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```Figure 1: Electromagnetic wave
An electric generator rotates a magnet to create electricity. An magnetic resonance imager
found in a hospital creates a magnetic field by passing a large electric current in a circle. A
changing electric field induces a magnetic field and a changing magnetic field induces an electric
field. The magnetic field is in each case normal to the electric field.
The fundamental equations that describe electromagnetic radiation are Maxwell’s equations.
The solutions to the equations are sinusoidal of form
⇣
~ = E~0 exp i~k · ~x
E
⇣
~ =H
~ 0 exp i~k · ~x
H
⌘
i!t
⌘
i!t
where E is the electric field and H is the magnetic field, the real component of ~k is the wavenumber
2⇡/ , ~x is the direction of wave propagation and ! is the angular frequency of the radition 2⇡⌫.
Some fundamental properties of the waves are that they do not diverge, that H and E are normal
to each other, and that radiation of one frequency or wavelength does not interact with radiation
of another frequency. The last point is particularly useful because it means we can analyse the
separate contributions of a range in frequency or wavelength to the total energy input from radiation
as it accumulates over time.
The symbols ✏ and µ stand for the electric permittivity and magnetic permeability, respectively.
They are merely properties of matter. Curiously, unlike say sound or water waves, E-M waves do
not require a medium and can travel in a vacuum. The permittivity and permeability of a vacuum
have the symbols ✏0 and µ0 . The instantaneous flux density of the electric field in the direction of
1
Figure 2: Electromagnetic wave quantities
the wave is defined by the Poynting vector
~=E
~ ⇥H
~
S
Averaged over time this is equal to the average Poynting Flux, which is given by
1
Sav = c✏0 E02
2
Sav has units of W m 2 (or J s 1 m 2 ) and is a function of wavelength. This is units of flux (quantity
per length squared per second). In atmospheric sciences, we more commonly use the symbol F .
The point here is that the flux of energy F is proportional to the square of the magnitude of an
electromagnetic wave. It is the flux density of electromagnetic waves that is of primary interest to
climate studies.
Given that
⇣
⌘
~ = E~0 exp i~k · ~x i!t
E
Assume that the wavenumber vector ~k is a complex number with a real and imaginary component,
i.e.
~k = k~0 + ik”
~
2
and ! is the angular frequency of the wave in radians per second. The wave period is T = 2⇡/!.
The consequence of k being complex is the following
~ = E0 exp
E
⇣
⌘
⇣ ⇣
~ · ~x exp i k~0 · ~x
k”
⌘⌘
!t
~ = 0 then the amplitude of the wave is constant.
Note that if k”
For the special case where there is no imaginary component (this is actually the case of no
absorption), then
⇣ ⇣
~ = E~0 exp i k~0 · ~x
E
⌘⌘
!t
the phase of the wave is given by
= k~0 · ~x
!t
At constant phase, e.g. following the crest or trough of the wave
d = 0 = k~0 · d~x
!dt
implying that the phase speed of radiation in the direction the waves propagate is given by
v=
d~x
!
=
dt
k~0
or alternatively
!
2⇡
where ⌫˜ is the wave frequency in Hertz. The point here is that frequency and wavelength are linked
through the phase speed of the wave. In a vacuum the phase speed is c = 3.0 ⇥ 108 m/s, the “speed
of light”, where
!
!
c = ⌫˜ =
=
2⇡
k
It is common practice in the atmospheric sciences to divide ⌫˜ by the speed of light to obtain the
wavenumber
⌫˜
⌫=
c
which is most commonly expressed in units of cm 1 . The only reason for dividing by c is that we
end up with numbers around 1000 that our poor little brains can handle more easily than 1011 .
v = ⌫˜ =
3
```
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