Beats, Interference, and Coherence

Beats, Interference, and Coherence

Wave phenomena are governed by superposition.

When two or more waves occupy the same region, the resulting field is the sum of the individual fields.

For two scalar waves,

(1)

That simple statement produces several distinct observable effects.

If two waves have the same frequency but different phase, the result is interference.

If two waves have nearby but unequal frequencies, the result is beating.

If the relative phase remains sufficiently stable, interference remains observable. This phase stability is called coherence.

These ideas apply broadly to:

• waves on strings,
• sound waves,
• water waves,
• electromagnetic waves,
• quantum wave amplitudes.

The mathematics is classical and general.

1 The superposition principle

For a linear wave equation, if

(2)

and

(3)

are solutions, then

(4)

is also a solution.

This follows because a linear differential operator

(5)

satisfies

(6)

If

(7)

and

(8)

then

(9)

Superposition is therefore a consequence of linearity.

2 Two sinusoidal waves of the same frequency

Consider

(10)

and

(11)

Their sum is another sinusoid at the same angular frequency:

(12)

The resultant amplitude follows from phasor addition:

(13)

Thus the amplitude depends on relative phase.

3 Equal-amplitude interference

If

(14)

then

y

= A cos(ωt) + A cos(ωt + ϕ).

(15)

Use

(16)

Then

(17)

The resultant amplitude is therefore

(18)

4 Constructive interference

For

(19)

(20)

Thus

(21)

This is complete constructive interference for equal amplitudes.

5 Destructive interference

For

(22)

(23)

Thus

(24)

This is complete destructive interference.

Complete cancellation requires equal amplitudes.

If the amplitudes differ, the minimum resultant amplitude is

(25)

6 Intensity interference

Measured wave intensity is usually proportional to the time average of the square of a field amplitude.

For two coherent waves, the total intensity is

(26)

This interference term is the intensity counterpart of amplitude superposition.

For equal intensities

(27)

(28)

Using

(29)

(30)

At constructive interference,

(31)

At destructive interference,

(32)

7 Why the maximum intensity can be four times one-source intensity

If two equal coherent waves are in phase, the field amplitude doubles.

Since intensity is proportional to amplitude squared,

(33)

Therefore the local maximum is four times the intensity due to either source separately.

This does not violate energy conservation.

Interference redistributes energy spatially.

Regions of enhancement are accompanied by regions of reduced intensity.

8 Fringe visibility

When the two source intensities are unequal,

(34)

and

(35)

Define the visibility

(36)

Then for perfectly coherent waves,

(37)

The maximum visibility

(38)

occurs when

(39)

9 Phase difference from path difference

Suppose two coherent waves have the same frequency and wavelength

(40)

but travel different path lengths.

Let

(41)

Then the propagation phase difference is

(42)

If the sources also begin with an intrinsic phase difference

(43)

then

(44)

10 Constructive and destructive path conditions

For sources in phase at emission,

(45)

Constructive interference occurs when

(46)

Destructive interference occurs when

(47)

11 Example 1: two-source sound interference

Two coherent acoustic sources emit the same frequency

(48)

in air with

(49)

The wavelength is

λ	=	(50)
	=	(51)
	= 0.686 m.	(52)

If the path difference is

(53)

then

(54)

For in-phase sources, the observation point therefore lies at a destructive-interference condition.

12 Interference intensity as a function of phase

For equal coherent source intensities,

(55)

(56)

This pattern is periodic in phase with period

(57)

13 Beats from nearby frequencies

Now consider two equal-amplitude waves with slightly different angular frequencies:

(58)

and

(59)

Their sum is

(60)

Using the sum identity,

(61)

Define

(62)

and

(63)

Then

(64)

The rapidly oscillating factor acts like a carrier.

The slowly varying factor controls the envelope.

14 Beat frequency

The signed envelope factor is

(65)

But a measured amplitude magnitude repeats whenever the cosine changes from

(66)

to

(67)

and back in magnitude.

Therefore the beat angular frequency is

(68)

In ordinary frequency,

(69)

This factor-of-two distinction between the signed envelope cosine and the observed beat rate is a common source of confusion.

15 Example 2: acoustic beats

Two tuning forks produce

(70)

and

(71)

Then

(72)

The rapid oscillation occurs near

(73)

A listener therefore hears a tone near

(74)

whose loudness rises and falls four times per second.

16 Beat period

The time between adjacent beat maxima is

(75)

For the previous example,

(76)

17 Unequal-amplitude beats

Let

(77)

and

(78)

The instantaneous squared resultant amplitude contains the phase-dependent combination

(79)

Therefore the envelope ranges between

(80)

and

(81)

Unequal amplitudes therefore produce incomplete beat minima.

18 Beats in the frequency domain

M06-10 showed that a signal can be represented by its frequency spectrum.

A simple beat signal contains two narrow spectral components at

(82)

and

(83)

The slow beat modulation does not require a third independent physical frequency source.

It arises from interference between the two nearby components.

This is an important connection between time-domain envelopes and frequency-domain structure.

19 Standing waves as spatial interference

A Standing Wave is another form of interference.

Consider equal waves traveling in opposite directions:

(84)

and

(85)

Their sum is

(86)

Thus standing-wave nodes and antinodes are produced by spatially varying interference between counter-propagating waves.

Beats and standing waves are mathematically related:

• beats arise from a small frequency difference,
• standing waves arise from a propagation-direction difference at the same frequency.

20 What coherence means

Interference requires more than simply overlapping waves.

The relative phase must remain sufficiently predictable during the measurement.

Two waves are coherent if their phase relation is stable enough that the interference term does not average away.

Perfectly monochromatic waves with a fixed phase difference are ideally coherent.

Real sources have finite bandwidth and finite phase stability.

Coherence therefore has a finite time and length scale.

21 Why random phase destroys interference

For two waves,

(87)

Suppose

(88)

changes randomly and rapidly during the measurement.

Then the time average approaches

(89)

Thus

(90)

This is why independent incoherent sources usually contribute intensities rather than stable interference fringes.

22 Temporal coherence

Temporal coherence describes phase correlation of a wave with itself at different times.

A source with a very narrow frequency spectrum maintains a predictable phase over a relatively long time.

A source with broad bandwidth loses phase predictability more quickly.

Define a characteristic coherence time

(91)

A rough inverse-bandwidth scaling is

(92)

The numerical coefficient depends on how bandwidth and coherence time are defined and on the spectral shape.

23 Coherence length

If the wave travels at speed

(93)

then the coherence length is of order

(94)

Using the inverse-bandwidth estimate,

(95)

Interference between paths with a difference much greater than the coherence length is strongly reduced.

24 Example 3: coherence estimate

Suppose an acoustic signal has effective bandwidth

(96)

A rough coherence-time estimate is

(97)

Using

(98)

the corresponding coherence-length scale is

(99)

This is an order-of-magnitude estimate rather than an exact universal equality.

25 Spatial coherence

Temporal coherence concerns correlations separated in time.

Spatial coherence concerns phase correlations across different transverse positions in a wave field.

An extended source generally has lower spatial coherence than a small source because different source regions contribute with different phases.

Spatial coherence determines whether two separated sampling points can produce high-contrast interference.

A complete treatment uses correlation functions, but the physical idea is simple:

Coherence measures how reliably phase information is shared between two measurements of a wave field.

26 Complex degree of coherence

For a more general description, introduce the complex degree of coherence

(100)

Its magnitude satisfies

(101)

For two fields, the interference intensity can be written

(102)

If

(103)

the fields are fully coherent.

If

(104)

the stable interference term vanishes.

27 Visibility and partial coherence

For equal source intensities and suitable phase scanning, the fringe visibility becomes

(105)

For unequal intensities,

(106)

Thus poor visibility can arise from:

• unequal source intensities,
• imperfect coherence,
• or both.

28 Coherence time, bandwidth, and wave packets

M06-10 showed that a narrow frequency spectrum produces a long spatial packet, while a broad spectrum produces a short packet.

Coherence follows the same Fourier logic.

A narrow spectral width

(107)

corresponds to a long characteristic time scale.

A broad spectral width corresponds to a short characteristic time scale.

Thus temporal coherence is another manifestation of the reciprocal relationship between time localization and frequency bandwidth.

29 Interference versus beats

It is useful to separate two related cases.

For equal frequencies,

(108)

the phase difference is constant in time.

The interference pattern is stationary.

For unequal frequencies,

(109)

the relative phase evolves:

(110)

The interference therefore cycles between reinforcement and cancellation.

That time-dependent interference is what produces beats.

30 Beats as moving phase difference

For equal amplitudes,

(111)

Using the identity

(112)

(113)

Thus a beat is simply interference whose relative phase continuously drifts.

31 Example 4: unequal coherent intensities

Let

(114)

For full coherence,

(115)

The minimum is

(116)

Therefore

(117)

Perfect coherence does not guarantee unit visibility when the source intensities differ.

32 Example 5: partial coherence

Let two equal-intensity fields have

(118)

Then the maximum possible fringe visibility is

(119)

The intensity can be written

(120)

Therefore

(121)

and

(122)

33 Connection to acoustics

For coherent sound sources, pressure amplitudes superpose before intensity is calculated.

This allows:

• interference maxima and minima,
• active noise cancellation,
• beats between nearby tones,
• standing-wave patterns in rooms and ducts.

For independent sound sources whose relative phases fluctuate rapidly, long-time average intensities approximately add instead.

This distinction is why acoustic phase information matters.

34 Connection to resonance

Beats can appear when two resonance frequencies lie close together.

Coupled oscillators from M06-04 provide a classic example.

If both nearby normal modes are excited, the observable motion may show periodic energy transfer and an amplitude envelope at the difference frequency.

Thus beating provides a time-domain signature of closely spaced spectral components.

35 Connection to Fourier analysis

Interference and coherence are naturally described in Fourier language.

A perfectly monochromatic component has a sharply defined frequency and indefinitely long phase memory in the ideal model.

A finite-bandwidth signal combines many nearby frequencies.

As those components acquire different phases, the field loses long-time phase predictability.

Therefore Fourier bandwidth and temporal coherence are reciprocal aspects of the same wave structure.

36 Common mistakes

1. Adding intensities directly when coherent amplitudes should be added first.
2. Forgetting that destructive interference requires phase opposition and equal amplitudes for complete cancellation.
3. Using I₁ + I₂ for coherent fields with a fixed phase relation.
4. Confusing phase difference with path difference.
5. Forgetting the factor 2π∕λ in converting path difference to phase difference.
6. Writing the beat frequency as (f₁ − f₂)∕2 instead of |f₁ − f₂|.
7. Confusing the average carrier frequency with the beat frequency.
8. Forgetting that unequal-amplitude beats do not fall to zero.
9. Assuming two overlapping waves are automatically coherent.
10. Treating coherence time as exactly 1∕Δf without specifying a bandwidth convention.
11. Confusing temporal coherence with spatial coherence.
12. Assuming poor fringe visibility always means poor coherence.
13. Forgetting that unequal intensities also reduce visibility.
14. Treating standing waves and beats as unrelated phenomena.
15. Forgetting that beats are interference with a time-varying phase difference.

37 Practice exercises

1. Derive the resultant amplitude of two same-frequency waves with amplitudes A₁ and A₂ and phase difference ϕ.
2. For equal amplitudes, derive the constructive and destructive interference conditions.
3. Starting from amplitude addition, derive the coherent intensity formula
   
4. Derive the visibility formula for two perfectly coherent waves of unequal intensities.
5. Convert a path difference Δr into a phase difference.
6. Determine the constructive and destructive path-difference conditions for two in-phase sources.
7. Derive the beat identity for two equal-amplitude nearby frequencies.
8. Explain why the observed beat frequency is |f₁ − f₂|.
9. Determine the beat period for two specified frequencies.
10. Derive the maximum and minimum envelopes for unequal-amplitude beats.
11. Show how a standing wave results from interference of equal counter-propagating waves.
12. Explain why rapidly fluctuating random phase causes the interference term to average toward zero.
13. Estimate coherence time and coherence length from a specified bandwidth.
14. For unequal intensities and specified |γ₁₂|, calculate the maximum fringe visibility.
15. Explain the Fourier connection between narrow bandwidth and long coherence time.

38 Summary

Linear waves obey superposition:

(123)

For two same-frequency waves,

(124)

For coherent intensities,

(125)

Path difference produces phase difference:

(126)

For two nearby frequencies,

(127)

The beat period is

(128)

For unequal beat amplitudes,

(129)

A rough temporal-coherence scaling is

(130)

with coherence length

(131)

Partial coherence can be represented by

(132)

Interference, beats, standing waves, and coherence are therefore not isolated topics. They are different consequences of superposition, phase evolution, and spectral content.

References

References