An Attempt to Illustrate Wave Mechanics Concepts.

Karl Peter Burr

May 27, 2001

Contents

1 Introduction. 1

2 Monochromatic waves. 1

3 Superposition of plane waves. 7

4 Dispersion of a localized initial disturbance. 10

1 Introduction.

The objective of the material below is to illustrate concepts and ideas central to understanding wavephenomena in uids. To reach a broad audience, we start with simple concepts, like the de�nition ofwave number and wave frequency, and progressively move on to more advanced concepts. Our approach isto illustrate these concepts through computer generated animations that are to a large extent interactive,so the user may easily explore and test the ideas and concepts introduced.

2 Monochromatic waves.

To start we consider linear wave problems with only one space variable x, and we denote time by t. Inan in�nite domain (�1;+1), if all coeÆcients of the governing equations are independent of x and t,then the wave problem admits as solution a superposition of sinusoidal wave trains. We �rst examine thephysics of a single sinusoidal wave train of the form:

�(x; t) = jAj cos(kx� !t+ �A) = <fAeikx�i!tg; (1)

1

where A = jAjei�A is the complex wave amplitude of magnitude A and phase �A. To simplify our notation,we omit the symbol <f: : : g \the real part of", so we write

�(x; t) = Aeikx�i!t (2)

First of all, a few de�nitions about sinusoidal waves in general are useful. We shall call

�(x; t) = kx� !t (3)

the wave phase. Clearly the trigonometric function is periodic in phase with period 2�. In the x; t plane,� has a constant value along a line of constant phase. In �gure 1, plots of lines with constant phaseequal to � = �n (n = 0; 1; 2; : : : ) are illustrated in the x; t plane. Values of � = 2�n (n = 0; 1; 2; : : : )correspond to the wave crests where � = jAj is greatest. On the other hand, � = (2n+1)� (n = 0; 1; : : : )correspond to wave troughs where � = �jAj is smallest. As illustrated in �gure 2, jAj is the half verticaldistance between adjacent crests and troughs and is called wave amplitude. A is also called the complexamplitude.

2

x axis (meters)

taxi

s(s

econ

ds)

-20 -10 0 10 200

2

4

6

8

10

12

14

16

18

20

0

−2π

−4π

−6π

−8π

−10π

−12π−14π−16π−18π

2ππ

π

−π

−11π−13π−15π−17π−19π

−3π

−5π

−7π

−9π

Figure 1: Lines with constant phase in the x; t plane. Full lines have phase � = 2�n (wave crests) anddotted lines have phase � = (2n+ 1)� (wave troughs) where n is an integer number.

Clearly @�@x

represents the change in phase � per unit distance, i.e., the rate of change of the phase � withdistance, at a given instant. It is called the wavenumber

wavenumber = k =@�

@x; (4)

3

which is related to the wave length as

wave length = � =2�

k: (5)

The wave length is de�ned as the distance between adjacent crests, or the distance between phase lineswith di�erence of 2�. The wavenumber k can be de�ned in terms of the wave length � as the number ofwave lengths in a distance of 2�. In �gure 1 the distance (x axis) between adjacent crests (troughs) is10 meters, which gives a wave length of 10 meters. Between neighboring crests the wave phase � changes2�, so the density of phase lines is 2�=10 radians per meter. In other words, the wavenumber k is 2�=10radians per meter.

At a given time instant, �gure 2 shows a snapshot of a sinusoidal wave train. We do not know thewave number k for this wave train, but we can evaluate it from this record. We can choose a crest, locatedat x1. The next crest is located, say, at x2. As we travel from x1 to x2, � changed exactly 2� (from onecrest to the next crest, the wave phase � change is always 2�), so the wavenumber is k = 2�=(x2 � x1)radians per meter.

4

x axis (meters)

wa

vee

leva

tio

n(m

ete

rs)

0 5 10 15 20

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Sinusoidal wave trainx1 = 5 x2 = 15

2A

Figure 2: Snapshot of a sinusoidal wave train. x1 = 5 meters and x2 = 15 meter, so the wavenumber isk = 2�=10.

On the other hand, �@�@t

represents the change in phase � at a �xed location x per unit time, i.e., thetime rate of change of the phase � at a �xed location x. It is called the wave frequency.

wave frequency = ! = �@�

@t; (6)

and it is related to the wave period as

wave period = T =2�

!: (7)

The wave period is de�ned as the time for two adjacent crests to pass a �xed location x, or the time fortwo phase lines with a di�erence in phase of 2� to pass a �xed x location. In terms of the wave period

5

T , the wave frequency ! is de�ned as the number of wave periods T in a time of 2�. We can see fromthe t axis of �gure 1 that the time for adjacent crests to pass, for example, the location x = �20 is2:53 seconds. This implies that the change of phase at the location x = �20 per unit of time is 2�=2:53radians per second. In other words, the wave train with crests and troughs illustrated in �gure 1 haswave period of 2:53 seconds and wave frequency of 2�=2:53 radians per second.

To stay with a particular line of constant phase, say a crest, one must have

d� = kdx� !dt = 0;

which implies that one must move at the phase velocity,

c =dx

dtj�= constant =

!

k: (8)

In �gure 1, the slope of the lines with constant phase is the phase speed c. This is consistent with equation(3), according to which the equation for the lines of constant phase is t = �=! � cx.

The wave frequency was de�ned as the time rate of change of the phase � at a �xed location x. Toillustrate, we generate an animation of a sinusoidal wave train, where we follow a particular wave crest(In the animation, we put a symbol over this particular wave crest). At the beginning of the animation,the wave crest next to the marked crest is exactly at the right end of the movie frame, which is the �xed xlocation. The animation shows the marked crest traveling towards the right end of the animation frame,and it stops when the marked crest reaches there. The elapsed time is t = 2:53 seconds, and during thistime, the value of the wave phase at the right end of the frame changed 2� (phase di�erence betweenadjacent crests). The wave frequency is then 2� divided by 2:53 seconds. To see the animation just clickhere.

In the animation the marked crest travels at a constant speed which is the phase velocity. To evaluatethe phase velocity we just need to divide the distance traveled by the marked crest by the elapsed time(2:53 seconds). The distance traveled is just the distance between adjacent crests, or the distance betweenphase lines with di�erence of 2�, which was de�ned as the wave length. Then, the phase velocity is justthe ratio �

T= !

k= 3:95 meters per seconds.

For a sinusoidal train represented by equation (1) to satisfy the appropriate equation of motion, itturns out that the wave frequency and the wave number must satisfy a dispersion relation. Examples oflinear governing equations and their dispersion relations are given below.

6

Table 1: Examples of Linear Partial Di�erential Equations and their Dispersion relations.

Equation Dispersion relation

�tt � c2r2� = 0 ! = ck

�tt � �2r2� � �2r2�tt = 0 ! = � �kp1+�2k2

�t + ��x + �xxx = 0 ! = �k � k3

The �rst equation in table 1 is the wave equation, which occurs, for example, in acoustics and in theshallow-water limit for water waves. In the context of acoustics, � is the pressure �eld and c is the soundspeed, and in the context of water waves in shallow water, � is the free surface elevation and c =

pgh,

where g is the gravity acceleration and h is the water depth. The second equation in table 1 appears inwater waves in the Boussinesq approximation for long waves with respect to the water depth. The thirdequation in table 1 applies for long water waves propagating in one direction; it is the linearized form of theKorteweg-de Vries equation. For these two last equations, the independent variable � is the free surfaceelevation (wave elevation), � =

pgh, �2 = h2=3 and =

pghh2=6, where g is the gravity acceleration

and h is the mean water depth. In the second column of table 1 we have the dispersion relation forthe equations mentioned above. What the last two equations have in common is that their dispersion

relation is a nonlinear function of the wavenumber (the wave frequency ! is a nonlinear function of thewavenumber k). For sinusoidal wave trains with nonlinear dispersion relation, the phase velocity is notconstant, it depends on the wavelength. In general, wave trains with phase velocity dependent on thewavenumber are called dispersive waves. The �rst equation in table 1 has a dispersion relation whichgives the wave frequency as a linear function of the wavenumber. The phase velocity of sinusoidal wavetrains satisfying this equation do not depend on the wavenumber, it is constant and equal to c. Wavessatisfying the �rst equation of table 1 are non-dispersive waves.

3 Superposition of plane waves.

An interesting physical feature for dispersive waves in general can be found by superposing two trainswith slightly di�erent frequencies and wave numbers:

�(x; t) = A�ei(k

+x�!+t) + ei(k�x�!�t)

�(9)

7

where

k� = k �4k with 4k =k+ � k�

2;4k� k: (10)

We also de�ne

!� = !(k�); ! =!(k+) + !(k�)

2and 4! =

!(k+)� !(k�)

2(11)

Now we can write the superposition of two trains (9) in the form

�(x; t) = A(x; t)ei(kx�!t) (12)

where

A(x; t) = 2 cos(4kx�4!t) (13)

The factor exp(ikx� i!t) is called the wave carrier and A(x; t) is called the envelope. Thus, the result isa sinusoidal wave train with a periodic envelope which has a very long wavelength 2�=4k � 2�=k andmoves at the velocity 4!=4k. In the limit 4k! 0,

4!

4k! Cg =

d!

dk(14)

where Cg is de�ned as the group velocity, which is in general di�erent from the phase velocity for dispersivewaves. The wave carrier has wave length 2�=k and period 2�=!, which are its characteristic length andtime scales. According to equation (10), the wavenumber k and the wave frequency ! are, respectively,the average values of the wave numbers k� and of the wave frequencies !�. On the other hand, thetime and length scales for the wave envelope are, respectively, its period 2�=4! and its wave length2�=4k. We de�ne the quantity � as the ratio between the wave carrier and envelope length scales. Forthe superposition of two wave trains we have

� =4k

k: (15)

The quantity 1=� provides an estimate of the number of crests (troughs) of the carrier wave in one wavelength of the envelope. If we approximate !� by

8

!� = !(k)� d!

dk4k +O(4k2) (16)

the envelope A(x; t) is approximated by the expression

A(x; t) = 2 cos(4k(x� (d!=dk)t)) (17)

This means that in the limit 4k ! 0, the envelope A(x; t) moves with the group velocity Cg.To illustrate the concepts presented in this section, we have animations of the superposition of two

wave trains. For the animations which follow, the carrier wave has wave number 2�=10 radians permeter. For the �rst animation, the envelope wave number is 2�=100 radians per meter and the value ofthe quantity � is 1=10. This means that we expect to see in the �rst animation around 10 crests (troughs)of the wave carrier in one wave length of the wave envelope. In this animation, we show the di�erencebetween the velocity of the carrier wave and the velocity of the envelope for dispersive waves. We mark aparticular crest of the carrier wave and see how it propagates with respect of the nodes of the envelope.We use the �nite depth water waves dispersion relation

!(k) =pgk tanh(kh); (18)

where g is the gravity acceleration (g = 9:8 meters per second) and h is the water depth (h = 159:15meters). For the dispersion relation (18), the phase velocity is larger than the group velocity, so we willsee the marked crest of the carrier wave approaching a node of the wave envelope.

In the second animation we demonstrate that the envelope travels with a velocity close to the groupvelocity at the wavenumber k. We consider now 4k equal to 2�=50. So the quantity � is 1=5, and weexpect to see around 5 crests (troughs) of the carrier wave in one wave length of the wave envelope. Thisanimation has two frames. The top frame shows the animation of the superposition of wave trains withthe envelope A(x; t) given by equation (13), and the bottom frame shows the superposition of wave trainswith envelope A(x; t) given by equation (17). The envelope in the bottom frame travels with the groupvelocity Cg at the wavenumber k, and the envelope at the top frame travels with velocity 4!=4k. Ineach frame we mark a particular wave envelope node, which in both frames starts at the same position.The marked node in each frame travels a distance approximately of 490 meters. The elapsed time is 246:4seconds. After an elapsed time of around 100 seconds, it is possible to notice that the wave record startsto be di�erent from one frame to the other. The marked nodes also start move apart from each other.This becomes clear at the last frame of the second animation, when the marked node in the top frameis at a position greater than 500 meters, but the marked node at the bottom frame is still at a positionsmaller than 500 meters.

9

To evaluate the di�erence in position between the marked envelope node in the top and bottom framesat the time instant of the last frame of the second animation (t = 246:4 seconds), we need to compute thedi�erence between the group velocity Cg and the exact envelope velocity 4!=4k. The wave trains have,respectively, wave numbers 2�=10�2�=50 radians per meter and 2�=10+2�=50 radians per meter. Fromthe dispersion relation (18) we obtain the corresponding wave frequencies, such that we can compute theenvelope speed

4!

4k= 1:9847 meters per second.

The approximate envelope A(x; t), given by equation (17), travels with the group velocity Cg at thewavenumber 2�=10 radians per meter. For the dispersion relation (18), the group velocity is given by theequation

Cg =g

4!(k)

�sinh(2kh) + 2kh

cosh2(kh)

�; (19)

and at the wavenumber 2�=10 the group speed is

Cg = 1:9747 meters per second.

The speed di�erence is 0:01 meters per second. Therefore, the marked envelope nodes will have a di�erencein position around 2:464 meters after an elapsed time of 246:4 seconds.

In summary, the second animation shows that for a short time or space scale (order of the envelopeperiod or envelope wave length) the approximation of the exact envelope speed by the group velocity ofthe carrier wave is a good approximation, but for time or space scale of the order of O(1=�) wave periodsor wave lengths of the wave envelope, this approximation gives an error in the position of the envelopeof the order of a wave length of the carrier wave.

4 Dispersion of a localized initial disturbance.

For dispersive medium, the solution for monochromatic waves already shows that waves of di�erentwavelengths move at di�erent velocities. Now the question is: What is the consequence of dispersion fora general initial condition? Initial conditions that are locally con�ned can be represented by a Fourierintegral, which amounts to a sum of in�nitely many sinusoids with a wide spectrum, so we are going toemploy the tools of the Fourier transform.

As in previous sections, �(x; t) represents the wave \elevation". We consider a locally con�ned initialcondition for the wave \elevation" in the form of a wave pulse with a Gaussian envelope:

10

�(x; 0) = f(x) =1

2�exp(�x2

2�2) cos(~kx) (20)

and with initial velocity

�t(x; 0) = g(x): (21)

The form of the dispersion relation ! = !(k) follows from the linear partial di�erential equation governingthe wave motion, as illustrated in table 1. The function g(x) is not speci�ed yet, but it will be chosensuch that �(x; t) is a right-going wave packet. The width of the wave packet given by equation (20) isof the order of O(�=�) and � is the wave length of the carrier wave. Let us de�ne the Fourier transformand its inverse by

u(k) =

Z +1

�1

e�ikxu(x)dx; u(x) =1

2�

Z +1

�1

eikxu(k)dk (22)

For any wave motion governed by a linear partial di�erential equation, the wave \elevation" at x; t canbe represented in terms of Fourier integrals

�(x; t) = <�

1

2�

Z +1

�1

�1(k)eikx+i!(k)tdk +

1

2�

Z +1

�1

�2(k)eikx�i!(k)tdk

�: (23)

The �rst and second integrals in the right hand side of the equation (23) represent waves propagatingalong the x axis. Each integral corresponds to a superposition of sinusoidal wave trains over the entirerange of wave numbers. Within the small range (k; k + dk) the wave amplitude in the �rst (second)integral is �1(�2). The functions �j(j = 1; 2) are called the Fourier amplitude spectrum for the waves

represented by the �rst (j = 1) and second (j = 2) integrals. The amplitudes �j are obtained from theinitial conditions (20) and (21).

The time rate of change of the wave \elevation" (the vertical velocity) is just the time derivative ofthe right hand side of the equation (23), given by

�t(x; t) = <�

i

2�

Z +1

�1

�1(k)!(k)eikx+i!(k)tdk � i

2�

Z +1

�1

�2(k)!(k)eikx�i!(k)tdk

�; (24)

To determine �j we apply the initial conditions (20) and (21). Equations (23) and (24) at time t = 0give

11

f(x) =1

2�

Z +1

�1

[�1(k) + �2(k)]eikxdk (25)

and

g(x) =i

2�

Z +1

�1

[�1(k)� �2(k)]!(k)eikxdk (26)

We apply the Fourier transform to equations (25) and (26) and solve the two resulting equations, whichgive

�1 =ig(k)

2!(k)+

f(k)

2and �2 =

�ig(k)2!(k)

+f(k)

2(27)

In terms of the Fourier transform of the initial conditions (20) and (21), the wave \elevation" is nowgiven as

�(x; t) =1

2�<(Z +1

�1

"�ig(k)2!(k)

+f(k)

2

#eikx+i!(k)tdk +

Z +1

�1

"ig(k)

2!(k)+

f(k)

2

#eikx�i!(k)tdk

)(28)

We did not completely specify the initial conditions yet, but from the expression (28) we now specifyg(x) in terms of f(x) to obtain, for example, only the right going wave. The Fourier transform of thefunction f(x) is a superposition of two Gaussians

f(k) =1

2�p2�

(exp

�1

2

(k � ~k)2

�2

!+ exp

�1

2

(k + ~k)2

�2

!); (29)

with maxima at k = �~k. Clearly, � is the scale for the width of the wavenumber spectrum of the functionf(x). We de�ne

f1(k) =1

2�p2�

exp

�1

2

(k � ~k)2

�2

!(30)

12

and

f2(k) =1

2�p2�

exp

�1

2

(k + ~k)2

�2

!(31)

such that we can write equation (28) in the form

�(x; t) =1

2�<(Z +1

�1

"�ig(k)2!(k)

+f1(k)

2+

f2(k)

2

#eikx+i!(k)tdk

+

Z +1

�1

"ig(k)

2!(k)+

f1(k)

2+

f2(k)

2

#eikx�i!(k)tdk

) (32)

Notice that the wave packets

1

2�<(Z +1

�1

f1(k)

2eikx+i!(k)tdk

)(33)

and

1

2�<(Z +1

�1

f2(k)

2eikx�i!(k)tdk

)(34)

are left propagating wave packets. The �rst wave packet (equation 33) is a superposition of wave trainswith wavenumber ranging e�ectively from ~k � � to ~k + �. Each sinusoidal wave train

f1(k)

2eikx+i!(k)t (35)

with wavenumber in the range k + dk has amplitude f1(k)=2. The function f1(k) is non-zero mostly forpositive values of k, and the phase of this sinusoidal wave train implies that for positive values of k itpropagates to the left direction. Therefore, the sinusoidal wave trains contained in the wave packet givenby equation (33) propagate to the left direction, and their superposition is a left propagating wave packet.Similar reasoning shows that the wave packet (34) propagates to the left direction as well. We choosethe function g(k) so as to cancel out the left propagating components of the wave \elevation" (32). Tothis end, the function g(k) has to satisfy the equations

13

f1(k)

2+

f2(k)

2� i

g(k)

2!(k)= f2(k) (36)

and

f2(k)

2+

f1(k)

2+ i

g(k)

2!(k)= f1(k) (37)

The function g(k) which satis�es both equations (36) and (37) is

g(k) = �i!(k)(f1(k)� f2(k)) (38)

The function g(x) is the inverse Fourier transform of the function g(k), and is given by the equation

g(x) =1

2�<��iZ +1

�1

!(k)(f1(k)� f2(k))eikxdk

�(39)

Now the wave \elevation" �(x; t) is given by the equation

�(x; t) =1

2�<�Z +1

�1

f2(k)eikx+i!(k)tdk +

Z +1

�1

f1(k)eikx�i!(k)tdk

�(40)

It is easy to see that f2(�k) = f1(k) and if we assume the dispersion relation an even function of thewavenumber, we can write the equation (40) for the wave \elevation" in the form

�(x; t) =1

�<�Z +1

�1

f1(k)eikx�i!(k)tdk

�(41)

To illustrate the e�ects of dispersion in the evolution of the localized in space initial condition, we evaluatenumerically the Fourier integral in the equation (41) for the wave \elevation" �(x; t), but �rst, we need tospecify the form of the dispersion relation !(k) with respect to the wave number. We chose the dispersionrelation for waves on water of �nite depth

!(k) =pgk tanh(kh); (42)

14

where g is gravity acceleration and h is the water depth. According to this dispersion relation, the wavefrequency ! is a nonlinear function of the wavenumber k for almost all the range of kh. The exceptionis the limit kh� 1, where the dispersion relation reduces to

!(k) =pghk: (43)

For shallow water (kh � 1), water waves are non-dispersive, since in this limit the dispersion relationturns out to be a linear function of the wavenumber. For kh > O(1), tanh(kh)! 1, which implies that

!(k)!pgk (44)

and the wave frequency is a non-linear function of the wavenumber. For kh � O(1), water waves su�er aqualitative change as the parameter kh approaches the limit kh� 1, they become less and less dispersive.

For large values of the parameter kh (kh > O(1)), the time evolution of a space localized initialcondition (equation (32) with f(k) de�ned by equation (29)) shows that the localized wave packet losesits initial shape as time goes on. It disintegrates with time. For not too small values of �, the spectrumof the initial wave packet (see equation (41)) contains a not too narrow range of wave numbers around~k. Each sinusoidal wave train

f1(k)eikx�i!(k)t

with wavenumber in the range k + dk has amplitude f1(k), which is non-zero mostly for positive valuesof the wavenumber k (see equation (30)). For positive values of the wavenumber k, the phase of thissinusoidal wave train implies that it propagates to the right direction with group velocity

Cg =d!

dk(k) (45)

at the wavenumber k. Therefore, sinusoidal wave trains with di�erent wavenumber propagate at di�erentvelocities. Since the initial wave packet can be represented as a superposition of sinusoidal wave trainswith wave numbers ranging e�ectively from ~k � � to ~k + �, as time evolves, sinusoidal wave trains withsmaller group velocity are left behind. As a result, the broadth (interval in the x axis where the waveelevation is non-zero) of the wave packet increases with time, while the amplitude decreases with time.We illustrate these e�ects of dispersion with an animation generated from the numerical integration ofthe Fourier integral (41). This animation has three frames, and we set � = 0:1 and ~k = 2�=10 radiansper meter in equation (30). From frame to frame, we vary the value of ~kh. For the top, middle and

15

bottom frames we have, respectively, ~kh = 100, ~kh = 1=2 and ~kh = 0:05. In the top frame the e�ectsof dispersion are stronger, and in the bottom frame, the wave packet su�ered almost no change as timesevolves, since for ~kh = 0:05 we are in the shallow-water limit (water waves in shallow water are non-dispersive). In the middle frame the e�ects of dispersion are mild, but still visible. In each frame, thewave packet approximately travels with the group velocity at the wavenumber ~k. The group velocity forwater waves is given by the equation (47). For the top middle and bottom frames of the animation, thevalues of the group velocity at ~k = 2�=10 are, respectively, 1:9754 meters per second, 2:486 meters persecond and 0:882 meters per second.

In the limit � ! 0, the spectrum of the initial wave packet (equation (29)) reduces to pair of deltafunctions at the wave numbers �~k. In this limit, the wave packet reduces to a single sinusoidal wavetrain with wavenumber ~k. For � � 1 we can approximate the dispersion relation !(k) by

!(k) = !(�~k) + d!

dk(�~k)(k � ~k) +O((k � ~k)2) (46)

where for water waves in �nite depth,

Cg(k) =d!

dk=

g

4!(k)

�sinh(2kh) + 2kh

cosh2(kh)

�: (47)

In this limit, equation (40) simpli�es to

�(x; t) =1

2�<�exp(i!(�~k)t+ iCg(�~k)~kt)

Z +1

�1

f2(k)eik(x+Cg(�~k)t)dk

+ exp(�i!(~k)t+ iCg(~k)~kt)

Z +1

�1

f1(k)eik(x�Cg(~k)t)dk

�;

(48)

Notice that Cg(k) given by equation (47) is an odd function in the wavenumber k, so Cg(�~k) = �Cg(~k),and that the dispersion relation (42) is an even function in the wavenumber k. These observations andthe fact that the inverse Fourier integrals in equation (48) can be evaluated in closed form imply thatto this order of approximation for the dispersion relation, the wave \elevation" �(x; t) is given by theequation

�(x; t) =1

2�exp(��2

2[x� Cg(~k)t]

2) cos(~kx� !(~k)t) (49)

16

This result represents a wave packet whose envelope travels at the group speed Cg(~k) and the carrierwave travels at the phase speed !(~k)=~k. In the limit � ! 0, the breadth of this wave packet tends to 1,and equation (49) represents basically a sinusoidal wave train with wavenumber ~k, as expected.

We can consider higher-order terms in the expansion (46) of the dispersion relation. If we keepquadratic terms in the expansion of the dispersion relation, the group velocity is

Cg(k) = Cg(~k) +d2!

dk2(~k)(k � ~k): (50)

To the order of this approximation, the wave elevation �(x; t) does not have a closed form solution and thewave packet does not keep its shape as it propagates. Speci�cally, if we assume d2!

dk2(~k) positive (negative),

from the approximation (50) for the group velocity, a sinusoidal wave train with wavenumber ~k+4k hasa group speed larger (smaller) than Cg(~k) by

d2!

dk2(~k)4k; (51)

and a sinusoidal wave train with wavenumber ~k�4k has a group velocity smaller (larger) than Cg(~k) bythe same amount (51). This implies that the size of the wave packet grows symmetrically as it evolves.For initial conditions with non-symmetric, but narrow, spectrum similar results apply; quadratic termsin the wavenumber expansion of the dispersion relation cause the wave packet to atten as it propagates.

For the symmetrical wave packet given by equation (20), � = 2�=~k is the carrier wave length scaleand �=� may be taken as the characteristic scale of the wave packet. The ratio � between the wavecarrier length scale and the wave packet length scale then is

� = � (52)

For the symmetric wave packet given by equation (20), the ratio between the carrier wave length scaleand the wave packet length scale is of the same order of magnitude as the width of the wave packetspectrum. This follows from the Fourier analysis and it is not true only for our symmetric initial wavepacket, but also for narrow wave packets of more general form.

We would like to estimate the length and time scales associated with the e�ects of dispersion. Theinitial wave packet can be represented as a superposition of sinusoidal wave trains with wave numbersranging e�ectively from ~k � � to ~k + �, and the group velocity of wave trains with these wave numbersare

Cg(~k � �) = Cg(~k)� d2!

dk2(~k)� +O(�2) (53)

17

The time then for these wave trains to have a di�erence in propagation distance of the order of O(�) canbe estimated from the equation

hCg(~k + �)� Cg(~k � �)

i4t = �: (54)

Now, if based on equation (53), we obtain

4t � 1

2

�d2!

dk2

��1�

�!4t � O(

1

�) wave periods or 4t � O(

1

�) wave periods; (55)

and the distance for the dispersion e�ects of this order of magnitude to be evident is just

4x = Cg(~k)4t!4x � O(1

�) wave lengths (56)

For a change in the wave packet size of the order of O(�=�), we proceed as in the previous case. Wesubstitute � by �=� in the right side of equation (54) to obtain

4t � O(1

�2) wave periods, (57)

and the distance for dispersion e�ects of this order of magnitude to show up is

4x = Cg(~k)4t!4x � O(1

�2) wave lengths: (58)

Based on these estimates, as the spectrum of the wave packet becomes more narrow, it takes a longertime for dispersion to a�ect the wave packet. We illustrate this fact with a set of animations and pictures.First, we consider two animations, where we show the wave disturbance pro�le traveling in space as timeevolves. At each instant, we update the wave pro�le, so we see a traveling wave disturbance. E�ects ofdispersion are evident if we compare the initial wave pro�le with the wave pro�le at later times. Second,we consider a set of pictures where the vertical axis is the time coordinate and the horizontal axis isthe space coordinate. So, each line represents the wave pro�le at the correspondent instant given in thevertical axis. For the wave pro�le at t = 0 seconds, the vertical axis also gives the values of the waveamplitude. In these pictures, it is easy to notice the di�erence in group and phase velocities. If we followa crest line in time, we see that the crest (phase velocity) moves faster than the wave packet (group

18

velocity) itself. It is also clear that, as time increases, both the wave packet size and the number ofcarrier waves inside the wave packet increase.

Now, we describe in detail the two animations mentioned above. Each of these two animations hasfour frames. In the top frame we give the animation generated from the numerical evaluation of theFourier integral in equation (41) for �(x; t). In the second frame from above we give the animationgenerated from the result (48) for �(x; t), where we approximated the dispersion relation to �rst orderin the wavenumber. In the third frame from above (bottom frame), we give the animation generatedfrom the numerical evaluation of the Fourier integral in equation (41) for �(x; t), but with the dispersionrelation !(k) expanded to second order (third order) in the wavenumber with respect to ~k. The valuesof the parameters � and ~kh and of the wavenumber ~k used in the animations are given in table 2 below.

Table 2: Values for the parameters � and ~kh and for the wavenumber ~k.

Value of � Value of ~kh Wavenumber ~k (rad=m) Link for the animation.

0.1 100 2�=10 to see animation click here0.025 100 2�=10 to see animation click here

In the �rst animation mentioned above (� = 0:1), a change in the wave packet size of the order ofthe wave length of the carrier wave (10 meters) due to dispersive e�ects should be evident after the wavepacket propagates a distance of 10 wave lengths of the carrier wave, around 100 meters. In the secondanimation, dispersion e�ects shown in the previous animation are not evident, since for these e�ects tohappen, we need the wave packet to propagate a distance of the order of 40 wave lengths of the carrierwave (� = 0:025), and the distance displayed in the animation x-axis is only 232 meters.

Next, we display the set of pictures mentioned above. In the caption of each picture we a give a linkto the animation that corresponds to the generation of the displayed picture. It shows the wave pro�lesbeing plotted as time evolves. We have a total of eight pictures. For the wave packet in the �rst andsecond set of four pictures, the value of the parameters � and ~kh and of the wavenumber ~k is given,respectively, in the �rst and second lines of table 2.

In the �rst four set of pictures, we have a narrow wave packet (� = 0:1) a�ected by dispersion as itevolves in time. In �gure 3 we use the full dispersion relation. In �gure 4 we approximated the dispersionto �rst order in the wavenumber. The dispersion relation was expanded to �rst order in the wavenumberwith respect to the central wavenumber ~k. In �gures 5 and 6 the dispersion relation was approximated,respectively, to second and third order in the wavenumber.

19

x axis (meters)

time

(se

con

ds)

0 50 100 150 2000

10

20

30

40

50

60

70

80

90

100

110

120

Full Dispersion Relation

Figure 3: Wave packet evolution generated from the numerical evaluation of the Fourier integral inequation (41) for �(x; t). Full dispersion relation. Parameters: � = 0:1, ~kh = 100 and of the wavenumber~k = 2�=10. To see the animation which illustrate the generation of this picture, click here.

20

x axis (meters)

time

(se

con

ds)

0 50 100 150 2000

10

20

30

40

50

60

70

80

90

100

110

120

Dispersion Relation approximated to first order

Figure 4: Wave packet evolution generated from the result (48) for �(x; t), where we approximated thedispersion relation to �rst order in the wavenumber. No dispersion e�ects. Wave packet does not changewith time. Parameters: � = 0:1, ~kh = 100 and of the wavenumber ~k = 2�=10. To see the animationwhich illustrate the generation of this picture, click here.

21

x axis (meters)

time

(se

con

ds)

0 50 100 150 2000

10

20

30

40

50

60

70

80

90

100

110

120

Dispersion Relation Approximated to Second Order

Figure 5: Wave packet evolution generated from the numerical evaluation of the Fourier integral in equa-tion (41) for �(x; t), but with the dispersion relation !(k) expanded to second order in the wavenumberwith respect to ~k. Dispersion e�ects present, but notice the di�erence from �gure 4. Parameters: � = 0:1,~kh = 100 and of the wavenumber ~k = 2�=10. To see the animation which illustrate the generation of thispicture, click here.

22

x axis (meters)

time

(se

con

ds)

0 50 100 150 2000

10

20

30

40

50

60

70

80

90

100

110

120

Dispersion Relation Approximated to Third Order

Figure 6: Wave packet evolution generated from the numerical evaluation of the Fourier integral inequation (41) for �(x; t), but with the dispersion relation !(k) expanded to third order in the wavenumberwith respect to ~k. Dispersion e�ects present, but notice the di�erence from �gures 4 and 5. Parameters:� = 0:1, ~kh = 100 and of the wavenumber ~k = 2�=10. To see the animation which illustrate thegeneration of this picture, click here.

23

Next, we consider a broad wave packet (� = 0:025). We show a set of four pictures that illustratesthe evolution of this wave packet with time. The e�ects of the dispersion are very small. For e�ects ofthe order the wavelength of the carrier wave (order of 10 meters) to be evident, the wave packet has topropagate a distance of 40 wave lengths. In �gure 7 we show the evolution of this wave packet with thefull dispersion relation. In �gures 8, 9 and 10, we show the evolution for the wave packet with time, butwith the dispersion relation approximated, respectively, to �rst, second and third order.

24

x axis (meters)

time

(se

con

ds)

0 50 100 150 2000

10

20

30

40

50

60

70

80

90

100

110

120

Full Dispersion Relation

Figure 7: Wave packet evolution generated from the numerical evaluation of the Fourier integral in equa-tion (41) for �(x; t). Full dispersion relation. Parameters: � = 0:025, ~kh = 100 and of the wavenumber~k = 2�=10. To see the animation which illustrate the generation of this picture, click here.

25

x axis (meters)

time

(se

con

ds)

0 50 100 150 2000

10

20

30

40

50

60

70

80

90

100

110

120

Dispersion Relation approximated to first order

Figure 8: Wave packet evolution generated from the result (48) for �(x; t), where we approximated thedispersion relation to �rst order in the wavenumber. No dispersion e�ects. Wave packet does not changewith time. Almost no di�erence between this picture and �gure 7. Parameters: � = 0:025, ~kh = 100 andof the wavenumber ~k = 2�=10. To see the animation which illustrate the generation of this picture, clickhere.

26

x axis (meters)

time

(se

con

ds)

0 50 100 150 2000

10

20

30

40

50

60

70

80

90

100

110

120

Dispersion Relation Approximated to Second Order

Figure 9: Wave packet evolution generated from the numerical evaluation of the Fourier integral in equa-tion (41) for �(x; t), but with the dispersion relation !(k) expanded to second order in the wavenumberwith respect to ~k. Almost no di�erence from �gure 4. Parameters: � = 0:025, ~kh = 100 and of thewavenumber ~k = 2�=10. To see the animation which illustrate the generation of this picture, click here.

27

x axis (meters)

time

(se

con

ds)

0 50 100 150 2000

10

20

30

40

50

60

70

80

90

100

110

120

Dispersion Relation Approximated to Third Order

Figure 10: Wave packet evolution generated from the numerical evaluation of the Fourier integral inequation (41) for �(x; t), but with the dispersion relation !(k) expanded to third order in the wavenumberwith respect to ~k. Little di�erence from previous �gures. Parameters: � = 0:025, ~kh = 100 and of thewavenumber ~k = 2�=10. To see the animation which illustrate the generation of this picture, click here.

28

To see a change in wave packet size of the order of one wave packet length scale, we need to displaythe wave packet after it had propagated a distance of the order of 1

�2wave lengths of the carrier wave. We

consider a set of animations and pictures showing dispersion e�ects of this order of magnitude. We usevalues for � displayed in table 3, but the x-axis in the movies and pictures frame runs from 0 meters to 300meters. The initial time is 0 seconds and the �nal time around 152 seconds. As before, we �rst considerone animation with four frames. In the top frame we give the animation generated from the numericalevaluation of the Fourier integral in equation (41) for �(x; t). In the second frame from above we give theanimation generated from the result (48) for �(x; t). In the third frame from above (bottom frame), wegive the animation generated from the numerical evaluation of the Fourier integral in equation (41) for�(x; t), but with the dispersion relation !(k) expanded to second order (third order) in the wavenumberwith respect to ~k. The values of the parameters used in the animations are given in table 3 below.

Table 3: Values for the parameters � and ~kh and for the wavenumber ~k.

Value of � Value of ~kh Wavenumber ~k (rad=m) Link for the animation.

0.2 100 2�=10 to see animation click here

Second, we consider a set of four pictures. The value of the parameters �, ~kh and ~k used is given intable 3.

In the set of four pictures, we have a narrow wave packet (� = 0:2). We display it until it hadpropagated a distance of the order of O( 1

�2) wave lengths of the carrier wave, and the change of its breath

is now of the order of O(��). In �gure 11 we use the full dispersion relation, and in �gures 12, 13 and 14

we approximated the dispersion relation, respectively, to �rst, second and third order in the wavenumber.The dispersion relation was expanded with respect to the central wavenumber ~k.

29

x axis (meters)

time

(sec

on

ds)

0 100 2000

20

40

60

80

100

120

140

Full Dispersion Relation

Figure 11: Wave packet evolution generated from the numerical evaluation of the Fourier integral inequation (41) for �(x; t). Full dispersion relation. Parameters: � = 0:2, ~kh = 100 and of the wavenumber~k = 2�=10. To see the animation which illustrate the generation of this picture, click here.

30

x axis (meters)

time

(sec

on

ds)

0 50 100 150 200 2500

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Dispersion Relation Approximated to First Order

Figure 12: Wave packet evolution generated from the result (48) for �(x; t), where we approximated thedispersion relation to �rst order in the wavenumber. No dispersion e�ects. Wave packet does not changewith time. Parameters: � = 0:2, ~kh = 100 and of the wavenumber ~k = 2�=10. To see the animationwhich illustrate the generation of this picture, click here.

31

x axis (meters)

time

(sec

on

ds)

0 50 100 150 200 2500

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Dispersion Relation Approximated to Second Order

Figure 13: Wave packet evolution generated from the numerical evaluation of the Fourier integral inequation (41) for �(x; t), but with the dispersion relation !(k) expanded to second order in the wavenum-ber with respect to ~k. Dispersion e�ects present, but notice the di�erence from �gure 12. Parameters:� = 0:2, ~kh = 100 and of the wavenumber ~k = 2�=10. To see the animation which illustrate thegeneration of this picture, click here.

32

x axis (meters)

time

(sec

on

ds)

0 50 100 150 200 2500

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

Dispersion Relation Approximated to Third Order

Figure 14: Wave packet evolution generated from the numerical evaluation of the Fourier integral inequation (41) for �(x; t), but with the dispersion relation !(k) expanded to third order in the wavenumberwith respect to ~k. Dispersion e�ects present, but notice the di�erence from �gures 12 and 13. Parameters:� = 0:2, ~kh = 100 and of the wavenumber ~k = 2�=10. To see the animation which illustrate the generationof this picture, click here.

33