Johnson R.S. A Modern Introduction to the Mathematical Theory of Water Waves

Подождите немного. Документ загружается.

54 1 Mathematical

preliminaries

Q1.31 Differentiation under the

integral

sign:

examples. Use the formula

given in QUO to

(a) find an expression for d//dx, where

I(x)= / —dj>, JC>0;

J y

(b) show that

where g is a twice differentiable function, is a solution of the

partial differential equation

0xx ~-<t>tt ~Z

(

=0,

for a certain value of the positive integer, «, which should be

determined. [Hint: integrate your expression for

<t>

by parts,

twice.]

Q1.32 An energy equation. An incompressible, inviscid flow with

F = —

is described by Euler's equation. Take the scalar pro-

duct of this equation with the velocity vector, u. Integrate the

resulting equation over the volume V, which is

fixed

in space, and

hence show that

— I

-pu-udv

= - I

l-pu-u

+ P + pQ

Jund^,

v s

where S bounds V.

Q1.33 Energy and a

uniqueness

theorem.

(a) The kinetic energy of the fluid occupying the volume V is

= -f

pnudv;

see Q1.32. For an

incompressible,

irrotational

flow,

show

that

-J

Exercises 55

(b) The result in (a) can be used to provide a uniqueness theo-

rem. Suppose that there are two possible flows, u\ = Vfa and

= V0

both satisfying the same given conditions on S.

Write U = Uj

—

and

= fa

—

(f>

and show that

/= fv-Vdv=

and hence that / = 0 if either 0 or u is prescribed on S. With

these boundary conditions, deduce that U = 0, so u

= u

the velocity field is unique.

Q1.34 Elementary

nondimensionalisation.

A two-dimensional flow, with

u = (w, w) and x = (x, z), is both incompressible and irrotational.

Nondimensionalise according to

u -> cu, x -> kx, z -> hz,

and hence obtain the nondimensionalisation of the stream func-

tion,

\jr,

of the velocity potential, 0, and of w. Write down the

nondimensional version of w = 0

Q1.35 A Reynolds number. Use the scheme described in Section 1.3 to

nondimensionalise the Navier-Stokes equation (Appendix A),

and hence obtain the nondimensional parameter which incorpo-

rates the viscosity, /x, and is based on the scales associated with

the horizontal motion.

[The reciprocal of this parameter is called the Reynolds num-

ber; knowledge of its size is of fundamental importance in the

study of fluid mechanics.]

Q1.36 Dynamic condition in cylindrical coordinates. Use the non-

dimensionalisation described in Section 1.3.1 to obtain the

corresponding boundary condition written in cylindrical coordi-

nates; cf. equation (1.64). Further, by suitably scaling the pres-

sure in terms of s, and by using an appropriate definition of the

Weber number, rewrite this condition and then approximate it

for s -> 0.

Q1.37 Nondimensionalisation of the pressure equation. Use the non-

dimensionalisation described in Section 1.3.1, followed by the

scaling adopted in Section

1.3.2,

to obtain the appropriate

form of the pressure boundary condition, equation (1.29).

(This will require a nondimensionalisation and scaling of the

velocity potential, 0; see Q1.34.)

56 1 Mathematical

preliminaries

Q1.38 Irrotational flow: approximations. Write down the non-

dimensional, scaled equations for irrotational flow in the absence

of surface tension (see Q1.37) and for a bottom boundary which

is independent of time. Hence obtain the approximate form of

these equations in the limit of small amplitude, e -> 0. Also write

down the corresponding approximate boundary condition when

surface tension is included.

Q1.39 The

classical wave

equation.

Use the method of characteristics to

derive d'Alembert's solution of the wave equation

= 0,

and hence obtain that solution which satisfies u(x, 0) = p(x) and

(x, 0) = q(x), —oo < x < oo.

Q1.40 Data on compact

support.

See Q1.39; now suppose that both/?(x)

and q(x) are zero for x < 0 and x > x

(> 0). Describe the form

of the solution for t >

/(2c).

Q1.41 Dispersion relation. Discuss the nature of the solution of the

equation

+ u

xxx

- u

= 0,

on the basis of its dispersion relation.

Q1.42 Dispersion relations compared. Compare the dispersion relations

for the two equations

+ u

xxx

= 0; u

+ u

- u

xxt

= 0,

particularly for long waves (k -> 0) and short waves (k -» oo).

Q1.43 Nonlinear wave equation. Obtain, explicitly, the solution of the

equation

+ u)u

= 0,

which satisfies

f ax, 0 < x < 1

u(x, 0) = | a(2 - JC), 1 < x < 2

[ 0, otherwise,

where a is a positive constant. Also, by using the characteristics,

sketch this solution at various times, t > 0, and include t = I/a.

Q1.44 An implicit solution. Find the (implicit) solution of the equation

+ uu

= 0,

Exercises 57

which satisfies u(x, 0) =

cos

TTX.

Show that u(x, i)

first

has a point

where u

is infinite at time t = n~

. What happens to this

solution if it is allowed to develop beyond t = n~

Q1.45 Asymptotic

expansions

(a) Obtain the first two terms in the asymptotic expansion of

/(JC;

e) = (l+ex — + e"^)"

, x > 0, s > 0,

for x = O(l) as s -> 0. Also obtain the leading order terms in

the expansions valid for (i) x =

O(e),

(ii) x =

O(s~

Show

that your expansions satisfy the matching principle.

(b) Obtain the first three terms in an asymptotic expansion of

/(*;

e) = (1 + ex + e

l/2

, x > 0, e > 0,

for x = O(l) as s -> 0. Show that your expansion is not

uniformly valid as x

oo. In the fvtw further asymptotic

expansions that are required, find the first two terms in

each and confirm that they satisfy the matching principle.

Q1.46 Asymptotic

expansions

II. The function

f(x; e) = (l-ex- s

- e"

x/£

)

1/2

e > 0,

is real for 0 <

(s).

Construct asymptotic expansions of

f(x; e), as e -> 0, as follows:

(a) x = O(l): first two terms algebraic in e, first exponentially

small;

(b) x =

O(s):

first two terms;

—

O(s~

first two terms.

Show that your expansions satisfy the matching principle and,

from your expansion obtained in (c), deduce that x

(s) ~ e~

—

as s -> 0

Q1.47

Long-distance

scale.

For the propagation equation

introduce the characteristic variable ^ = x

—

t, and the long-

distance variable X = ex, and hence obtain the appropriate

Korteweg-de Vries equation which describes the first approxima-

tion to u (as s -> 0) in the far-field. (You may assume that u -> 0

as |£| -> oo.)

58 1 Mathematical

preliminaries

Q1.48 Left-going

wave.

See Q1.47; for this equation, find the Korteweg-

de Vries equation (as the first approximation for e -> 0) in the

far-field defined by £ = x + t = O(l) and r = et = O(l).

Q1.49 Left- and right-going waves. See Q1.47 (and also Q1.48); intro-

duce the characteristic variables £ = *-/, £ = x + t, and the

long-time variable r = et. Seek a solution

for £, f and r all O(l) as e -> 0, in which / and g separately

satisfy appropriate Korteweg-de Vries equations. (You may

assume that/(g) decays as |§| -> oo (as |f| -• oo).) What, then,

is the solution for u{\

Q1.50 Nonlinear

Schrodinger

equation.

A wave is described by the equa-

tion

"«

-U

= 8{{U

)

- MM**}.

Use the method of multiple scales with

£ = x

—

t, f = e(x

—

t), x = s

and seek an asymptotic solution in the form

oo n+\

0 m

as £ -> 0, which is uniformly valid as |£| -> oo. Here,

ii = exp(ifc£) and

(> 1) is a given (real) number. Find c

(k)

and c

(fc) (and confirm that c

= d(kc

)/dk), and show that

= 0.

Q1.51 Wave

hierarchies

I. A wave, which satisfies u -> 0 as x -> -f

oo,

described by the multiwave speed equation

.— (C

£ U)— £ -

where cx,c

and c are constants. Show that, if q < c < c

then on the time scale e~

the wave moving at speed c\

decays exponentially in time, to leading order as e -> 0

. (To

accomplish this, you will find it convenient to introduce

Exercises 59

§ = x

—

t = O(l), r = s

t = O(l).) Now show that this same

property is exhibited by the wave moving at speed c

Q1.52 Wave

hierarchies

II. See

Q1.51;

show that, on the time scale s~

the wave moving at speed c has diffused over a distance O(e~

)

about its wavefront. In particular, show that this wave is

described by an equation of the form

(Df

~\~

fflffly

X(bvy,

to leading order as e -> 0

. (Similar to

Q1.51,

it is useful to

introduce X = s

(x - ct) = O(l), T = s

t = O(l).) Determine

the constant X, and confirm that X > 0 provided c\ < c < c

Find the solution of this leading-order problem which describes

a steady wave and which satisfies w^-0asA

^-oo,w->l as

X-+

-oo.

[This equation, here expressed in terms of

</>,

is a famous and

important equation: it is the Burgers equation, which can be

linearised by the Hopf-Cole

transformation

</>

= — 2Xd(lnO)/dX.]

Q1.53 A nonlinear wave equation. A wave motion is described by the

equation

.2 o2 o4

Introduce £ = x - t and r = st, and hence show that the leading

approximation (as e -> 0) satisfies the equation

+ 2uu# + u\ + u

= 0, (*)

where § = O(l), r = O(l).

Now introduce two characteristic variables

§ - x - t + sf(x + t, r); r/ - x +

+ £g(x - t, r),

and seek a solution

= F(|, r) + Gfo, T)

as £ -> 0, where F satisfies (*) and G satisfies the corresponding

equation for left-running waves. Confirm that the results are

consistent when only left- or right-running waves alone are

present.

Q1.54 Bretherton's equation. A weakly nonlinear dispersive wave is

described by

+ u

xxxx

+ u = su

60 1 Mathematical preliminaries

Introduce the variables X

— EX,

T = et and 0, where

k(x,

T),

-co(x,

and seek an asymptotic solution

which is uniformly valid as \0\ -> oo. Write

and obtain the equation for A which ensures that U\ is periodic

in 0. Introduce the dispersion relation, relating co and k, and

hence show that

co\k)A

^-A\A\

co\k)A,

2co

and then re-express this by writing A = a^ (for a, 0 real).

[This model equation for the weakly nonlinear interaction of

dispersive waves was introduced by Bretherton (1964).]

Q1.55 Steady travelling waves. Seek a solution of each of these equa-

tions in the form u(x,t)=f(x

—

ct), where c is a constant,

satisfying the boundary conditions given:

(a) u

—

6uu

+ u

xxx

= 0 with u, u

-> 0 as \x\ -> oo;

(b) u

+ uu

= u

with u -> 0 as x -> oo, u -^ w

(> 0) as

x -> —oo.

[The solution to (a) is the solitary wave of the Korteweg-de Vries

equation, and (b) gives the Taylor shock profile of the Burgers

equation.]

Some classical problems in water-wave theory

Yet let us hence, and find or feel a way

Thro'

this blind haze

The Passing of Arthur

The study of problems in water-wave theory, particularly under the

umbrella of the linear approximation, goes back over 150 years. In the

intervening time, many different problems - and extensions of standard

problems - have been discussed by many authors. In a text such as ours,

it is necessary to make a selection from this body of classical work; we

cannot hope to describe all the various problems, nor all the subtle var-

iants of standard problems. Our intention is, of course, to include the

simplest and most fundamental results (such as, for example, the speed of

waves over constant depth and the description of particle paths), but

otherwise we choose those topics which contain some interesting and

relevant mathematics. However, since we shall not present all that

some readers might, perhaps, expect or prefer, we endeavour to remedy

this by introducing additional examples through the exercises. The suffi-

ciently dedicated reader is therefore directed to the exercises, particularly

if a broader spectrum of water-wave theory is desired.

The material here is presented under two separate headings. The

first

linear

problems, where, apart from the elementary aspects mentioned

above, we single out those topics that are attractive and which will

prove relevant to some of our later discussions. Thus we describe

waves

sloping

beaches,

as well as the phenomenon of

edge

waves.

We shall

also develop some rather general ideas associated with ray theory, and

apply the results to

variable

depth, ship waves, and waves on currents.

Under the second heading,

nonlinear

problems,

we extend the application

to waves on a sloping beach in order to include the effects of nonlinearity.

We also describe the Stokes

expansion

(which produces higher approx-

imations to the classical linear wave), and introduce the fully nonlinear

solitary wave

- a very famous wave. Other nonlinear waves that we shall

describe include the

hydraulic

jump and bore, and we shall explain the

62 2 Some classical problems in water-wave theory

analogy between nonlinear water waves and (nonlinear) gas dynamics;

this leads us to introduce the notion of simple waves and the role of the

Riemann invariants.

I Linear problems

Our hoard is little, but our hearts are great.

The

Marriage

Geraint

The linear equations (defined by s -> 0, keeping all other parameters

fixed) have been described, for an inviscid fluid, in Section 1.3.1. These

equations, expressed in Cartesian coordinates, are

with

(2.1)

and

w = ub

+ vb

on z = b.

Correspondingly, written in cylindrical coordinates, these equations

become

= -p

\ v

= - -p

= -p

; - (ru)

+ - v

+ w

= 0,

with

1 1

(2.2)

r ir

and

w = ub

+ -b

on z = b(r, 6).

Most of the problems that we present, however, will be based on

rectangular Cartesian geometry.

2.1 Wave propagation for arbitrary depth and wavelength

We consider, first, the simplest problem of

all:

the propagation of a plane

harmonic wave in the x-direction over constant depth. The depth, in

nondimensional variables, is 1

—

b (> 0), but we may choose b = 0

Wave propagation

for

arbitrary depth

and

wavelength

(since the actual depth is subsumed into the length scale in the z-direction;

see Section

1.3.1).

The governing equations, (2.1), therefore reduce to

with \ (2.3)

w = rj

p = rj

—

Wrj

on z = 1; w = 0 on z = 0

The surface wave is described by

= A&

-**

)

+ c.c, (2.4)

where A is a complex constant; this represents a wave whose initial form

(at t = 0) is

+ c.c

where k is the (nondimensional) wave number.

It is convenient to write

E = exp{i(A;jc -

cot)},

and then to seek a solution (upon the suppression of the complex

conjugate) in the form

u = U(z)E, w = W(z)E, p = P(z)E. (2.5)

To avoid the obvious confusion, the Weber number is rewritten here as

; the equations (2.3) now give, presented in the same order as above,

^U =

P' =

ico8

JV;

W + ikU = 0 (2.6)

(where the prime denotes the derivative with respect to z), with

W{\) =

-icoA;

P(l) =

)A;

W(0) = 0. (2.7)

From equations (2.6) we see, directly, that

W" = -ikU = --P' =

so the general solution for W(z) is

W = BQ

8kz

where B and C are arbitrary constants. The two boundary conditions for

W(z) (given in (2.7)) then yield the solution

(2.8)