A: Dispersive and nondispersive waves
1.1
1.1.1
Background theory— nondispersive waves
Oscillations
Oscillations (e:g:, small amplitude oscillations of a simple pendulum) are
often described by an equation of the form
Figure 1: Characteristics of a simple oscillation.
d2
+
dt2
2
(1)
=0
where t is time and is some system variable (angular displacement in the
case of the simple pendulum). (1) has solutions of the form
(t) = Re Y ei!t = Y0 cos(!t
)
(2)
where frequency ! =
, amplitude Y0 and phase are real constants, and
i
Y = Y0 e is a complex amplitude. Thus, as shown in Fig. 1, an oscillation
is characterized by three constants: amplitude, frequency, and phase.
1
1.1.2
Nondispersive waves
Unlike such simple oscillations, waves are functions of both time and space.
The simplest wave equation is of the form
@2
@t2
c20
@2
=0
@x2
(3)
where c0 is some constant. Thisn equation. could represent the electric or
magnetic …elds in vacuo, with c0 the speed of light; or pressure perturbations
in a compressible ‡uid, with c0 the sound speed, or waves on shallow water.
We can …nd solutions to (3) by separating the variables, writing
(x; t) = Re [A(t)B(x)] :
More speci…cally, if we look for “wave-like” solutions for which B(x) = eikx ,
where k is a real wavenumber (so 2 =k is wavelength)1 , then d2 B=dx2 =
k 2 B, so
@2
d2 B
=
Re
A(t)
(x) = k 2 Re [A(t)B(x)] ;
@x2
dx2
and (3) becomes
d2 A
+ k 2 c20 A = 0 :
dt2
This has solutions like
A=
where + and
satis…es
+e
i!t
; A=
e+i!t
are constant (complex) amplitudes and the frequency !
! 2 = k 2 c20 :
(4)
The full solution is
(x; t) = Re
i(kx !t)
+e
+
ei(kx+!t) :
(5)
Each of the two terms in (5) describes a progressive wave (Fig. 2):
at any instant, it is just a sinusoidal wave disturbance, of wavelength
2 =k
1
In general, any function of x can be expressed as a Fourier integral of such waves.
2
Figure 2: Characteristics of a progressive wave.
at any …xed location x = x0 , it is just an oscillation of the form De
where D = eikx0 is its complex amplitude, of period T = 2 =!
i!t
,
it propagates with phase speed c = !=k = c0 . Note from (5)
that the …rst term of
is constant along characteristics with kx
!t =constant, i.e., x = !k t+constant. The second term is constant
along x = !k t+ constant. Thus, (5) represents two solutions, propagating in opposite directions with phase speed c0 .
Eq. (4) is the dispersion relation for the wave: for a given wavenumber
k, it tells us the wave’s frequency. This form is particularly simple, as shown
in Fig. 3.
[Note that this is drawn for positive k only— we may de…ne k positive,
without loss of generality, as long as we do not try to constrain the sign of !.]
The phase speed c = !=k = c0 ; the waves can propagate in either direction.
These waves are nondispersive, i:e:, their phase speed is independent
of wavenumber. Thus, all waves, of any wavenumber, propagate at the same
speed (in either direction), which means that non-sinusoidal disturbances
propagate without change of shape. In fact, any function
(x; t) = F (x
3
c0 t)
(6)
Figure 3: The dispersion relation for 1-D shallow water waves.
is a solution to (3)2 . Eq. (6) just describes any shape of disturbance, including a localized one, that propagates at speed c without changing its shape
(Fig. 4).
1.1.3
Two-dimensional waves
In two dimensions (x; y), (3) is replaced by
@2
@t2
where r2
form
@2
@x2
+
@2
.
@y 2
c20 r2 = 0
We can now look for plane wave solutions of the
(x; y; t) = Re A0 ei(kx+ly
2
To see this, note that if X = x
@
@x
@2
@x2
@
@t
@2
@t2
=
=
=
=
(7)
!t)
:
c0 t, then F = F (X) and the chain rule gives us
@F
dF @X
dF
=
=
;
@x
dX @x
dX
@
dF
@X d2 F
d2 F
=
=
;
2
@x dX
@x dX
dX 2
@F
dF @X
dF
=
= c0
;
@t
dX @t
dX
2
@
dF
@X d2 F
2d F
c0
= c0
=
c
:
0
@t
dX
@t dX 2
dX 2
So, (3) is satis…ed by (6).
4
(8)
Figure 4: Nondispersive waves: arbitrary disturbances propagate without
change of shape.
Then
r2 =
2
Re A0 ei(kx+ly
!t)
p
where = k 2 + l2 is the total wavenumber. Therefore, substituting into
(7) gives the dispersion relation for this case
!2 =
2 2
c0
:
(9)
[Note that the one-dimensional case we discussed above is just a special case
of the two-dimensional problem, with l = 0.]
Eq. (8) describes a plane wave because is constant along lines of constant phase kx + ly !t = constant, so at any instant in time, kx + ly =
constant; see Fig. 1.1.3. The wave pattern moves at right angles to the phase
lines, with speed c.
5
Plane waves are a special, and particularly simple, form of 2-D waves.
Exactly what shape the wavefronts have will in general depend on the geometry of the system and of the process that generated the wave. If the source
is very localized (e:g:, a stone dropped into water), the wavefronts will be
circular, as shown in Fig. 5. Note that, far from the source (in the dashed
rectangle), the wavefronts will look almost plane.
Figure 5: Circular wave fronts radiating from a localized source.
1.2
1.2.1
Background theory— dispersive waves
Dispersion
The general dispersion relation for the frequency ! of 1-D waves of wavenumber k can be written
! = !(k)
(10)
and the phase speed is
!(k)
:
(11)
k
Clearly, c is independent of k for all k only if !(k) =constant k— this is the
nondispersive case we discussed earlier and, as we saw, it implies that all
disturbances, including localized ones, propagate without change of shape.
This can be thought of in terms of Fourier components. Any non-sinusoidal
disturbance can be described a sum of components of di¤erent wavenumber;
c=
6
if all these waves propagate at the same speed, so will the disturbance itself,
and its shape will not change.
For many kinds of wave motion, however (including surface waves on
deep water, internal gravity waves, and Rossby waves), c varies with k, in
which case the di¤erent wavenumber components will have di¤erent speeds,
a phenomenon known as wave dispersion. Therefore the way they interfere
with one another will change with time— so the shape of the disturbance will
change.
1.2.2
Group velocity
There is one particularly important aspect of dispersion, which concerns the
way that modulations propagate on a wave train.
A monochromatic plane wave of the form
(x; t) = Re A0 ei[k0 x
!(k0 )t]
has a single wavenumber k. This is a special case of a more general form
Z 1
(x; t) = Re
A(k) ei[kx !(k)t] dk
(12)
1
where, in the plane wave case, the wavenumber spectrum has the simple form
of a -function, A(k) = A0 (k k0 ) [Fig. 6(a)]. The wave behaves in the
way we have discussed, propagating with a speed c = !(k0 )=k0 . Consider
Figure 6: Wavenumber spectra for (a) a monochromatic wave, and (b) an
almost-monochromatic wave.
now an almost monochromatic wave, with a narrow spectrum [Fig. 6(b)]. In
7
this case, the spectrum has …nite but small width such that A(k) is nonzero
only for wavenumbers in the close vicinity of k0 . Writing k = k0 + k, we can
rewrite (12) as
Z 1
(x; t) = Re
A(k0 + k) ei[k0 x !(k0 )t] ei[ k x ! t] d ( k) ;
1
where ! = !(k0 + k)
!(k0 ) ' (@!=@k) k. At t = 0, this is simply
(x; t) = Re F (x) eik0 x
where
F (x) =
Z
1
A(k0 + k) ei
kx
d ( k)
1
is the modulating envelope of the wave train, which has carrier wavenumber
k0 , as illustrated in Fig. 7. Now, for t > 0, the wave packet behaves as
Figure 7: An almost-monochromatic wave packet, comprising many wavelengths. The phase of the carrier wave propagates at the phase speed, but
the modulation envelope propagates at the group velocity.
(x; t) = Re
Z
1
= Re F (x
1
A(k0 + k) ei[
cg t) ei[k0 (x
where
cg =
@!
@k
8
k(x cg t)]
ct)]
;
d ( k) ei[k0 (x
ct)]
(13)
(14)
k=k0
is the group velocity. Thus, while the carrier wave propagates at the phase
speed, the modulation envelope propagates at the group velocity. This is an
important concept, as it is the latter velocity that governs the propagation
of information, as we shall see.
Nondispersive waves have ! = c0 k, with constant phase speed c0 , and
so their group velocity is the same as their phase velocity. But the group
velocity of dispersive waves di¤ers from the phase speed, so in a wave packet
like that shown in Fig. 7 the wave crests will move at a di¤erent speed than
the envelope. If c > cg (which, as we shall see, is the case for deep water
waves), new wave crests appear at the rear of the wave packet, move forward
through the packet, and disappear at its leading edge. We shall see some
examples of this below.
In general, it is easy to get a feel for both phase and group propagation graphically from the dispersion relation, as shown in Fig. 8. For any
Figure 8: The disperion relation !(k). The phase velocity at k = k0 is tan ,
the group velocity tan .
wavenumber k, the phase velocity is just given by the slope of a line joining
the point (!; k) to the origin, while the group velocity is given by the slope
tan of the tangent to the curve at (!; k).
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1.2.3
Group velocity in multi-dimensional waves
For future reference, we note here that for 2- or 3-dimensional waves, the
general dispersion relation is of the form
(15)
! = !(k)
where k = (k; l; m) is the (2-D or 3-D) vector wavenumber. The phase velocity3 is
! ! !
c=
; ;
;
(16)
k l m
and the group velocity
cg =
@! @! @!
;
;
@k @l @m
:
(17)
As we shall see (and a comparison of (16) and (17) implies), not only may
cg and c be of di¤erent magnitudes, they may also be in di¤erent directions.
3
The phase velocity is in fact not a vector, even though it has magnitude and direction.
It does not transform like a vector under rotation— this stems from the fact that phase
propagation has no meaning along the phase lines.
10