1. Representation of waves
1
REPRESENTATION OF WAVES
1.1 Introduction
One of the properties of matter is that is can support transfers of mechanical energy,
whether the matter is solid, liquid, or gas, without any net movement of the molecules
involved. Such transfers are referred to as "wave motion", or a transitory displacement of
atoms within the matter that passes on kinetic energy. This chapter discusses the
physics of mechanical wave motion. Light also involves wave motion, but not of a
mechanical sort, and will not be discussed in this course.
We are all familiar with wave motion. The sounds that we hear are waves passing
through the atmosphere, with the air compressing and decompressing in an oscillating
fashion, passing the sound energy outward at hundreds of metres per second. We are
all familiar with the waves that expand outward after we drop a big rock in a pond or the
waves that we create by whipping a rope tied to a fence up and down.
There are two kinds of waves: "compression" or "longitudinal" waves, and "transverse"
waves (Figure 1.1). In a compression wave, the molecules move back and forth in the
direction of the wave motion. Sound is a compression wave. In a transverse wave, the
molecules move back and forth at a right angle to the direction of motion. A rope being
shaken with an up-and-down motion generates a transverse wave, and the ripples we
see spreading across a pond are also transverse waves.
Figure 1.1 Two fundamentally different modes of wave propagation.
In the atmosphere wave motion takes a special place among the multitude of motions
and many different waves can be distinguished. Wave motion can be described with the
aid of a linear or linearized system of differential equations, which is relatively easy to
treat mathematically. This approach is valid because wave motion is important and
covers the entire spectrum of atmospheric motions.
If we consider the large-scale flow in the free atmosphere then in the mid-latitudes we
will regularly see a change from zonal to meridional flow and vice versa. This has led to
the notion of troughs and ridges as a part of planetary waves with a wave number of 4-6.
As a rule these waves travel eastward with a speed of 6 degrees longitude per day. For
the weather in the mid-latitudes they are of the utmost importance. On a slightly smaller
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1. Representation of waves
scale the baroclinic waves are important. They usually have a larger wave number (a
smaller wavelength) and are controlled by the vertical windshear and the static stability.
Beyond a critical windshear the waves become unstable leading to cyclogenesis where
potential and internal energy of the atmosphere is transformed into kinetic energy.
Waves also occur on smaller scales. If stably stratified air flows over an obstacle (e.g. a
mountain ridge) then lee waves will develop downstream of the obstacle. These waves
have a wavelength of 5 to 10 km and larger. The displacement of the air the is in the
vertical. The nature of these waves is determined by the vertical profiles of wind and
temperature and they are an example of internal gravity waves. Lee waves can reach
into the stratosphere. Together with a strong vertical windshear in the upper atmosphere
these waves may become unstable and generate what is called “Clear Air Turbulence”
(CAT).
The simplest forms of gravity waves are those that occur on the discontinuity between
two homogeneous fluids of different density, where due to Earth’s gravity the denser
fluid lies beneath the lighter, less dense fluid. The largest amplitude will occur on the
surface of the discontinuity. These waves are called internal gravity waves. If the density
of the upper fluid decreases to, almost, zero then the surface of discontinuity will be the
upper boundary of just one fluid and perturbations of this surface are called external
gravity waves. Ripples on the water surface of a pond (Figure 1.2) are a perfect example
of such waves.
Figure 1.2 An example of external gravity waves.
Waves in the atmosphere transport physical quantities such as momentum and energy.
A study of atmospheric waves is therefore important because it will give us a better
understanding of the physics of the atmosphere.
Together with wave propagation in the atmosphere we will also dedicate some time on
the propagation of sound waves. Sound waves do not play an important role in largescale weather but must be considered because of the stability of numerical modelling
used for weather forecasting. In order to obtain numerical stability, waves which
propagate with the speed of sound must be filtered out of the system. Therefore we
need to consider sound waves as well.
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1. Representation of waves
1.2 Representation and properties of waves
One-dimensional waves
Consider a particle exhibiting a periodic movement in space i.e. an oscillating particle.
The energy of the periodic movement is transferred to neighbouring particles which will
start to oscillate: the oscillation is propagating in space. If this propagation is in one
direction only we are dealing with a one-dimensional (1D) wave. The mathematical
representation of such a wave is given by
θ ( x, t ) = θ 0 cos(kx − ν t ) .
(1.1)
This is a one-dimensional (1D) wave propagating in the x-direction (although the
oscillations may be in another direction), see Figure 1.3.
Figure 1.3 Representation of a simple one-dimensional wave (After Pichler, 1997).
From Figure 1.3 we have the following quantities:
- θ0 is the amplitude; it is half the difference of the perturbation between a crest and a
trough,
- λ is the wavelength (in m) it is the distance between two crests (or troughs) of a
wave,
- τ is the period (in s) it the amount of time needed for one complete oscillation and it
must equal the time required for the wave to travel one wavelength τ =
λ
c
,
- k is the wave number (in radians m-1 or usually m-1); it is the number of waves per 2π
unit length, hence
k=
-ν =
2π
τ
2π
λ
,
(1.2)
is the angular (or circular) frequency: 2π times the number of crests passing
a point in unit time (in radians s-1 or usually Hz = s-1).
If we follow an individual crest or trough we are traveling with the phase speed. For a
wave traveling in the x-direction only we have:
3
1. Representation of waves
c=
λ ν
=
τ k
(1.3)
where the phase speed is in m s-1. It is the rate at which the crests and troughs of the
wave propagate.
In the one-dimensional case with propagation along the x-axis we can rewrite (1.1) as
θ ( x, t ) = θ 0 cos(kx − ν t ) = θ 0 cos[k ( x − ct )]
(1.4)
Where we have used
ν = kc .
Often it is easier to represent waves with the aid of Euler’s formula
e iφ = cos(φ ) + i sin (φ ) ,
(1.5)
this leads to:
θ ( x, t ) = θ 0 e i (kx −ν t ) = θ 0 e ik ( x −ct ) .
(1.6)
Only the real part of (1.6) represents the periodic behaviour.
Three-dimensional waves
In general, wave propagation is not restricted to one dimension only but it is in all three
dimensions. The general expression for (1.6) in three dimensions is
θ (r , t ) = θ 0 e i ( µ⋅r −ν t ) = θ 0 e i (kx +ly + mz −ν t ) .
(1.7)
In this case r represents the spatial vector r = xi + yj + zk and
µ = ki + lj + mk = µ1i + µ 2 j + µ3 k
(1.8)
is the three-dimensional wave number vector. Here i, j and k are vectors of length 1 in
the x, y and z direction, respectively. The values k, l and m are the components of the
vector µ and represent the wave number in the respective coordinate directions (x,y,z).
The wave number vector µ is perpendicular to the wave front (i.e. all points with the
same phase). The wave propagation is in the direction of the wave number vector µ .
From (1.8) the length of the wave number, or just the wave number (µ) is given by:
µ = µ = k 2 + l 2 + m2 .
(1.9)
The wave number and the wavelength are, just as in (1.2), related by
µ=
2π
λ
.
(1.10)
From (1.3) the phase speed now is given by
4
1. Representation of waves
c=
ν
ν
=
µ
k 2 + l 2 + m2
(1.11)
For each component of the wave number vector we have, similar to (1.10)
k=
2π
; l=
λx
2π
; m=
λy
2π
λz
.
(1.12)
The different wavelengths are now defined as the distances between crests in the three
coordinate directions (see Figure 1.4).
Figure 1.4 The propagation of a two-dimensional wave. (a) Two lines of constant phase (e.g. two
wave crests) at time t1. The wave is propagating in the direction µ. with wavelength λ. (b) The
same line at two successive times a period τ apart. The phase speed is the speed of
advancement of the wave crests in the direction of travel, and so c = λ / (t 2 − t1 ) = λ / τ . The
phase speed in the x-direction is the speed of propagation of the wave crests along the x-axis
and c x = λ x / τ = c / cos α (After, Vallis 2006)
From equations (1.9)-(1.12) it follows that for the wavelength we have:
1
λ
2
=
1
λ
2
x
+
1
λ
2
y
+
1
λ2z
.
(1.13)
This shows that the wavelength (λx, λy, λz) in each coordinate direction is larger than the
wavelength (λ) measured in the direction of propagation. From the definition of the
phase speed (1.3) we now find that the phase speed has a direction and has values
along each axis:
cx =
λ x λ x 2π ν
=
=
τ
2π τ
k
; cy =
λy ν
=
τ
l
; cz =
λz ν
=
τ
m
(1.14)
where cx, cy and cz are all > c. Note that these are NOT the components of a vector!
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1. Representation of waves
Hence
c 2 ≠ c x2 + c 2y + c z2
but rather
1
1
1
1
= 2 + 2 + 2.
2
c
cx c y cz
(1.15)
This is depicted in Figure 1.5.
The phase speed (c) vector (it has a direction and a magnitude) is actually given by
c=
ν
µ
µ2
(1.16)
Figure 1.5 Phase speed c and its components
in the x- and y-directions. Thin lines represent
wave crests.
From equation (1.5) the wave front is a plane surface; therefore the waves are called
plane waves. Waves emanating from a point source are called spherical waves; in that
case the wave front consists of spherical surfaces (Figure 1.6). However, far from the
source these spherical surfaces approach plane surfaces and (1.5) can be used once
more.
Figure 1.6 Wave front (a) from a spherical wave.
In point A this front approaches the form of a plane
wave front (b) (From Pichler, 1997).
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1. Representation of waves
.
1.3 The wave equation
It is easy to verify by direct substitution that (1.5) is the solution of the following
differential equation:
∂ 2θ
= c 2 ∇ 2θ
2
∂t
(1.17)
if the amplitude and phase speed, or wave number, remains constant. This equation is
called the wave equation. It is a linear partial differential equation and therefore a
combination of two particular solutions is also a solution of this equation. We will
consider two waves propagating in the same direction but with a slight difference in
frequency and consequently also a slight difference in wave number. A superposition of
these two waves gives:
θ = θ1 + θ 2 = θ 0 ei [( µ + ∆µ )⋅r −(ν + ∆ν )t ] + θ 0 ei [( µ − ∆µ )⋅r −(ν −∆ν )t ]
(1.18)
and leads to the formation of wave groups. From (1.18) we will have
θ = θ1 + θ 2 = θ 0 [e i [∆µ⋅r −∆ν t ] + e −i [∆µ⋅r −∆ν t ] ] e i [ µ⋅r −ν t ]
(1.19)
which, with the help of the relation 2 cos ϕ = eiϕ + e − iϕ leads to
θ = θ1 + θ 2 = 2θ 0 cos(∆µ ⋅ r − ∆ν t ) e i [ µ⋅r −ν t ] .
(1.20)
1.4 Phase velocity, group velocity and dispersion
The superposition θ = θ1 + θ 2 can be regarded as a wave of which the amplitude also
shows a periodic change (AM: amplitude modulation). This is depicted in Figure 1.7.
Figure 1.7 Representation of wave groups as a result of adding two slightly different
waves traveling in the same direction (From Pichler, 1997).
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1. Representation of waves
The velocity of the wave groups, the group velocity (cg) need not be the same as the
phase speed of the two original waves. If we go from ∆ν → dν and from ∆µ → dµ then
we find for ν = ν ( µ )
dν =
∂ν
∂ν
∂ν
dk +
dl +
dm = ∇ µν ⋅ dµ .
∂k
∂l
∂m
(1.21)
With the help of (1.21) we find for the phase of the amplitude modulation:
(dµ ⋅ r − dν t ) = dµ ⋅ [r − (∇ µν )t ]
(1.22)
which leads to the group velocity (cg is a vector just as any velocity):
cg =
dν
,
dµ
(1.23)
or in vector representation
∂ν
∂ν
 ∂ν ∂ν ∂ν  ∂ν
c g = (cg , x , cg , y , c g , z ) = 
,
,
i+
j+
k.
=
∂l
∂m
 ∂k ∂l ∂m  ∂k
(1.24)
Using (1.13) we can write for the x-component of cg
cg ,x =
∂c
dc
∂ν
∂
=
cx k = cx + k x = cx − λ x ,
dλ
∂k ∂k
∂k
(1.25)
and likewise for the y- and z-components. For the magnitude cg of the vector cg we have
similarly
cg = c + µ
∂c
dc
=c−λ
.
∂µ
dλ
(1.26)
Equation (1.26) means that if the phase speed depends on the frequency or the
wavelength then the group velocity and the phase speed will be different. This
phenomenon is called dispersion. The following three possibilities can be discerned:
dc
dλ
dc
dλ
dc
dλ
> 0 ⇒ c g < c normal dispersion,
< 0 ⇒ c g > c anomalous dispersion,
= 0 ⇒ c g = c no dispersion.
Sound waves are an example of non-dispersive waves: high and low tones travel with
the same group/phase speed. Waves on a water surface are an example of dispersive
waves (normal dispersion, see Figure 1.8 and also e.g. Holton Figure 7.4).
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1. Representation of waves
Figure 1.8 A superposition of two traveling waves illustrating the difference
between the phase speed c of the wave crests and the group velocity cg of the
envelope of the waves. In this case cg< c indicating normal dispersion
(From Gill, 1982).
In the case of normal dispersion the wave with the largest wavelength has a larger
phase speed than the wave with the smaller wavelength. In the case of anomalous
dispersion the opposite occurs.
1.5 Unstable and damped waves
Up to now we only considered waves where the amplitude remained constant during
time i.e. the amplitude did not grow or damp out. However, in nature both possibilities
occur. For cyclogenesis the growth of the amplitude of a (baroclinic) wave is essential.
With a growing or damping amplitude we need to have a complex representation of the
angular frequency:
ν = ν r + iν i .
(1.27)
where ν r = Re(ν ) is the real part and ν i = Im(ν ) is the imaginary part of the complex
frequency ν .
If you substitute (1.27) into the wave formula (1.7) then it follows directly that
θ (r , t ) = θ0eν t ei ( µ⋅ r −ν
i
r
t)
.
(1.28)
For ν i > 0 the amplitude will grow with time, the wave is unstable. For ν i < 0 the
amplitude will diminish with time approaching zero, the wave is stable or damped. For
ν i = 0 the amplitude will remain constant in time, this is also a case of a stable wave. In
general we will study wave propagation using ν = ν r ± iν i . Using the principle of
superposition we will get the following wave representation
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1. Representation of waves
θ (r , t ) = θ01eν t ei ( µ⋅ r −ν
i
r
t)
+ θ 02e −ν i t ei ( µ⋅ r −ν r t ) .
(1.29)
On of the two amplitudes in (1.28) will grow with time, so that if ν i ≠ 0 we will always
have an unstable wave. Finally, we must realize that the amplitude factors θ0, θ01 and θ02
themselves can be functions of the spatial vector. We will have to use this for the
propagation of certain waves in the atmosphere e.g. θ 0 = θ 0 ( z ) .
References
Gill, A.E., 1982: Atmosphere-Ocean Dynamics. International Geophysics series, Volume
30. Academic Press, Inc. London, 662 pp.
Holton, J.R., 2004: An introduction to Dynamic Meteorology (4th edition). International
Geophysics Series Vol. 88. Elsevier Academic Press, Amsterdam, 535 pp.
Pichler, H., 1997: Dynamik der Atmosphäre , 3 aktualisierte Auflage. Spektrum
Akademischer Verlag, Heidelberg, 572 pp.
Vallis, G.K., 2006: Atmospheric and oceanic fluid dynamics, Cambridge University
Press, 745 pp.
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