Electromagnetic Waves and Wavelength: Frequency, Propagation, and Spectral Description

(Topic)

Electromagnetic Waves and Wavelength: Frequency, Propagation, and Spectral Description

Electromagnetic radiation can be described classically as propagating electric and magnetic fields.

The most important quantities for describing a simple electromagnetic wave include:

amplitude;
wavelength;
frequency;
period;
phase;
propagation speed;
direction of propagation;
polarization.

For astronomy, optics, radio-frequency engineering, and photometry, wavelength and frequency are especially important because they provide the coordinate used to describe an electromagnetic spectrum.

A central relation is

|--------| |v = fλ, | ---------

(1)

where v is the phase speed of the wave in the medium, f is ordinary frequency, and λ is wavelength in that medium.

In vacuum,

|--------| -c-=-fλ,-|

(2)

where

c = 299 792 458 m s−1

(3)

exactly in SI.

This article develops the physical meaning of these quantities and connects the general language of wave mechanics to the electromagnetic spectrum, blackbody radiation, and the spectral quantities used later in astrophysical photometry.

1 Where this article fits in PhysicsLibrary

Several existing PhysicsLibrary articles approach light from complementary directions.

The article Electromagnetic Radiation gives a broad overview of light as an electromagnetic wave and as photons.

The article Electromagnetic Spectrum organizes electromagnetic radiation into astronomical bands such as radio, infrared, visible, ultraviolet, X-ray, and gamma-ray radiation.

The electromagnetic-wave sequence beginning with EM01 develops the field language needed to represent quantities such as

E (r,t).

(4)

The Wave Mechanics sequence develops wavelength, frequency, phase, Wavenumber, and traveling waves in a general setting.

The present article serves as a bridge among those topics.

Its central progression is

|---------------------------------------------------------------------------------------------------------------------------| |oscillation −→ wavelength and frequency − → electromagnetic wave −→ electromagnetic spectrum −→ spectral measurement. | ----------------------------------------------------------------------------------------------------------------------------

(5)

This bridge becomes especially important when a later photometry article writes a quantity as a function of wavelength, such as

fλ(λ).

(6)

Before interpreting such a function, one must understand precisely what the wavelength coordinate means.

2 A field that varies in space and time

An electromagnetic wave is not a material object oscillating up and down as it moves forward.

Instead, the electric and magnetic fields vary with position and time.

A simple plane electromagnetic wave propagating in the positive z direction can be written schematically as

|--------------------------------| E (z,t) = ˆxE cos (kz − ωt + ϕ ),| -------------0----------------0---

(7)

together with

|--------------------------------| B (z,t) = ˆyB cos (kz − ωt + ϕ ).| -------------0----------------0---

(8)

For a vacuum plane wave,

|----------| | E0 | |B0 = ---.| --------c--

(9)

The Electric Field, magnetic field, and propagation direction are mutually perpendicular:

|------------------------------| E ⊥ B, E ⊥ ˆk, B ⊥ ˆk.| --------------------------------

(10)

Figure 1. In a simple vacuum plane wave the electric and magnetic fields are transverse to the propagation direction and to each other.

3 What wavelength means

Freeze time at some instant t = t₀.

The wave then becomes a spatial pattern:

E(z,t0).

(11)

The wavelength λ is the smallest positive spatial distance over which the complete wave pattern repeats.

For a sinusoidal wave,

|--------------------------------------------------------------------| -λ-=-distance-between-corresponding--points in-adjacent-spatial cycles.

(12)

Examples include:

crest to next crest;
trough to next trough;
one upward zero crossing to the next upward zero crossing;
any point to the next point with the same field value and the same spatial phase.

Mathematically,

|----------------------| E-(z-+-λ,t0) =-E-(z,-t0).-

(13)

Wavelength is a distance.

Its SI unit is the meter:

[λ ] = m.

(14)

Useful electromagnetic wavelength units include:

1 mm	= 10⁻³ m,	(15)
1 μm	= 10⁻⁶ m,	(16)
1 nm	= 10⁻⁹ m.	(17)

Figure 2. Wavelength is a spatial period. It is measured between equivalent phase points in neighboring cycles, not merely between any two locations having the same field value.

4 What frequency means

Now hold position fixed at

z = z0.

(18)

The wave becomes a time history:

E(z0,t).

(19)

The period T is the time required for one complete cycle:

|----------------------| E (z0,t + T ) = E (z0, t).| ------------------------

(20)

Frequency is the number of cycles per unit time:

|--------| | 1 | |f = --.| ------T--

(21)

The SI unit is the hertz:

|--------------| -1-Hz-=--1 s−-1.

(22)

Astronomy and spectroscopy often use the Greek letter ν for ordinary frequency:

|------| |ν = f.| --------

(23)

This article uses f when discussing general wave mechanics and ν when emphasizing astronomical spectral notation.

Figure 3. Frequency is a temporal quantity measured at one location. Wavelength is a spatial quantity measured at one instant.

5 Wavelength and frequency are different kinds of repetition

Wavelength and period are analogous, but they are not the same quantity.

Wavelength answers

|----------------------------------------------------------| |How far must one move in space for the pattern to repeat? | ------------------------------------------------------------

(24)

Period answers

|--------------------------------------------------------------| How---long-must--one-wait-at-one-point-for-the-pattern-to-repeat?--

(25)

Frequency then answers

|--------------------------------------------| How--many---temporal--cycles occur-per-second?-

(26)

The distinction between spatial and temporal repetition is essential.

A wave can have a very short wavelength but a very high frequency because the pattern propagates rapidly.

6 Deriving the wave-speed relation

Suppose a wave crest moves at constant speed v.

During one period T, the crest advances by exactly one wavelength:

Δx = λ.

(27)

Therefore

Δx-- λ- v = Δt = T .

(28)

Since

f = 1-, T

(29)

we obtain

|--------| -v-=-fλ.-|

(30)

This equation is a kinematic relation.

It connects the spatial repetition scale, temporal repetition rate, and propagation speed.

7 Electromagnetic waves in vacuum

Maxwell’s Equations in empty space imply wave equations of the form

|----------------| | 2 ∂2E | ∇ E = μ0𝜖0---2 , -------------∂t---

(31)

and

|-----------------| | 2 ∂2B-- | ∇ B = μ0𝜖0 ∂t2 .| -------------------

(32)

Compare these with the standard three-dimensional wave equation

2 1-∂2-ψ ∇ ψ = v2 ∂t2 .

(33)

The vacuum propagation speed is therefore

|-------1----| |c = √-----. | -------μ0𝜖0--|

(34)

Thus vacuum electromagnetic waves satisfy

|--------| -c-=-fλ.-|

(35)

This is one of the central connections between Electromagnetism and wave mechanics.

8 Angular frequency and wavenumber

Ordinary frequency counts cycles per second.

Angular frequency measures phase advance in radians per second:

|---------| ω-=--2πf.--

(36)

Likewise, wavelength describes meters per spatial cycle, while angular wavenumber measures phase advance in radians per meter:

|--------| |k = 2π-.| ------λ---

(37)

The sinusoidal phase is then

|------------------| -Φ-=-kz-−--ωt +-ϕ0.|

(38)

For a vacuum electromagnetic wave,

|------------| | ω- | -c =-k-=-f-λ.-

(39)

The Wave Mechanics articles on phase and wavenumber develop these quantities in greater detail.

9 Phase difference and path length

Wavelength is also the natural length scale for phase.

If two otherwise identical waves travel paths differing by

Δr,

(40)

the corresponding phase difference is

|--------------------| | 2π | |Δ ϕ = kΔr = ---Δr. | ---------------λ------

(41)

Therefore:

Δr	= λ	Δϕ	= 2π,	(42)
Δr	=	Δϕ	= π,	(43)
Δr	=	Δϕ	= .	(44)

This relation appears throughout interference, diffraction, antennas, interferometry, and coherent signal processing.

10 Example 1: visible-light wavelength to frequency

Consider vacuum wavelength

−7 λ = 500 nm = 5.00 × 10 m.

(45)

Then

f	=	(46)
	=	(47)
	≈ 5.996 × 10¹⁴ Hz.	(48)

Equivalently,

|----------------| |f ≈ 599.6 THz. | -----------------

(49)

Visible wavelengths therefore correspond to extremely rapid electromagnetic oscillations.

11 Example 2: radio frequency to wavelength

Consider

f = 1.57542 GHz.

(50)

This is close to the GPS L1 carrier frequency.

The vacuum wavelength is

λ	=	(51)
	=	(52)
	≈ 0.1903 m.	(53)

Thus the wavelength is approximately

|--------| |19.0 cm. | ----------

(54)

The same relation applies across the electromagnetic spectrum.

12 Vacuum wavelength versus wavelength in matter

The relation

v = f λ

(55)

always uses the wave speed and wavelength in the same medium.

In vacuum,

v = c.

(56)

In a material medium, an idealized refractive index can be defined by

|----c-| n = -.| -----v--

(57)

Then

c- v = n.

(58)

At a stationary interface, the wave frequency is determined by the time dependence of the source and remains continuous across the interface.

Therefore the wavelength changes:

|------------------------| | v- c-- λ0-| λmedium = f = nf = n ,| --------------------------

(59)

for the idealized case where n is the refractive index at that frequency and

λ = c- 0 f

(60)

is the vacuum wavelength.

This distinction is important in optics.

When astronomical electromagnetic wavelengths are quoted without qualification, they are normally understood as vacuum wavelengths or frequencies defined independently of terrestrial material propagation.

13 Dispersion

In many materials the refractive index depends on frequency:

n = n (f ).

(61)

Then the phase speed also depends on frequency:

--c-- vp(f) = n(f ).

(62)

This phenomenon is called dispersion.

Different frequency components of a broadband signal can therefore accumulate different phases or travel with different group delays.

Dispersion is central in optics, plasma propagation, radio science, and astronomical signal propagation.

The vacuum relation

c = f λ

(63)

remains the clean reference relation for labeling an electromagnetic spectrum.

14 Frequency and wavelength describe the same vacuum spectral coordinate

For vacuum electromagnetic radiation,

|-------| | c | ν = λ. | --------

(64)

Therefore a wavelength and its corresponding frequency contain the same information.

But the mapping reverses order:

|--------------------------------------| -short-wavelength-⇐-⇒--high-frequency,-|

(65)

and

|------------------------------------| -long-wavelength-⇐-⇒--low-frequency.-|

(66)

Figure 4. Vacuum wavelength and frequency are reciprocal spectral coordinates connected by the speed of light. Moving toward shorter wavelength means moving toward higher frequency.

15 The electromagnetic spectrum

The electromagnetic spectrum is the continuous range of electromagnetic frequencies or wavelengths.

Common descriptive bands include:

radio;
microwave;
infrared;
visible;
ultraviolet;
X-ray;
gamma ray.

These names are useful classifications, not fundamental discontinuities in the laws of physics.

A wave does not undergo a sudden change in its basic electromagnetic nature merely because its wavelength crosses a conventional band boundary.

The same relations

c = f λ

(67)

and

E = hf γ

(68)

continue across the spectrum.

Figure 5. The electromagnetic spectrum is continuous. Conventional band names identify useful wavelength and frequency ranges rather than distinct kinds of classical electromagnetic field.

16 Approximate wavelength scales

Representative vacuum wavelength scales are:


Band	Representative wavelength scale	Representative frequency scale

Radio	meters to kilometers and beyond	kilohertz to hundreds of megahertz
Microwave	meters to millimeters	hundreds of megahertz to hundreds of gigahertz
Infrared	millimeters to about 700 nm	hundreds of gigahertz to hundreds of terahertz
Visible	about 400 to 700 nm	about 750 to 430 THz
Ultraviolet	shorter than visible to tens of nm	above visible frequencies
X-ray	roughly nm to hundredths of nm	very high frequency
Gamma ray	shorter still	highest frequencies

The boundaries vary somewhat by discipline.

For quantitative work, the numerical wavelength or frequency is more fundamental than the band name.

17 Electromagnetic waves and photons

Classical electromagnetic theory describes waves and fields.

Quantum physics describes electromagnetic radiation in terms of photons.

For a photon,

|---------| E-γ-=-hν,--

(69)

where h is Planck’s constant.

Using

c ν = -, λ

(70)

we obtain

|----------| |E γ = hc. | -------λ---|

(71)

Therefore

|------------------------------------------| |short wavelength ⇐ ⇒ high photon energy. | -------------------------------------------

(72)

Figure 6. Photon energy increases with frequency and decreases with wavelength. The classical wavelength coordinate therefore also organizes photon energies.

18 Example 3: photon energy at 500 nm

For

λ = 500 nm,

(73)

E_γ	=	(74)
	≈ 3.973 × 10⁻¹⁹ J.	(75)

Using

1 eV = 1.602176634 × 10−19 J,

(76)

the photon energy is

|--------------| |Eγ ≈ 2.48 eV. | ----------------

(77)

Thus the same radiation can be characterized by:

λ	= 500 nm,	(78)
ν	≈ 5.996 × 10¹⁴ Hz,	(79)
E_γ	≈ 2.48 eV .	(80)

These are three linked descriptions of the same spectral location.

19 Monochromatic, narrowband, and broadband radiation

An ideal monochromatic electromagnetic wave has one frequency:

ν = ν . 0

(81)

Equivalently, in vacuum it has one wavelength:

λ = λ . 0

(82)

No real finite-duration source is perfectly monochromatic, but the approximation can be excellent for sufficiently narrow spectral lines or stabilized lasers.

A narrowband signal occupies a small interval around a central frequency or wavelength.

A broadband signal contains a substantial range of frequencies or wavelengths.

Starlight is generally broadband.

A blackbody is the canonical broadband thermal spectrum.

This distinction becomes crucial in photometry because a broadband detector or filter does not measure only one wavelength.

20 A spectrum is a distribution over wavelength or frequency

Suppose an instrument separates incoming radiation according to wavelength.

One can then ask how some measured quantity is distributed across wavelength.

Examples include:

spectral energy density;
spectral intensity;
spectral irradiance;
spectral flux density;
spectral luminosity.

The notation

Qλ(λ )

(83)

means that the quantity is expressed per unit wavelength.

Similarly,

Q (ν ) ν

(84)

means per unit frequency.

This distinction matters because spectral density is not the same thing as an ordinary total quantity.

A later standalone PhysicsLibrary article will develop spectral flux density carefully.

21 Why per-wavelength and per-frequency spectra are not numerically identical

Suppose a total quantity Q is represented either by wavelength or by frequency:

∫ ∫ Q = Qλ(λ )dλ = Q ν(ν)dν.

(85)

The corresponding infinitesimal amounts must match in magnitude:

|----------------| Q λ|dλ| = Q ν|dν |.| ------------------

(86)

Since

ν = c, λ

(87)

we have

| | ||dν-|| -c- |dλ | = λ2.

(88)

Therefore

|------------| | -c- | -Qλ-=-Q-νλ2-,-

(89)

or equivalently

-------------- | 2 | |Qν = Q λλ--.| ----------c---

(90)

This Jacobian factor is fundamental in spectroscopy and photometry.

A spectrum plotted per unit wavelength and the same spectrum plotted per unit frequency can have different shapes.

22 Blackbody radiation and wavelength

A blackbody spectrum is broadband.

When represented per unit wavelength, Planck’s law can be written

2hc2 1 B λ(λ,T ) = --5--------------------- . λ exp (hc∕(λkBT )) − 1

(91)

The wavelength at which B_λ reaches its maximum satisfies Wien’s displacement law:

|----------| λmaxT--=-b,-

(92)

where

b ≈ 2.898 × 10−3 m K.

(93)

Hotter blackbodies therefore peak at shorter wavelengths.

This connects directly to the PhysicsLibrary article on Wien displacement law and the existing blackbody-temperature example.

Figure 7. Increasing blackbody temperature shifts the peak of the per-wavelength spectrum toward shorter wavelength while also increasing the emitted spectral radiance.

23 An important subtlety about spectral peaks

The wavelength-space spectrum

B λ(λ,T )

(94)

and frequency-space spectrum

B (ν,T ) ν

(95)

represent the same total radiation but use different spectral coordinates.

Because the transformation includes a Jacobian, the maximum of B_λ does not map to the maximum of B_ν simply by applying

ν = c∕λ

(96)

to the wavelength of the B_λ peak.

This is not a contradiction.

The two functions describe density per different coordinate interval.

The same issue will appear again when converting between

f λ

(97)

and

fν.

(98)

24 From blackbody surface flux to stellar power

A blackbody surface emits a total energy flux

4 F = σT .

(99)

For a spherical star of radius R, the surface area is

4πR2.

(100)

Therefore the luminosity is

|--------------| |L = 4πR2 σT 4.| ----------------

(101)

This is the relation used in the existing PhysicsLibrary example on calculating the power of a star.

The present article supplies part of the conceptual bridge:

|----------------------------------------------------------------------------------------------------| -electromagnetic--spectrum--−-→--thermal--spectral distribution-−-→-integrated-flux--−→--stellar-luminosity.--

(102)

25 Wavelength is not color

Visible color is related to the spectral response of the human visual system.

A single visible wavelength can be associated with a spectral color under suitable viewing conditions, but most physical light sources contain many wavelengths.

An astronomical broadband color such as

B − V

(103)

is not one wavelength.

It is constructed from measurements through two broad response bands.

Thus one should distinguish:

|--------------------------------| -wavelength-⁄=--photometric-color.-

(104)

The PhysicsLibrary article on color in astrophysics develops this distinction further.

26 Bridge to photometric passbands

A telescope and detector generally do not respond equally to every wavelength.

Let

SX (λ )

(105)

represent the response of a photometric band X.

If the incoming spectrum is described by a spectral flux density

f (λ), λ

(106)

then a schematic band signal has the form

|------∫-----------------| |FX ∝ fλ(λ)SX (λ)dλ. | -------------------------|

(107)

The point of the present article is not yet to derive the exact photometric calibration convention.

Instead, notice what wavelength is doing:

|------------------------------------------------------------------------------------------| -λ =-the-coordinate-over which-the-source-spectrum--and--instrument--response-are-compared.--|

(108)

The next conceptual steps are therefore:

|--------------------------------------------------------------------------------------------------------| |electromagnetic wavelength −→ spectral flux density −→ photometric passband − → magnitude and color.| ----------------------------------------------------------------------------------------------------------

(109)

Figure 8. A broadband measurement samples a range of wavelengths. The detected contribution depends both on the source spectrum and on the wavelength-dependent system response.

27 Common wavelength-frequency conversions

For vacuum radiation,

λ = c. f

(110)

Useful conversion scales include:

1 GHz	↔0.2998 m,	(111)
100 GHz	↔2.998 mm,	(112)
1 THz	↔299.8 μm,	(113)
500 THz	↔599.6 nm.	(114)

These conversions are particularly useful when moving between radio engineering, infrared astronomy, and optical astronomy.

28 Common mistakes

Treating wavelength as a time instead of a distance.
Treating frequency as a propagation speed.
Using c = fλ inside a material without replacing c by the appropriate phase speed.
Assuming frequency changes when a wave crosses a stationary material interface.
Forgetting that wavelength changes when phase speed changes but frequency remains fixed.
Confusing ordinary frequency f with angular frequency ω.
Confusing wavelength λ with angular wavenumber k = 2π∕λ.
Treating electromagnetic-spectrum band boundaries as sharp changes in physical law.
Assuming a broadband source has one unique wavelength.
Treating photometric color as identical to wavelength.
Assuming a spectrum per unit wavelength is numerically identical to a spectrum per unit frequency.
Converting the peak of a per-wavelength spectrum into the peak of a per-frequency spectrum merely by using ν = c∕λ.
Forgetting that photon energy increases as wavelength decreases.
Assuming the amplitude of the electric field is itself an energy.

29 Connections to other PhysicsLibrary articles

This article connects directly to the existing article Electromagnetic Radiation, which provides the broader wave and photon picture.

It connects to Electromagnetic Spectrum, which classifies astronomical radiation by wavelength and frequency.

It connects to Electromagnetism and the EM series, where Maxwell’s equations produce electromagnetic wave propagation.

It connects to EM01, where a one-dimensional traveling wave is generalized into a vector field such as

E (r,t).

(115)

It connects to the Wave Mechanics articles on oscillation in time, wavelength, phase, wavenumber, traveling waves, and wave speed.

It connects to Wien displacement law through

λmaxT = b.

(116)

It connects to the blackbody-temperature example by turning a measured peak wavelength into a temperature.

It connects to the stellar-power example through

L = 4πR2 σT 4.

(117)

Finally, it prepares for the next photometric concepts:

|------------------------------------| f λ(λ ), fν(ν), SX (λ ), FX .| --------------------------------------

(118)

30 Summary

An electromagnetic wave is a propagating time-dependent electric and magnetic field.

Wavelength is its spatial period:

|------------------------------| λ = one complete spatial cycle. --------------------------------

(119)

Period is one complete temporal cycle:

|----------------------------------| |T = one complete temporal cycle.| -----------------------------------

(120)

Frequency is

|--------| | 1 | |f = --.| ------T--

(121)

Wave speed connects the spatial and temporal descriptions:

|--------| -v-=-fλ.-|

(122)

In vacuum,

|--------| -c-=-fλ.-|

(123)

Angular frequency and angular wavenumber are

|----------------------| |ω = 2πf, k = 2π-.| --------------------λ--|

(124)

Photon energy is

|--------------| E γ = hν = hc.| ------------λ---

(125)

The electromagnetic spectrum is a continuous distribution that can be labeled by wavelength or frequency.

This provides the foundation for describing an astronomical source with spectral quantities such as

fλ(λ)

(126)

and for integrating those quantities through wavelength-dependent photometric response functions.

References

[1] D. J. Griffiths, Introduction to Electrodynamics, 4th ed., Cambridge University Press, 2017.

[2] S. J. Ling, J. Sanny, and W. Moebs, University Physics, Volume 2, OpenStax, 2016.

[3] R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Volume II, Addison-Wesley, 1964.

[4] F. S. Crawford, Jr., Waves, Berkeley Physics Course, Volume 3, McGraw-Hill, 1968.

[5] E. Hecht, Optics, 5th ed., Pearson, 2017.

[6] G. B. Rybicki and A. P. Lightman, Radiative Processes in Astrophysics, Wiley, 1979.

[7] B. W. Carroll and D. A. Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Pearson, 2007.

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Electromagnetic Waves and Wavelength: Frequency, Propagation, and Spectral Description