Towards the Kelvin wake and beyondAndrej Likar and Nada Razpet Citation: Am. J. Phys. 81, 245 (2013); doi: 10.1119/1.4793510 View online: http://dx.doi.org/10.1119/1.4793510 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v81/i4 Published by the American Association of Physics Teachers Related ArticlesWhy do bubbles in Guinness sink? Am. J. Phys. 81, 88 (2013) The Electric Whirl in the 19th and 21st Centuries Phys. Teach. 50, 536 (2012) Beyond the Van Der Waals loop: What can be learned from simulating Lennard-Jones fluids inside the region ofphase coexistence Am. J. Phys. 80, 1099 (2012) The Enigma of the Aerofoil: Rival Theories in Aerodynamics, 1909–1930. Am. J. Phys. 80, 649 (2012) Simple, simpler, simplest: Spontaneous pattern formation in a commonplace system Am. J. Phys. 80, 578 (2012) Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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Towards the Kelvin wake and beyond

Andrej Likara)

Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, SI—1000 Ljubljana,Slovenia

Nada Razpetb)

Faculty of Education, University of Primorska, Cankarjeva 5, SI—6000 Koper, Slovenia

(Received 6 February 2012; accepted 12 February 2013)

The difference between wave propagation in dispersive and non-dispersive media can be

effectively demonstrated by observing the wave patterns invoked by uniformly moving surface

disturbances. Although the dispersion relation of surface waves on water is complicated, there are

some frequency intervals where the phase velocity of the waves reduces to the simple power law

behavior cp / xj. Among these cases are gravity waves short compared to the depth of the water

(j ¼ �1), short capillary waves (j ¼ 1=3), and long waves in shallow water (j � 0). Making use

of this power-law behavior, we vary the exponent and visualize the smooth transition from a non-

dispersive to a dispersive medium. VC 2013 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4793510]

I. INTRODUCTION

Waves on the surface of water have complex dispersionproperties. The dispersion relation between angular fre-quency x and wave number k is given by1–4

x2ðkÞ ¼ gk þ rk3

q

� �tanhðkhÞ ; (1)

where g is the gravitational field strength, r the surface ten-sion, q the density of water, and h the water depth. This is arather involved relation and is seldom appropriate fordetailed treatment in undergraduate courses. The expressionfor group velocity cg ¼ @x=@k is even more involved, and itis difficult to gain any physical insight from such compli-cated equations. Figure 1 shows the dispersion relation ofEq. (1) in the more convenient form of phase and groupvelocities as a function of frequency for water with a depthof one meter. The complexity of the relation is clearly seen.However, there are three frequency intervals where the dis-persion relation reduces to a simple power law so that thephase velocity cp ¼ x=k and group velocity are given by

cp ¼ c0

xx0

� �j

; (2)

cg ¼cp

1� ðx=cpÞðdcp=dxÞ ¼cp

1� j; (3)

where c0 and x0 are constants. (Usually x0 is chosen to be afree parameter while c0 depends on the specific limitingcase.) The limiting cases with j ¼ 0, �1, and 1/3 are indi-cated in Fig. 1.

For gravity waves short compared to the depth of thewater, we take limh!1 tanhðkhÞ ¼ 1 and neglect capillarywaves (r ¼ 0) to find

x2 ¼ gk; or cp ¼ffiffiffig

k

r¼ g

x: (4)

In this case, we have c0x0 ¼ g. Such waves stretch behindmoving objects and form the Kelvin wake, as shown in Fig. 2.

One striking feature of the Kelvin wake is that the wake(half-)angle HK is independent of the object’s velocity.

For capillary waves in deep water, we neglect gk in com-parison with rk3=q to get

x2 ¼ rk3

q; (5)

giving a phase velocity of

cp ¼ffiffiffiffiffirk

q

s¼ xr

q

� �1=3

: (6)

In this case, we have c30=x0 ¼ r=q. The capillary waves in

front of an obstacle in a stream of water appear as a station-ary wave pattern; they can be used to demonstrate interfer-ence phenomena from two sources, as shown in Fig. 3.

Finally, for a water depth that is shallow compared to theobserved wavelengths, we write tanhðkhÞ � kh and neglectcapillary waves to get

cp ¼ffiffiffiffiffigh

p; (7)

so that j ¼ 0 (and cg ¼ cp).As is evident from Fig. 1, these three regimes apply to cer-

tain clearly visible frequency intervals. In order to discussdispersion phenomena, we study the wakes produced by a

Fig. 1. The dispersion relation of Eq. (1) is used to plot the phase cpðxÞ and

group cgðxÞ velocities as a function of frequency for water with a depth of

h¼ 1 m. The frequency intervals where Eqs. (2) and (3) (approximately)

hold with j ¼ 0, �1, and 1/3 are indicated.

245 Am. J. Phys. 81 (4), April 2013 http://aapt.org/ajp VC 2013 American Association of Physics Teachers 245

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uniformly moving disturbance on an idealized surface wherethe simplified dispersion relation cp ¼ c0ðx=x0Þj holds forany frequency x. This simplification has the obvious advant-age that we can study wakes for any value of j and is espe-cially attractive for j � 0 because changing the sign of jchanges the relation between phase and group velocity fromcg < cp for j < 0 to cg > cp for j > 0. Additionally, byvarying j towards j ¼ �1 and beyond we can elucidate theformation of the celebrated Kelvin wake.

In this paper, we show that a wake can be seen as an inter-ference pattern of waves traveling in different directions.This result is found by applying the Hamiltonian method inthe limit of large times to calculate the wakes. The resultingpattern can then be visualized by any program that is capableof displaying a two-dimensional data field. One such pro-gram, useful for many activities in teaching physics andmathematics, is DERIVE.5

The remainder of the paper is organized as follows. InSec. II, we present the Hamiltonian method and then discussthese results in Sec. III. In Sec. IV, we present results forwakes in non-dispersive and weakly-dispersive media, whilein Sec. V, we describe a gradual transition towards stronglydispersive waves. Finally, we provide some brief conclusionsin Sec. VI.

II. THE HAMILTONIAN APPROACH TO THE

CALCULATION OF WAKES

Recently, it was demonstrated that the Hamiltonianmethod can be elegantly used to calculate the wave fields of

Cherenkov radiation6 and an oscillating electrical dipole.7,9

The essence of the method in our case is to expand the wavefield in harmonic functions on a large square area of sidelength L. We consider linear and purely transverse waves tokeep the discussion as simple as possible. For the case of anon-dispersive medium, the wave equation for a field u sub-ject to a driving force f is given by

@2u

@t2¼ c2r2uþ f ð~r; tÞ; (8)

where c is the speed of the waves. The amplitude of the driv-ing force is of no interest here; for an infinitesimally “thin”disturbance moving with speed v in the y-direction, it isapproximated by

f ð~r; tÞ / dðxÞdðy� vtÞ; (9)

where d is the Dirac delta function.We expand the wave field u in a Fourier series

uð~r; tÞ ¼Xk;i

qkiðtÞAkið~rÞ; (10)

where the expansion coefficients qkiðtÞ are functions of time,and the harmonic functions Aki (for i¼ 1, 2) are given by

Ak1 ¼ffiffiffi2p

Lcosð~kk �~rÞ; Ak2 ¼

ffiffiffi2p

Lsinð~kk �~rÞ: (11)

These functions form a complete orthogonal set for wavevectors ~kk that satisfy the so-called square quantization

~kk ¼2pL

nx;2pL

ny

� �T

; (12)

with ðnx; nyÞ integers. The factorffiffiffi2p

=L in Eq. (11) representsthe appropriate normalization so that the orthogonality con-dition is expressed asð

S

Aki Alj dS ¼ dkldij; (13)

where S indicates integration over the two-dimensionalsquare.

The equations for qki are obtained from the wave equation(8). The Laplace operator r2 acting on the eigen-amplitudesAki gives

r2Aki ¼ �k2kAki; (14)

allowing us to write Eq. (8) asXk;i

qkic2k2

kAki þXk;i

€qkiAki ¼ f ð~r; tÞ: (15)

Multiplying both sides of this equation by Alj and integratingover the square, we get

€qlj þ x2lqlj ¼

ðS

f ð~r; tÞAlj dS; (16)

where x2l ¼ c2k2

l. Using our simplified [see Eq. (9)] andproperly normalized driving function f ð~r; tÞ we finally obtainFig. 3. An interference pattern that results from two capillary wave fronts.

Fig. 2. The wake behind a duck is an example of short (compared to water

depth) gravity waves. The wake (half-)angle HK is indicated.

246 Am. J. Phys., Vol. 81, No. 4, April 2013 Andrej Likar and Nada Razpet 246

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€ql1 þ x2lql1 ¼ cosðkyvtÞ; (17)

€ql2 þ x2lql2 ¼ sinðkyvtÞ: (18)

These equations represent driven harmonic oscillatorswith natural frequencies xl. The solutions are given by

ql1 ¼cos xpt� cos xlt

x2l � x2

p

(19)

and

ql2 ¼xl sin xpt� xp sin xlt

xlðx2l � x2

pÞ; (20)

where xp is given by

xp ¼xl

cv cos#; (21)

# being the angle between, say, the y-axis and the wave vec-tor ~kl.

Up until now, the wave velocity c was considered constantfor all modes. We can introduce dispersion into theHamiltonian method by assuming different phase velocitiescl for different modes l (see Ref. 8). We therefore introducethe dispersion relation

kl ¼xl

cl; (22)

with cl ¼ clðxlÞ.The wave field u is now given by summing over all modes

l to get

u /X

l

hql1 cosð~kl �~rÞ þ ql2 sinð~kl �~rÞ

i: (23)

The mode l is defined by integers nx and ny so the numberof modes in the intervals Dnx and Dny is simply DnxDny.Expressing DnxDny with intervals of frequencies Dxl andinterval of angles D# of ~kl, we obtain the number of modeswithin the interval as

dNl /L2xl

c2l

DxlD#: (24)

Instead of summing over the modes we can therefore inte-grate, knowing that the number of modes per frequencyinterval dxl and angle interval d# is

dNl /L2xl

c2l

dxld#: (25)

We therefore have

u /ð1

0

xl

c2l

dxl

ðp=2

0

d# ½ql1 cosð~kl �~rÞ þ ql2 sinð~kl �~rÞ�:

(26)

For our purposes, we can drastically simplify this expression.The structure of both ql1 and ql2 suggests that the main

contribution to the integral will come from the pole atxl � xp. We can take advantage of this feature by writingql1 and ql2 in a more useful way as

ql1 ¼ �2

x2l

sin½ðxlt=2Þð1þ bl cos#Þ�1þ bl cos#

�sin½ðxlt=2Þð1� bl cos#Þ�

1� bl cos#; (27)

ql2 ¼2

x2l

cos½ðxlt=2Þð1þ bl cos#Þ�1þ bl cos#

�sin½ðxlt=2Þð1� bl cos#Þ�

1� bl cos#; (28)

where bl ¼ v=cl. A common factor in both terms of theabove equations is

sin½ðxlt=2Þð1� bl cos#Þ�1� bl cos#

; (29)

which, for large values of t, tends to dð1� bl cos#Þp=2.Integrating over # then leads to

u /ð

dxl1

xlc2l sin#l

sinðxlt� ~kl �~rÞ; (30)

where the propagation angle is defined by cos#l ¼ 1=bl.Note that the integration here extends only over frequenciesfor which cl=v < 1 because the propagation angle obeys

cos#l ¼1

bl¼ cl

v(31)

and ~kl is given by (see Fig. 4)

~kl ¼xl

cl½sin#l; cos#l�T : (32)

A comment is in order here. Because the common factorshown in Eq. (29) fails to tend to a delta function for very

Fig. 4. Propagation of partial wave with frequency xl and corresponding

phase velocity cl forming propagation angle #l with the direction of motion;

the propagation angle obeys cos#l ¼ cl=v. The wave is represented by the

line perpendicular to ~kl where the phase xlt� ~kl �~r ¼ 0:

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low frequencies xl � 0, the wave field calculated from Eq.(30) may, in some cases, be correct only up to an additiveconstant. But we know that the field far from the disturbancevanishes so the constant can be trivially determined. Tomimic real situations, the frequency interval over which theintegral in Eq. (30) extends should include the intervalswhere the simplified dispersion relation with specific j isvalid.

We stated that wakes can serve as an intelligible and self-evident demonstration of wave dispersion. To show this ex-plicitly, we introduce the coordinates a and b along andacross the wake (see Fig. 5) and express the partial wavessinðxlt� ~k �~rÞ in Eq. (30) in terms of a and b. To be clear,we assume weak dispersion with j � 0; cl � c0, andcos#0 ¼ c0=v. The wave field u can then be written as

uðs; bÞ /ð

dxl

xlsin

xl

c0

½sðcl � c0Þ � b�� �

; (33)

where s ¼ a=ðc0 tan#0Þ is the time needed for non-dispersive waves to travel from point A to point B (see Fig.5).

This result is precisely what one expects to find for a wavefield in a one-dimensional dispersive medium cl ¼ clðxlÞwith an initial excitation in the form of step function. Indeed,for a non-dispersive medium with cl ¼ c0 we get

uðs; bÞ / �ð

dxl

xlsin

xl

c0

b

� �/ �HðbÞ þ 1

2; (34)

where HðbÞ is the unit step function (with step at b ¼ 0). Inthe case of stronger dispersion (say j ¼ �1), the qualitativepicture remains the same. The wake can, therefore, be inter-preted as a set of u(t, x) diagrams taken along the coordinatea which we usually present in the case of a one-dimensionaldispersive medium.

III. KELVIN WAKE AND CAPILLARY WAVES

We calculate the wave pattern in the Kelvin wake usingEq. (30). In the limit of gravity waves in deep water, thephase velocity is given by

cl ¼g

xl: (35)

Propagation wave angles #l are now

cos#l ¼g

vxl; (36)

and wave vectors ~kl are given by

~kl ¼xl

cl½sin#l; cos#l�T ¼

g

v2cos2 #l½sin#l; cos#l�T :

(37)

The resulting wakes for two different velocities v areshown in Fig. 6. Here, the results are plotted using normal-ized coordinates

n ¼ x

vtand g ¼ y

vt(38)

so that the disturbance is located at n ¼ 0 and g ¼ 1.We repeat the calculation of the wave pattern for capillary

waves as well. In Fig. 7, we show a typical pattern obtainedusing Eq. (30) and an actual photograph of ripples in front ofan obstacle in flowing water. The patterns show good quali-tative agreement.

Fig. 5. The coordinate a along the wake is proportional to the time s it takes

the partial waves to travel from point A to point B. The coordinate bexpresses the section across the wake.

Fig. 6. Calculated wakes for j ¼ �1 behind a moving disturbance with ve-

locity v (top) and 2 v (bottom).

248 Am. J. Phys., Vol. 81, No. 4, April 2013 Andrej Likar and Nada Razpet 248

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IV. DISPERSIVE MEDIA WITH j CLOSE TO ZERO

Gravity waves as well as capillary waves in deep watershow very strong dispersion. It is not easy to demonstratethe spread of the wave packet and to show convincinglythat longer waves are faster than shorter ones. As men-tioned, the non-dispersive case with j ¼ 0 can serve as areference point to show the qualitative difference in wave

propagation between different dispersive media. We there-fore calculate the non-dispersive wake exactly using theGreen’s function technique, where the driving force is

fGð~r; tÞ / dðx� x0Þdðy� y0Þdðt� t0Þ: (39)

The Green’s function G is easily obtained from theHamiltonian approach:

Gðx� x0;y� y0; t� t0Þ ¼(fc2ðt� t0Þ2� ½ðx� x0Þ2þðy� y0Þ2�g�1=2

for ðx� x0Þ2þðy� y0Þ2 < c2ðt� t0Þ2; t� t0

0 for ðx� x0Þ2þðy� y0Þ2 > c2ðt� t0Þ2; t< t0:(40)

The response to a sliding force f / dðxÞdðy� vtÞ is thenobtained by noting that f ¼

ÐfG dt0 with x0 ¼ 0 and y0 ¼ vt0

so that

u /ðt

0

Gðx; y� vt0; t� t0Þ dt0: (41)

This integral can be solved and when written using our nor-malized coordinates n and g takes the form

u / arcsing� b�2 þ ðb�2 � 1Þsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b�2ðg� 1Þ2 � ð1� b�2Þn2q

su

sl

; (42)

where b ¼ v=c. The lower sl and upper su > sl limits definethe interval where the argument of the arcsin function isbelow 1.

The resulting field for a disturbance starting at the origin(n ¼ 0; g ¼ 0Þ with v¼ 2c is shown in Fig. 8. Notice that a“Cherenkov triangle,” analogous to the “Cherenkov cone” inthree dimensions, is formed. We see that the wake appears asa steep wall in the form of a wedge; this is similar to the sit-uation that is encountered in a “sonic boom.”

We now proceed to slightly dispersive media with j closeto zero. The situation for a medium with j ¼ 0:1 is shown inFig. 9 and for j ¼ �0:1 in Fig. 10. The steep wall, formed ina non-dispersive medium with j ¼ 0, now spreads into manyindividual wakes, each with a slightly different propagationangle #l. Note that in Fig. 9 the shorter waves precede thelonger waves, while in Fig. 10, the opposite is true.

Partial waves with different phase velocities and corre-sponding propagation angles #l for selected time t are shownschematically in Fig. 11. Again, zero phase lines representthe partial waves. Interference of all partial waves producesthe wake.

Because the dispersion relation for water waves involvesthe depth of the water h, one might think that by regulatingthe depth of water in a ripple tank,2,10 it would be possible todemonstrate all the features that our simplified relation withj � 0 offers. Unfortunately, very short capillary waves

Fig. 7. Capillary wake in front of a moving disturbance for j ¼ 1=3 (top)

and an actual photograph of the ripples in front of an obstacle in flowing

water (bottom). The qualitative agreement is quite good.

Fig. 8. The wake behind a moving disturbance in a non-dispersive medium

(j ¼ 0).

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cannot be avoided, although they can be reduced to someextent by lowering the surface tension r. Thus, the surface ofwater always shows some features of a medium with j > 0.

For waves in shallow water [where tanhðkhÞ � kh] the sur-face, apart of capillary waves, shows non-dispersive behav-ior in a rather broad frequency interval with a constant phasevelocity c0 ¼

ffiffiffiffiffighp

. The wake in this case is not the Kelvinwake as it shows a velocity dependence for the angle #0

given by

cos#0 ¼c0

v: (43)

To demonstrate this kind of wake in a single picture, anaccelerated disturbance (v¼ at) on 1-cm deep water wasobserved as shown in Fig. 12. Neglecting the capillary wavesin front of the moving vertical rod, the wake generated is evi-dently curved due to the changing velocity of the rod. Thepropagation angle #0 changes from #0 � 0 at the bottomsides of the picture to about #0f ¼ 60 near the position ofthe rod at the moment when the picture was taken. Sincec0 � 0:3 m/s in 1-cm deep water the final velocity of the rodis estimated to be

vf �c0

cosð#0f Þ� 0:6 m=s: (44)

It is not possible to prepare the water surface with a smallnegative j over a sufficiently large frequency interval.Therefore, to show dispersion phenomena convincingly tostudents one must rely on the simplified relation introducedabove.

Fig. 9. The wake in a slightly dispersive medium with j ¼ 0:1.

Fig. 10. The wake in a slightly dispersive medium with j ¼ �0:1.

Fig. 11. Schematic depiction of partial waves and corresponding propaga-

tion angles in a dispersive medium at time t. The first wave (at #l< for

xl < x0, long dashed line) is slower then the wave with phase velocity c0

and propagation angle #0 (solid line), while the second (at #l> for xl > x0,

short dashed line) and third (at #l for xl x0, dotted line) waves are

faster.

Fig. 12. The wake in shallow water with the depth of 1 cm shows a changing

propagation angle with an accelerated disturbance.

250 Am. J. Phys., Vol. 81, No. 4, April 2013 Andrej Likar and Nada Razpet 250

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V. TOWARDS THE KELVIN WAKE AND BEYOND

Having our simplified dispersion relation cl ¼ c0ðxl=x0Þjin mind and using our calculations for creating wave patterns,we can now change the power j from 0 to �3/2 and experi-ence an unusual journey towards the Kelvin wake as seen on adeep water surface; Fig. 13 shows the resulting wakes. The

values of j used to calculate the patterns are j ¼ �0:25;j ¼ �0:75, and j ¼ �1:5, in conjunction with the exactexpression [Eq. (26)]. In each of these plots the disturbancebegins at g ¼ 0 and travels the distance vt up to g ¼ 1. Weused the exact expression to calculate the field in order toshow the gradual formation of the wake due to the disturb-ance; using Eq. (30), as we did in Fig. 6, the transient at g � 0could not be seen. It is quite evident that with increasing dis-persion, the waves form wakes with sharper half-angle HK , aclear consequence of the decreasing group velocity comparedto the phase velocity:

cg ¼cl

1� j: (45)

To determine the angle of the Kelvin wake for arbitrary j,we consider a group of waves in the vicinity of some chosenxl and determine where this group would arrive starting fromthe origin at time t. From Fig. 14, it is clear that instead of arriv-ing at point W, a distance clt from the origin, the group insteadarrives at W0, a distance w ¼ ðcltÞ=ð1� jÞ from the origin.Using the geometry of Fig. 14, we find that the partial wakehalf-angle HKl for this selected group of waves is given by

tan HKl ¼w sin#l

vt� w cos#l(46)

or, noting that cos#l ¼ cl=v,

tan HKl ¼sin#l cos#l

1� j� cos2 #l: (47)

We see that HKl does not depend explicitly on either v or t.An important question is which value of HKl to take for

the Kelvin wake as a whole. A plot of Eq. (47) for, sayj ¼ �1, shows that HKl has a well defined maximum(defined as HK) as a function of #l, where the partial wavesobviously accumulate. Taking this value of #l to representthe Kelvin wake we find, by setting the derivative of Eq.(47) equal to zero, that

Fig. 13. Wakes for different values of j, gradually decreasing from j¼ �0:25 (top) down to j ¼ �1:5 (bottom).

Fig. 14. Group wake from a selected partial wave at #l. The group arrives at

W0, the phase at W.

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sin2 #l ¼j

2j� 1; (48)

and for the wake half-angle

HK ¼ arctan1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

jðj� 1Þ

s !: (49)

Using j ¼ �1, we find

HK ¼ arctan1ffiffiffi8p � 0:34 rad; (50)

which is the celebrated 19:5 as originally derived by LordKelvin in Ref. 11.

If we instead use j ¼ �1:5 we find

HK ¼ arctan1ffiffiffiffiffi15p � 0:25 rad; (51)

or a not-so-famous 14:5, but in good agreement with ourcalculated wake in Fig. 13(c).

From Fig. 13(a), we see that for j ¼ �0:25 the Kelvinwake is still well established. For very small j, this kind ofpattern loses its meaning because it only becomes fullydeveloped at longer and longer times as j! 0 and itbecomes less and less pronounced. The same is true forwakes behind very fast boats on deep water (j ¼ �1) wherevery long partial waves are generated. For j! 0, the Kelvinwake occurs at HK ¼ p=2. For positive j no such accumula-tion of waves exists, so the Kelvin wake cannot be observed.

VI. CONCLUSION

To conclude, we believe that the pedagogic treatment ofdispersive phenomena can be enriched by taking a closer

look at wakes behind moving disturbances. Waves on awater surface can initiate discussion among students, but at acertain point one has to switch to the simple dispersion rela-tion to keep the discussion clear and comprehensive. It is ourbelief that the simplification we propose here is attractiveand, at the same time, simple enough for teaching needs. Thesimple and intuitive expression for generating waves mayserve as a relevant exercise. For a more advanced studentproject, one may use the exact expressions given here andcompare the results obtained by the power law and by theoriginal dispersion relation.

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