On the Kramers-Kronig relations with special reference gravity waves

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KONINKL. NEDERL. AKADEMIE VAN WET ENSOHAPPEN

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Reprinted from Proceedings, Series B, 66, No. 5,

ON THE KRAMERS-KRONIG RELAT1ONS. WITH SPECIAL REFERENCE TO GRAVITY WAVES

BY

L. VAN WLJNCIAAIRDEN

(Communicated by Prof. W. P. A. VAN LAMMBREN at the meeting of Sept. 28, 1963)

SUMMARY

It has been argued by BROER [1], that the dispersion equation for a

progressive wave satisfies the Kramers-Kronig relations only when, apart from other conditions, the wave phenomenon involves a finite velocity of propagation.

In this paper it is shown that such a condition is not requisite. In the case of gravity waves on water of finite depth the K-K relations do not

hold. As this cannot be explained by Broer's argument, this _{case is}

reexamined here. It appears that the reason why the K-K relations are

not valid in this case, is that the dispersion equation comprises also all steady motions in the region between bottom and free surface. 1. The Krarners-Kronig rei ations

A wave propagating in x direction with frequency w/2v and wave

length 2r/k can be represented by

(1) _{exp i(kx-wt).}

The wave is dispersive when the phase velocity 0)/k is a function of w. A relation, giving k as a function of w is called a dispersion equation. The real part of k(w) (henceforth denoted by Re k(w)) 1etermines the phase, the imaginary part Im k(w) determines the attenuation of the_wave. When certain conditions are satisfied, Re k(w) and Im k(w) _{are not} independent, but correlated by relations known as Kramers-Kronig (K-K) relations.

These relations are essentially equivalent to Cauchy's Theorem, since

they express the following. Consider a complex _{w plane, where w = w + ice.}

The dispersion equation gives us k on the positive w axis.

Suppose that k has a continuation in the w plane, which is analytic without singularities in the whole upper half.

Then, for a point w0 in the upper half plane, we have by Cauchy's

Theorem Ic(wo) =

-2n

where the integration is along any closed contour in the upper half plane

around W.

Now we require further

k(w) -- O

for w

Choosing a contour consisting of the real axis and a large semi circle

around the origin, (2) reduces to

+00

1C(WO) = _.L

I

2,n

j

-00

since the integral along the semi circle vanishes on account of (3), when the radius tends to infinity.

Fig. 1. Integration in w plane.

The final step is to take w0 on the real axis. Then the integrand in has a pole in w = w. We circumvent this point as shown in fig. i and obtain, since the integral along the small semi circle around wo yields one half of the residu in wo,

+00

i

k(w0) ciw.

.1

ww0

-00

The stroke through the integration sign indicates that the Cauchy

Principal Value is meant.

Separating real and imaginary parts, leads to

+00

Re k(wo)=

4

+

Im /c()

dw.

'r

j

-The relations (5) and (6), born from Cauchy's -Theorem, have been baptized in different fields with appropriate names.

In MUsKHELIsIJvIIJ's book on singular integral equations [2] they appear as Plemelj formulae.

In physics, (5) and (6) are known as KramersKronig (KK) relations

(see LANDAU and LIFsHrrz [3]), because these physicists used a relation of this type to express the connection between phase and absorption of light.

In the theory of regulating systems a minimum phase network is a network with a transfer function, which has no singularities in some half plane. For such a network the relations (5) and (6) exist between the phase shift and attenuation characteristics (see BODE [4]).

The formulation in terms of network analysis _{was done by Bode} (see ref. [4], chapter 14). Workers in this field use accordingly the name Bode Theorems.

The KramersKronig relations are often associated with the principle

of causality, for the following reason (see LANDAU and LIFsHITz _{[3], or}

VAN KAMPEN [5]).

Take the case where k(w) is the Fourier transform of some phenomenon,

which we denote by K(t) and which is caused by an action which starts

at time t = O. Then the principle of causality tells us that K(t) -_ O for t<í).

Consider

+

k(w)

'

etK(t)dt.

-As w = w + i, we conclude that the right hand side of (7) _converges

for t> O in the upper half of the w plane, due to the factor expöct.

As K(t) - O for t< O, _{the integral converges also for t <O in the upper}

half plane and k(w) is therefore analytic in the upper half plane. When also the condition (3) is satisfied, the KK relations hold for Ic(w) and can be considered to reflect the principle of causality.

From (7) we deduce

k(w) =k(w),

where the bar over k(w) denotes the complex conjugate. Using (8), we can write (5) as

Re k(wo) = 2 j wlmk(w)

_j

It was in fact a relation of the type (9), which was used by Kramers and

Kronig (see VAN KAMPEN [5]).

2. On a paper by Broer

In a recent paper BROER [1] considered various kinds of waves and remarked that waves, displaying dispersion but no absorption, do not satisfy the KK relations. As examples of this type of waves, he mentioned transverse waves in an elastic beam and gravity waves on water of finite depth. In the former case the dispersion equation is

k = cco

where c is a constant determined by the properties of the beam, while in the latter case the dispersion equation reads

w2 =g k tanh (k h),

g being the acceleration of gravity and h the depth of the water.

Broer argued that, should a dispersion equation satisfy the KK

relations, a finite velocity must be involved. In both wave phenomena, the velocity of propagation is not finite. In the case of (10) the velocity of propagation co/k tends to infinity for k - oc.

In the case of gravity waves, Broer observed that the physical plane, the x, y-plane say, is occupied by an incompressible fluid, so that,

although there is in the x, t-plane wave propagation with a phase

velocity which is always less than (gh). there is no finite velocity in the sense that a disturbance starting at t=O in x= 0, y=0, is felt at a finite

distance from the origin after a time t1> 0.

Broer concluded that in the absence of a finite velocity, in the sense mentioned above, the KK relations do not hold for the wave motions associated with (10) and (11).

These conclusions, however, can be criticized. It is not clear why dispersive waves without absorption cannot satisfy the KK relations. Further the requirement of a finite velocity does not seem necessary. The first point appears to be a matter of definition. Professor Broer

informed the author that he had, in writing down this conclusion,

equation (9) in mind, whereas in general (see e.g. LANDAU and LIFSHITZ [3])

the KK relations are defined by (5) and (6).

The relation (9) is more restrictive, since it is necessary that also (8) holds, which is the case when k(w) is the Fourier transform of a real function. Concerning the requirement of a finite velocity, consider the

following wave motion. A flat plate of infinite breadth and width, executes

in its own plane, the y, z-plane say, smafl oscillations, while being immersed in a viscous incompressible fluid.

Let the velocity Vo of the plate be in y direction and given by

0=A exp (iwl), where A is the amplitude and w the angular velocity of the oscillation. Because the fluid has only a velocity component y in y

i

i

w

direction, independent of y and z, the Navier-Stokes equations reduce to

ÒV ò2v

being the kinematic viscosity of the fluid. Seeking a solution of the form (1), we obtain

The diffusion of vorticity from the plate into the fluid involves just as in the case of thermal conduction, no finite velocity. This is easilyverified

by considering

=2vk Rek(w)

which expression becomes infinite for k

-

We consider

k (w)

= (1 +i) (2vw) w

where k(w) has been divided by w in order to secure the behaviour at infinity. This is of no further consequence, because, when k(w) is analytic in the upper half plane, the same is, with the possible exception of the origin, true for k(w)/w.

The function (w) at the right hand side of (13) is except at the origin analytic without singularities (holomorfic). When we introduce a cut in the w plane along the negative axis and in integrating along the real axis avoid the origin by means of a small semi circle, then the conditions for the K-K relations are satisfied. Therefore (5) and (6) should hold here. On the real axis Im (1 +i)w ==Iw sgn w, Re (1 +i)w =IwL. We insert these expressions in the right hand sides of (5) and (6).

The integrals are readily evaluated by observing that these are Hubert

transforms.

With help of tables of integral transforms edited by ERDÉLYI [6J, we obtain

w'Sgn w'

= IwI,

j-

,dw'=jwJsgnw.

Hence the K-K relations are satisfied.

We observe that, since (8) holds for k(w)=(1 +i)w. also the relation (9) is valid for this case.

We have thus shown by means of a counter example that the conditions for the validity of the KK relations do not involve the requirement of a finite phase velocity.

This is also clear when thinking of the KK relations in terms of the principle of causality since this principle says that the effect cannot precede the cause in time.

It does not, however, forbid the effect to follow the cause instantaneously.

Using the same procedure it can be shown that also for the transverse elastic waves in a beam the KK relations are valid (when applied to

k(w)/w).

:3. Gravity wave,s

In this _{ecion we consider the propagation of gravity waves on an}

incompressible fluid of depth h. For further reference it is useful to give an outline of the derivation of the dispersion equation (11).

We choose a cartesian frame of reference in which gravity is directed along the negative y axis, the free surface is located aty = O, the bottom

at y= h.

We assume that the oni _{velocity coniponents are u and y, in x and y} direction respectively, indi.; adent of z and small enough to allow the neglect of their squares ait ¿nets. In this linearized approximation the motion is irrotational,

o that we can

_{define a potential by}

u=ò/òx, v=ò/òy.

Then, denoting the surface elevatiou by _{r, the density of the fluid}

by , we have to satisfy (see LAMB 71)

(14 = O,

throughout the x, y-plane with the boundary conditions

aty=O,

+g71=O

aty=O,

=O

at y=-h.

The condition (16) expresses in the linearized approximation (a term (u2 - y2) has been neglected) the condition of constant pressure at the free surface.

Looking for solutions of (14)(17) of the type exp i(kxwt) times a

function of y, one obtains

coshk(y±h) const. exp i(kxwt)

with 'the dispersion equation (11), which _{we write down again for}

convenience

,w2=gktanh(kh), or

/ oi=(gIctanh(kh))

Due to the presence of the hyperbolic function in (19), k is as a function

of w many valued. To the negative w axis corresponds for instance

Re k<O, Im/c=O and Re k=0. /2<Im /c<; Re k-=O, 3v/2<Im k<2,

etc.

The KK relations are not valid here since k(w) is not holomorflc in the upper half plane.

This cannot, as we discussed in the foregoing section, be due to the

fact that there is no finite velocity in the sense that a disturbance,

starting at t=O in x=0, y=O, say, is felt at some other point when a time it> O has elapsed. A disturbance is in fact instantaneously felt by the whole fluid since the potential satisfies (14).

As, however, such a finite velocity is not a condition for the validity

of the KK relations, we must reexamine this case. The reason why the KK relations are not valid here, becomes clear when we consider the case w = (i.

Inserting this into (19) we obtain

(19)

(20) fl7(i

h'

These values of k are of no interest when propagating waves are con-sidered, but they are of importance in steady motion.

Then (14)(17) reduce to

V2ç = O,

=0, at y=O, u=h.

These equations determine steady potential flow between two parallel

plates, located at y=0 and y= h. Any potential flow can be build

up from sources, sinks and dipoles.

As an example we give consideration to a dipole with axis in x direction,

situated in x=0, y= a(a<h).

The boundary conditions (23) can be satisfied by constructing the

images of the dipole with respect to both planes and repeating this

process infinite times. In this way one obtains two rows of dipoles, at

= a ± 2 n h and at y = - a ± 2 nh. These rows of singularities are

associated with the points given by (20). Substitution of (20) into (18) gives

Because u and y must tend to zero for xJ

-

continuous at x =0. we take

00

nr

(24) x O : = A, e cos -r--,

x<O:q=-AneTcos.%..

The coefficients An must be determined from the properties of the dipole in (O, -a). Across a dipole layer the potential changes discontinuously, the difference between the values at both sides being equal to the dipole strength of the layer, in our case a delta function.

Referring to the method of images, we observe that the solution is the same as in the case where there are two infinite rows of dipoles, each row with period 2 h, and obtained from repeated reflections of the dipole in (O, -a) with respect to both plates.

The locale dipole strength at the y axis can therefore, the strength

of the dipole in (O, - a) being u, be represented by the generalized function

rn 00

{ò(y+u-2mh)H5(y-a--2mh»,

Pt -00

which has (see e.g. LIGHTHTLL [8]) the Fourier expansion

u 2u °° _nuj

nu

-F- -- cos

cOs -.

n-1 ¿

From (24) and (25) we obtain

00

ny

(28) q=+o - 92x-.-o =

2A COS

Since this must be equal to the local dipole strength, we obtain by

comparison with (27)

fL flU

A0, An=rTcos_-.

This example illustrates that the values of k given by (20) lead to solutions, periodic in y direction, which by the method of images can be shown to represent the steady potential flow determined by source and

dipole distributions between the plates.

We conclude that the solutions of (14)-(17) given by (18) and (19), represent not only propagating waves, but also all steadymotions between

rigid parallel plates at y =0 and y

= -

The singularities associated with these steady motions are the cause of the fact that the K-K relations do not hold here. In the example given above, we might allow the plates to move to infinity, thedipole remaining

fixed. Then the images disappear. Therefore we inspect whether for

h

-

The dispersion equation reduces to k=w2/g.

We consider /c/w3=1/wg. We have divided by w3 in order to secure

that the function considered tends to zero for w -* oc.

The function l/w has a pole in w= 0, but is otherwise analytic. The presence of the pole requires a slight modification of (5) and (6).

It is easily verified that in the case where k(w) has a pole in w=0,

(5) and (6) become (29) Re k(wo) +00 Im Ic(w) dw + Re

(\sl.

_wcia

wwo

where a is the residu of k(w) in w=0.

For the function 1/w both relations hold, since the _{integrals are zero} and the remaining equations are identities. The dispersion equation for waves on water of infinite depth satisfies therefore the modified KK relations (29) and (30).

The author is indebted to Professor R. Timman of the Technical University, Delft, and consultant to the N.S.M.B. for _{his stimulating} interest in this work.

Netherlands Ship Model Basin, Wagemingen.

REFERENCES

i. BROER, L. J. F., Golven en golfgroepen, Ned. T. Natuurk. 27 _(1961). MUSKIIELISHVILI, N. I., Singular Integral Equations. P. Noordhoff, Groningen

(1953).

LANDAU, L. D. and E. M. LIFSHITZ, Statistical Physics. Pergamon Press, London-Paris (1959).

BODE, H. W., Network Analysis and Feedback Amplifier Design. D. v

NOSTRAND (1949).

KAMPEN, N. G. VAN, Causaliteit en Kramers-Kronig relaties. _{Ned. T. Natuurk.} 24 (1958).

Bateman Manuscript Project, Tables of integral transforms, _{Vol. II, ed. by} A. ERDÉLYT, McGraw-Hill Book Company (1954).

LAMB, H., Hydrodynamics. Cambridge, At the University Press (1952). LIGwrHu.L, M. J., An Introduction to Fourier Analysis _{and Generalised}