely mathematical Fourier property. It appears in classical wave theory as well as in quantum mechanics.

8 Parseval relation

The Fourier transform also preserves a quadratic norm. With the transform convention used here,

(36)

This is Parseval’s relation.

It connects total quadratic content in position space with total quadratic content in spectral space. In wave mechanics, closely related relations underlie modal energy decompositions.

9 A wave packet

A localized wave packet centered near wave number

(37)

can be written as

(38)

If

(39)

is sharply peaked near

(40)

then the field resembles a rapidly oscillating carrier multiplied by a slowly varying envelope.

The carrier and envelope need not move at the same velocity.

10 Two nearby waves

Take two waves with nearby wave numbers and frequencies:

(41)

and

(42)

Their sum is

y1 + y2	= 2A cos
	× cos .	(43)

Define

(44)

The slowly varying envelope has constant phase when

(45)

Therefore its velocity is

(46)

In the narrowband limit,

(47)

11 Phase velocity

For one Fourier component,

(48)

Differentiating,

(49)

Therefore

(50)

This is the velocity of a constant-phase point such as a crest of an ideal sinusoidal component.

12 Group velocity

The narrowband envelope moves at

(51)

Graphically:

• v_p is related to the slope of the line from the origin to the point (k,ω),
• v_g is the slope of the tangent to the dispersion curve at that point.

13 Carrier-envelope form

Let

(52)

where

(53)

is small for a narrow spectrum.

Expand the frequency:

ω(k)

= ω0 + ω0′q + ω0′′q2 + ,

(54)

where

(55)

(56)

and

(57)

If the quadratic and higher terms are neglected, the packet becomes

(58)

where

(59)

Thus the carrier oscillates at

(60)

while the envelope translates at the group velocity.

14 Nondispersive medium

If

(61)

for positive

(62)

then

(63)

Also,

(64)

All Fourier components advance with compatible phase evolution. An ideal packet can translate without dispersive broadening.

The ideal stretched string in M06-08 is nondispersive within its linear model.

15 Dispersive medium

A medium is dispersive when

(65)

is nonlinear.

Then, in general,

(66)

and

(67)

Different Fourier components acquire different relative phases as time evolves. This can change the packet shape.

16 Second-order dispersion and packet broadening

For a narrow spectrum, retain the quadratic term:

ω(k)

≈ ω0 + vgq + ω0′′q2.

(68)

The linear term translates the envelope.

The quadratic term produces spreading.

For a Gaussian spectral amplitude

(69)

the amplitude envelope remains Gaussian under this quadratic approximation.

If its initial amplitude width is

(70)

then

(71)

The sign of

(72)

affects the spectral phase chirp, while the width depends on its magnitude in this simple Gaussian result.

17 Example 1: Gaussian spreading

Suppose

(73)

Then

(74)

Let

(75)

After

(76)

σx(t)	= 0.500	(77)
	≈ 0.943 mm.	(78)

The packet has nearly doubled its characteristic amplitude width.

18 Example 2: monatomic oscillator chain

M06-06 found

(79)

for

(80)

Take

(81)

At

(82)

(83)

The angular frequency is

ω	= 40 sin	(84)
	≈ 28.28 mrad∕s.	(85)

The phase velocity is

vp	=	(86)
	≈ 1.80 mm∕s.	(87)

The group velocity is

vg	= a cos	(88)
	= 2.00 cos	(89)
	≈ 1.41 mm∕s.	(90)

Thus

(91)

19 Spectral evolution is especially simple

Suppose the initial spectrum is

(92)

For a linear time-invariant wave system with dispersion relation

(93)

each component simply acquires a phase factor:

(94)

The spectral magnitude is unchanged:

(95)

Dispersion changes the relative spectral phases. Those phase changes alter the waveform after transforming back to position space.

20 Initial-value propagation by Fourier transform

A common wave-analysis workflow is:

1. Fourier transform the initial field to obtain its spectral amplitudes.
2. Multiply each spectral component by its propagation phase factor.
3. Inverse transform to reconstruct the field at the desired time.

Schematically,

(96)

This is the continuous-wave analogue of propagating independent normal coordinates in a finite-dimensional system.

21 Stationary-phase interpretation of group velocity

At large distance and time, rapidly varying phases tend to cancel under integration. The strongest contribution often comes from wave numbers where the total phase varies slowly.

For phase

(97)

stationary phase requires

(98)

Therefore

(99)

Hence the dominant wave number contributing along the ray

(100)

satisfies

(101)

This gives a deeper asymptotic interpretation of group velocity.

22 Group velocity and energy transport

In many conservative weakly dispersive systems, wave-packet energy is transported approximately at the group velocity.

This statement is powerful but should not be used mechanically in every possible medium. Absorption, strong inhomogeneity, active media, and unusual dispersion can require a more careful energy-flux analysis.

For the ideal conservative systems developed in this mechanics sequence, group velocity is the natural velocity associated with a narrowband packet envelope and usually with its transported energy.

23 Negative group velocity and sign conventions

The derivative

(102)

can be negative on some branches or for some sign conventions.

That does not mean the mathematical framework has failed. It means the local slope of the dispersion curve is negative.

The physical interpretation must be made using the full wave field, energy flux, and the chosen representation of positive and negative wave numbers.

24 Finite bandwidth matters

The approximation

(103)

works best when the spectrum is sufficiently narrow that the dispersion curve is nearly linear across the occupied bandwidth.

If the spectrum is broad, one group velocity cannot describe the entire packet. Higher derivatives such as

(104)

become important.

25 Connection to normal modes

On a finite string, the wave numbers are discrete and the field is expanded in normal modes:

(105)

On an infinite or very long translationally invariant medium, the spectrum becomes continuous:

(106)

The mathematical transition is

(107)

The physical idea is the same: decompose the motion into independent linear modes.

26 Common mistakes

1. Confusing Fourier-series coefficients with Fourier-transform amplitudes.
2. Mixing transform conventions without carrying the correct factors of 2π.
3. Treating a localized pulse as a single wave number.
4. Assuming a narrow spatial pulse has a narrow wave-number spectrum.
5. Confusing the phase velocity ω∕k with the group velocity dω∕dk.
6. Assuming v_p = v_g in every medium.
7. Calling every medium dispersive merely because it supports waves.
8. Forgetting that dispersion is controlled by the nonlinearity of ω(k).
9. Assuming a wave packet cannot change shape if its spectral magnitudes are constant.
10. Forgetting that relative spectral phase is enough to reshape the packet.
11. Using one group velocity for a packet whose bandwidth is too broad for linearization of ω(k).
12. Forgetting that ω′′ controls leading-order packet spreading near k₀.
13. Interpreting the Fourier localization inequality as exclusively quantum mechanical.
14. Forgetting conjugate symmetry c_−n = c_n^∗ for a real field.
15. Assuming a negative group velocity can be interpreted safely without checking the full energy-flow problem.

27 Practice exercises

1. Derive the real Fourier coefficients from sine and cosine orthogonality.
2. Derive the complex Fourier coefficient formula.
3. Show how Δk = 2π∕L approaches a continuous spectrum as L →∞.
4. Compute the Fourier transform of a Gaussian and identify its spectral width.
5. Verify Parseval’s relation for a Gaussian using the transform convention in this article.
6. Derive the two-wave envelope formula and the finite-difference envelope velocity Δω∕Δk.
7. Take the narrowband limit and obtain v_g = dω∕dk.
8. Derive the carrier-envelope form by expanding ω(k) about k₀ to first order.
9. Show that a linear dispersion relation gives v_p = v_g and no second-order dispersive spreading.
10. Derive the Gaussian packet-width formula using a quadratic expansion of ω(k).
11. Reproduce the Gaussian spreading example.
12. Reproduce the monatomic-chain phase and group velocities in Example 2.
13. Show that linear propagation preserves |A(k)| while changing spectral phase.
14. Derive the stationary-phase condition x∕t = dω∕dk.
15. Explain how Fourier decomposition on an infinite domain generalizes the normal-mode decomposition of a finite string.

28 Summary

A periodic waveform can be expanded as a Fourier series,

(108)

while an infinite-domain waveform can be represented by the Fourier transform pair

(109)

and

(110)

A wave packet is a superposition

(111)

The phase velocity is

(112)

and the group velocity is

(113)

A linear dispersion relation is nondispersive. A nonlinear dispersion relation generally causes different spectral components to accumulate different relative phases.

For a narrowband Gaussian packet, the second derivative

(114)

controls the leading-order broadening.

Fourier analysis therefore unifies normal modes, traveling waves, localized packets, and dispersion within one spectral framework.

References

References