Rayleigh waves and resonance phenomena in elastic bodies

RAYLEIGH WAVES AND RESONANCE PHENOMENA IN ELASTIC BODIES (VOLNY

RELEIA

I REZONANSNYE

IAVLENIIA

V UPRUQIKH

TELAKH)

PMM Vol. 29, No. 3, 1965, pp. 516-525

R. V. GOL’DSHTEIN (Moscow) (Received

It is chuacteristic changes

of stationary

when the velocity

the velocity

of propagation

As an example, body

we shall

is stress

free everywhere, boundaries.

finite

stresses

its shape, city

at its end.

as well

approaches than

forces

The length

to or exceeds

the crack

of an infinite

in front of the wedge. where

a smooth

properties.

velocity,

of the wedging

end region,

ensure

of the crack

It follows,

qualitative

equal

cohesive

closing

depends

forces

of crack

act between

boundaries

on the velocity

shown

[l],

that as the wedge

length

tends

to zero and the stresses

can form with wedging

and

of the wedge

It was

that no crack

elastic

Its surface

and

veloin

at a velocity

Rayleigh’s.

Other examples ary

problem

in a small

to have

becomes

waves.

is formed

except

theory,

source

A crack

cohesive

the Rayleigh infinite.

of elastic

the plane

[l].

as on the material

the body become higher

The

problems surface

consider wedge

5, 1965)

of the distrnrbing

of Raylaigh

by a thin semi-infinite

opposite

dynamic

of motion

February

are the stationary

problems

of moving

loads

and dies

along

the bound-

of a halfspace. The investigation

the velocity ments

at all points

transition ments. change

in [I] o f available

above

in the body become

Rayleigh’s

In the stationary

velocity

problem

load,

In the present in stationary

the half-plane the semi-infinite

is found paper

on the surface side

infinite.

there

This

[2-71

velocity, was

also

showed,

observed

of sign

loads,

leads,

in particular,

above

Rayleigh’s,

With velocities

and displace-

in [g]. With the

is a change this

that if

the stresses in the stresses

and displaceto a very unusual

the material,

under

to swell.

an explanation

contact

solutions

the Rayleigh

of moving

in the form of the free surface.

a compressive

mena

described

of the load or die approaches

problems. of which,

of the boundary,

is offered Thus,

beginning moves

of the above-mentioned

we consider

first

at some

uniformly

resonance

the nonstationary

time,

a load,

with a velocity

distributed below

pheno-

problem

of

along

that of sound.

Xayleigh

In any bounded

region

solution

of this

problem

enables

it to be investigated.

of the space,

In the following, of plane

waves,

tends,

moving

with

we consider

resonance

time,

phenomena

together

609

with the front end of the loading,

to the corresponding

a homogeneous,

isotropic

stationary

elastic

solution,

medium

the

and

under

conditiona

deformation.

1. Consider compressive

an elastic

half-plane

load of uniform

y = 0. This

load begins

load velocity

yQ0,

to move

is considered

free of stress.

q be applied

intensity with

constant

to be lower

At a time t = 0, let a normal

to the segment

velocity

x ( 0 of the boundary

V in the positive

than the velocity

of the slow,

x direction.

transverse,

The

sound

waves. Thus,

we see4

zero initial

a solution

and the following

of the dynamic boundary

equations

conditions

formulated

It is known theory

(see,

are satisfied,

are equal

can be solved

for example, if the stress

[3,6])

used

that the equations

components

and I/J are the scalar

for

by the method

and vector

function)

with

: (1.1)

y = 0

in [9, IO].

of the dynamic

elasticity

cr ,a% and T and displacements Y xY

u and v

potentials,

of the

to

Here 4, wave

problem

of elasticity

is Heavyside’s

T&/=0

uu = -qH(Vt-LIT), The

of the theory

(H (0

which

are the solutions

equations

Here p is the material transverse

waves,

A Laplace boundary

density

and e, and cr are the velocities

and

respectively.

time transformation

conditions,

using

of the wave

the zero initial

na,

z

equations

conditions,

P2Q,

n’y

for the potentials leads

=

Cl2 Tx,J= 0,

2, = -

2 and T denote

the Laplace

and the

to

P’u’ Cz2

_4exp P

Here cb, v,

of longitudinal

-

PH V

(4

representation

for

y = 0

of the corresponding

(1.3)

quantities

R. V. Gol’dshtein

610

and p is the parameter As in [9,

of the transformation.

lO],we

seek

of (1.3) in the form

the solution

@ = \ p (5) exp (@lx -I

P

1/L2i-

c~-~Y)d5

L

-a3

(1.4)

s 00

Q (5) exp (i&x +

Y=

P

my)

d<

-co

Here the integration are functiona

found

We construct (-

icJ

to ($

integrals

cuts,

(1.4)

will

using

along

axis

and determine

= 1. Also

dl

[, and P (0

and Q (6)

from point

($;I)

the branches

the exponential

and

of the

multipliers

in

0.

(1.4) and relations

for P (5)

in the plane

(1.3).

the imaginary

ice) , respectively,

die out for y f

equations

conditions

the condition

From the representation integral

on the real axis

in the rplane, and (-

ioe) roots

(P -f cl;V

is performed

from the boundary

(1.2) and (1.3).

we have

the following

and Q (4)

Co .V’ (6) 2iC 1/C.” 9

s -co

cIea -

Q (5) (2P -P c~-~)I exp (ip5x) df = 0

(1.5)

co s -00

IP (5) (2P

+

ore2) -f

Q (f) 2i5 1/C” f

c~-~I exp (ipcx)

d5 = (1.6)

K =

p9

exp

pxH V

(x) (K=

3)

Assuming

p we satisfy

Q (5) = 2i5

(6) = (X2 -J- cz-2) R (6),

equation

(1.5)

tion of the new unknown

. 1

identically, function

v/5*+

Q’ R (6)

and from (1.6) we get an equation

(1.7)

for the determina-

R (0

00

II (5) F,

Here

(5) exp (ipcx)

F, (5) = (X2 + c,-92 is Rayleigh’s the surface

dc = $exp

-

““v” (x)

(1.8)

--co

function. Rayleigh

Its only waves.

aeros

will

452 V(52 be 5 = f

+

s-2)

iVR-t,

(L2 + where

cz-2)

VR is the velocity

of

Royleigh

Closing halves

the contour

waves,

of integration

of the 5 plane,respectively,

resonance

in (13)

we note

for x > 0 and x <0

that

and

lower

is satisfied

if (1.9)

‘nipYL+(iV-l)(l;-_iV-l)

for x < 0 if

KLR (5)

FR

(5)

Here L+C() and L_ (4) the

in the upper

for x > 0 the equation

KL+ (5)

R (5)FR (0 = and

611

phenomena

upper

and

lower

Thus,

X_(r) For

X+(r)

is a function

and

Aj

analytic of the

Finally

(-

are analytic

point (l-4),

without

any axis

zeros

of the

finite

plane,

in

= x_ (E;)

(1.11)

half-plane.

i.e.,

at y = 0, representing

X_ (5) =- i (I’-’ $

or singularities

in the upper

of X_(l)

(1.10)

we have

(E- G-l) p;--IyE)

A and B are determined

we have

(E -f 0)

i&)

On the real

continuation

at every

(5 +

functions

(E + i&j = -

integrals

(5) it)

respectively.

analytic

-!- U. Constants

X- (Wl).

2nip3L_

&&

is the

the convergence

X_ (5) =

-

half-plane,

(8 =

Xt

=

Therefore

an entire

function.

@ and v’, we need

from the

known

values

X_ (-

i&)

F) and (1.12)

Xl (252 +

P(5) = 2nP3 (5 + ie) (5 -

W1)

K, = (K / V). From (1.12)

where

/ 5 13 cc, i.e.,

it follows

the

above

expressions

for P (0

potentials

4 and $.

we find the

stress

we will

converge

that

(1.4)

following

one of the

(5)’

integrals

theorem,

In the

F,

iK,c v/5’ + C~-~ v (‘) = np3 (5 + ie) (5 - ib’-l) F,

the

Substituting version

v and

cz-?)

P (b) -

everywhere,

need

the expressions

components,

for example

ce2, and except

and Q (5)

into

Q (5) z

at the (1.4)

and

for x > 0 of the vertical ox.

( v1 (u) =

The required

c-2

point

relations

(t) when

x = 0. y - 0.

using

the in-

displacement are

J, (u) + J, (u) -t- J, (u))

t1*13)

00

(1.14)

Here

R.V.

612

K1 (252 f

IV, (5) = f (4

for

= 1

~-2)

2ng (5 cl-1

32 -

Gol’dshtein

1/52 + cl-Z

iv-l)

F,

y fc2-z

(5)

-

cl_2

<

u

The expraasion

for 0,

V,(U) by substituting

‘g

in equation

the functions

n (q - ii’-‘) F, (rl)

f (u)

c&,

<

1 for %<%I, c1 p

gj+

KIT V/t12 + CI+

Nz (rl) =

’

for other u

0

for;<%

(;)=o

for ox, is obtained

(1.14),

N1 and N1 with

=

functions

from the equation

of the same

of

parameters

MI

and MI, where

Q (252 + G-2) (Zf” + 2c1-2 -

iJfl(5) = M,

Equations

Next,

(q2

+

(1.15)

C2--2)

F, (q)

of the stress

and displacement

we shall

consider

the interrelation

and the corresponding

solution

the stress

Here x ‘and

stationary

ox at any ooint

ax=5

[

I3

i

2---arctan-% hy

‘)

i

of the point moving

to [3, I], in the stationary

is given

-C

of the nonstationary

solution

According

of the half-plane

to the front end of the uniformly equal

of the above

problem.

y ‘determine the position

are, respectively,

by XI

+

-arc

tan

in the system

loading

G?

&$?z’-

of coordinates

and the coefficients

(1

-q,

m

>-

-IL c2

k12

equal

in particular, body,

motion

L=

means

that when

happens

1-

$,

of the loading

to zero and changes

The same

con-

A, B and C

f)

(I_

B =

c:1/1--m'1/1-~)m2,

For theuniform

(1.16) )1

to

A = vcq/--

becomes

can be written

fashion.

problem

nected

components

cz-2)

iv-‘) F, (5)

2qq y’(q2 + c1-2) nV (q - iv-‘)

(q) = -

for the remaining

in an analogous

2nV5 (5 -

sign

k,2

=

1 __

V = VR the stress

velocity,

exceeds

or becomes

to all the other components

&)

’

$

with the Rayleigh as the velocity

(If

of stress

the coefficient

the Rayleigh

infinite

at all points

and displacement.

A

one. This, in the

Rayleigh In other words, tion of stress is different

when the load moves

and displacement

2. We shall

consider

discussed

interval.

This

explain

the loads

move with

vestigate

the change

syetem

Consider, point,

Assume

of all three

distribution

y ‘: z =

values

with respect

components

load,

Using

2’

solution

distributed

in the nonstationary along

the semi-infinite

of the stationary

the relations

in an arbitrary

f

v1,

of section

y’

=

of t, the expressions

We shall

when in-

of the

of the moving

I/.

(1.14)

consider

solution I, we shall

fixed neighborhood

and (1.15)

to the front end of the load (i.e.,

is similar.

distribu-

the load velocity

exists.

for the nonexistence

velocity.

a stationary

the front end of the load to be the beginning

x ‘and

for large

stationary

the reason

in the stress

of coordinates

velocity,

If, however,

of the stationary

1, of the moving

the Rayleigh

front end of the load.

distribotion

the formulation

in section

will

with the Rayleigh

a stationary

613

phenomena

in the body is not possible.

from Rayleigh’s,

problem,

wavaa, resonance

for ox at an arbitrary

fixed x’and

the derivation

y

‘). The form

of one of them.

We

have

L= C&

since

5,

(ctr)

=

c_ (ctr).

For large

r (2.21

i.e.

c+

(t)

+

as I + OO. But with r) = (i/v)

(i / v)

under the integral,

has

V = VR. Therefore

we shall

either

a zero of order one, split

the integral

(2.1)

the denominator

of the fonction

MI (T)),

if V f VR, or a zero of order two, when into two, isolating

the pole

where

Ml” (iv-‘) It can be shown In the first velocity

that

case,

when

agreeing

with the hnown

In the second culations

show

of the remaining system

case

that tl

2l7-2 +

Q)

(-

0 (1 / t). Th erefore

=

the load velocity

L, has

of the expression

ox in the vicinity

moving

L,

VR, the quantity

components stress

= - z$ (--

a finite

limit.

2v-2

V does not coincide The results

for ox are the same.

stationary

Thus,

solution

% at where

u = u (V,,

in the expression

cl,

-

cz_2)

to calculate

I!,,.

with the Rayleigh

of calculating we obtain

the remaining

that with

with time,

V f VR the

to a finite

limit,

(1.161.

when the load moves with the Rayleigh

parameters

25-Z

we only need

of the front end of the load tends, [3,1I

+

cl,

q,

x’,

7’).

velocity

(V = V,),

Calculating

for ux, we get, with coordinates

cal-

the behavior fixed in the

K. k’. GoPdshtein

614

x

(-

2V,-a

+ cz-2) (_

2!1’, -2 .+. &-”

(r’)Z vR-2

-(y’)”

It can

thus

be seen,

that

moving

the stresses

with

the

In other

words,

the motion

never

become

steady.

Therefore,

loses

its meaning The

with

increase

Rayleigh

surface

-

Actually, addition

front

and therefore

also

front

end of the

With velocities sarface

waves.

be reached This this

case,

load

will

with

transmitted brings

and

is caused

The

pagation

waves,

of the front

of the load

end to

end will

stationary

problem

velocity.

of the front

end of the

case,

the energy

loading

can

be represented and times

overload

surface

gives

waves.

load,

moving

transmitted

by the

with

relative

of which

at the instant

of its

the velocity

of propagation

Therefore,

infinity

rise,

as a successive

of application

By assumption,

the velocity

waves.

than

the Rayleigh

time,

of the transverse,

the fronts

to an arbitrary

fixed

of the longineighborhood

if the

load

arising

waves

waves

below

is moving front

will

same

end of the

Thus,

exactly

with

happens

load

the

end of the load

have

a common

with

an equilibrium

phase

This

of the

occurs.

in the vicinity

Rayleigh

the will

velocity.

at different

front.

in the neighborhood

in the same

accumulates

is analogous

times front

front

As a result,

of the

to resonance

of the disturbing

waves

of the elastic

solntion, and

the

front

In

and

moves end

of the

the energy

end of the

load.

in stress.

described

of the internal

of the front

at the

end of the load.

velocity,

of energy.

of its motion,

by the velocity

velocities

O (1)

proportional

in this

than

outflow

the increase

phenomenon

Rayleigh

such

and

and

of surface

3. In the stationary for load

Each

approach

change

the front

about

neighborhood

asymptotically

the locations

is smaller

inflow

by the surface

This

overloads,

or higher

in the direction

a superposition

the

in the neighborhood

the

the surface

propagating together

Finally,

picture

t +

load.

lower

between

with

transverse

waves

(y’)r c,-r

in the neighborhood

because,

of the longitudinal

and transverse

cz-2)

region.

of the loading.

end of the loading

_

-

of the corresponding

of the semi-infinite

to longitudinal,

of the front

of the

place

end of small

on the motion

application,

fixed

in the neighborhood

in this

the motion

to its

tudinal

takes

(r’)2 I’,-*

increase

the solution

in the stresses

accumulates

velocity,

moves

(i’R-2

= (Z’lr + (Y’l?

of the medium

the load

velocity,

waves

depend

when

‘tv’,-2

in an arbitrary

Rayleigh

time.

Cz-Z)

cl-” ((V

of the load,

_

above

as can the

source half-plane,

be seen Rayleigh

in the usual

being i.e.,

from (1.16). velocity

[l].

equal

the surface the The

vibrating

to the velocity

stress same

Rayleigh

systems of prowaves.

is of opposite happens

to the

sign

Rayleigh

displacements

[l].

usual concepts, above

In particular,

the material

is tensile,

problem.

Based

v of the surface

motion of a compressive

we shall

we have the following

moving with a velocity

Also,

=

+

-

&

+)’

($

(4 +

lJr FI-

-

+ lG $

for the vertical

(‘$

(-

-

w2 (41

:< I(-

2 g

+

c2_12),

2 J$- +

&)

-g

U’2 (u)

-- 4 g w2

Figures

($

-

(+

velocity,

=

-

-

-$)”

(3.1)

(9

-

&)“’

(!

($

-

+.-’

x

for Cl-’ Z \c u < cz-&

$)I-’

&&

1

-

‘It u (;

CT)

-

for

-!J]-’

+

1

-l

du/dt

velocity

of time, when the load is moving

at an arbitrary with a velocity

point A (y = 0, below

and above

respectively.

du/dt

b

a

FIG. 1 From the time of arrival velocity

begins

to increase.

infinite

velocity

is caused

instant

the load is applied

at point A of the longitudinal Also,

for a velocity

by the surface at its front

wave,

below

Rayleigh’s

at tR =

end. A negative

wave front, t, VR-lT,

infinite

(Fig.

= la),

originating

velocity

will

x

u >, Ct_l z

(u < c2-b4

(u) = 0

la and 2a show the vertical

x > O), as a function the Rayleigh

displacement

cut

I (u >

WI (u) = 0

on the other hand,

T

c,-1.X c,-1.X

2 ‘5

to the

again turn to the nonstationary

expression

. s s[w,

v=

x I(-

loading

Contrary

point fy = 0, z > 0)

t

Wl (u)

changes.

contracts.

of this phenomenon,

on (1.13).

615

phenomena

under the load is found to swell.

the material

For an explanation

resonance

the shape of the free surface

under a stationary

Rayleigh’s,

if the loading

waves,

cl-lz

the

a positive at the occur

later

K. V. Gol’dshtein

616

at point

A, at the instant

the

a velocity

Rayleigh’s

For infinite ative

infinite Let

the

is connected velocity

us further

Rayleigh

at subsequent

at that

there

to the segment

(-

end of the

(do/dt). ative

small

the cases

of velocities

of load

an infinite

load

and

and the neg-

below

application

load

additions.

of disturbances

bo, 0) of the x-axis

instant, load,

Similarly,

infinite

vicinity

only

interaction

loading,

wave.

the instant

is suddenly

: here the positive

2~)

end of the

The

at subsequent

applied motion

which

to the body, of the medium

arise

as

times

and above

is characterized

theload

while can

is ap-

at the moving

front

load.

At the initial front

are

(Fig.

front

surface

between

problem,

there

of the

of the

of the

(to =m=I ‘-I_,~).

arrives

is reversed

arrival

difference

instant

as a result

end of the

the

In the present

instants

be regarded

the picture the

the arrival

explain

that,

end of the loading

with

with

value.

by the fact

plied

above

velocity

front

ahead

the particles

velocity.

of the

the particles

directly

front

of the medium of it, acquire

directly

Figure

behind

3 shows

end of the

load,

the

an example

at a time

situated

a positive front

on the surface infinite

end of the load

of the velocity

close

near

vertical

acquire

distribution

to the initial

one.

the

velocity a negin the

Roth

the positive

a

FIG. infinite (part

discontinaity 2) will

at the

instant

end of the of lower front

to (3.11,

surface

wave

velocities

of this

therefore

than

before

the

than

interaction

Rayleigh’s, is,

that

(Fig.

which

interact

Rayleigh’s,

arises

3) and

the negative

at the

front

on opposite

in a different

with

sides

manner,

the disturbances

one

end of the load of the

front

for the two cases

generated

by the

of its motion. the load.

disturbances

discontinuity

distribution wave,

(1) and (2) are situated

3). They

overtakes

infinite lower

Parts

aa a result

later

initial

the surface

velocities

(1) is situated

ing positive

result

(Fig.

higher

end of the load, Part

1) of the

along

it is applied.

loading

and

according The

(part

be carried

2

For the

case

do not change

front

end of the

of velocities its

load

and brings

of the particle

vertical

meets

end of the load

the

front

the

front

lower

form and velocity

end of the load,

velocity

when

to point (Fig. and

than

Rayleigh’s,

of propagation.

A the correspoudla).

Part

interacts

it reaches

point

(2).

with

for

it. The

A, brings

Rayleigh

waves,

resonance

a change initial

in velocity

For the case

the initial

arrival

of velocities

distribution. such

according

point

value.

of particle lags

to (1.2).

The

(1.4)

function

and is positive

A of the surface,

Actually,

using

on the part (1) of

to a superposition

velocity behind

(Fig.

of

from (3.1),

the

of the sur-

infinite

disconti-

20). The surface

wave,

the front end of the load and brings

infinite

discontinuity

of velocity,

distribution.

velocity

properties

correspond-

as t + oa tends

of Laplace

transforms,

to a const-

we have,

and (1.12)

the function

The displacement

leads

with a positive

the vertical

known

under the integral since

Rayleigh’s,

as can be seen

be associated

the negative

At an arbitrary

above

This

that,

ing to part (2) of the initial

ant negative

as part (2) of the

of the front end of the load at any point will

however, FIG. 3

nature

the front end of the load overtakes

disturbances,

nuity

of the same

distribution.

other hand,

face

617

phenomena

dies

out at infinity

under the integral

o at any point

of the surface

as (-2,

i.e.,

the integral

converges

is positive. dies

out to ( - m) as t + m, independent

of the load velocity. In contrast, change

as can be seen

of displacement

the surface

wave

For the case the instant

For the case brought point

arrive

about

A, while

depends

of velocities

the surface

from the above

at the surface

wave

investigation

of velocity

at the instant

the

behavior,

the

front end of the load and

on the load velocity. below

arrives

of velocities

point

Rayleigh’s,

above

Rayleigh’s,

by the front end of the load, the discontinuity

the displacement

and decreases

brought

the discontinuity

corresponds

about

at point A increases

at

when the front end of the load arrives. in the vertical

to an increase

by the surface

wave

velocity,

of displacement

corresponds

at

to a de-

crease. Knowing quently,

the vertical

also

the displacement

tion of vertical Figure

velocity

displacements

lb shows

the shape

(du/dt)

as a function

v at any point along

of time (Fig.

of the sarface,

the surface

of the half-plane

at any given surface

la and 2a), and, conse-

we can construct

the distribu-

time.

at some

given

time,

for a velocity

H. V. Gol’dshtein

619

below

Rayleigh’s.

Fig.

The

To obtain front

the stationary

end of the

load

between

the

and the surface

load Thus,

the

of the

in front

of it. For

pression

in front

stationary

is shown

shape

end will

of the

a similar

after

will

still

the

front

also

at that

II,

lb),

distance

within

swelling with

those

front

the end of

infinite.

an arbitrary

the load under

of the

larger

the

becomes

under

agreement

the

we find

the material

we have

in the vicinity

selected,

time,

be compressed

in full

the velocity

The dies

or loads

presence

waves

can

appear

with

and will

the

load

swell

and a de-

obtained

from the

excitation

because

waves

properties.

that

that

on the surface

is that are

similar

26

the dis-

time.

Thereinterval

< Vt and,

therefore,

of the load.

when,

in passing

(for example,

in passing

[II]

by boundary type

media

velocities

d ea 1s with liquid

contact

are not connected

in other with

along also. close

problems with

:

conditions

of wave,

the resonance

the surface

which

surface

These

effects

to those

of sur-

mechanism

by a turbulent

velocity,

of critical

of separation

they caused

of separation,

of a heavy

phenomena

of stationary

of another

is not one critical

a surface

ahead arise

phenomena

phenomena

there

along

is changed

of disturbance

but a spectrum

moves

material

of equation

the appearance

of dispersion body,

Rayleigh’s,

problems

to resonance

Note

at the particular

in stationary

or a surface

medium.

that

v in the above

boundary

source

26 we can see

rmt

phenomena

elastic be expected,

a charge

of the

given

of gravity

homogeneous It can

and

leads

the

of Fig.

the displacements

velocity

free

of a free

the curve

above to the

length

Actually,

x,,J. Here zmt is the coordinate

is a maximum

resonance

at Rayleigh’s

boundary

from

Fig.

(m,

to a finite

of the load.

dynamics),

These

the motion

1. From interval

relative

the type in gas

of equations.

propagate,

in the

is lifted

ahead

above

on a semi-infinite

is applied

by subtracting

velocities

singularities

moving

the

26). for a velocity

is distributed

to the material

subtraction,

in the investigated

presence

waves

load

(Fig.

Rayleigh’s,

a distance

For

end,

the load

above

over

and

velocity,

of the

The

left

of sound

in the type

that

the displacement

to the

critical

front

is obtained

to the left

monotonically.

difference

fact

relative

problem

at which

qualitative

some

the

its

a velocity

be lifted

end of the ‘finite’

through

near

monotonically

the motion increase

through

dielectric

with

but moved

point

Normally,

waves.

are

load

with

in this

v increases

of the surface

changes

results

load,

surface

curve,

placement

with

will

Rayleigh’s,

a compressive

If the moving front

fore,

of it. These

(Fig.

load

the time

as t + bo, this

Rayleigh’s

end of the

is not connected

interval. 1, its

below

the motion

larger

at which,

In the limit,

a velocity.above

lift udder

hayleigh’s,

when

Hayleigh’s,

solution.

The

case

above

investigate The

of time.

on the surface

wave.

front

we must

values

points

for a velocity

neighborhood

face

for a velocity

curve,

solution,

for large

distance

the

analogous

2t.

as in the cabe

wind.

of

In this

of an isotropic

velocities.

occur

in the theory

between

of Cherenkov

two dielectrics

with

radiation different

Rayleigh

The author

is sincerely

and the supervision

waves,

grateful

resonance

to G.I. Barenblatt

of the work, and also

619

phenomena

for the statement

to V.M. Entov

and R.L.

of the problem

Salganik

for help

in the

solution.

BIBLIOGRAPHY

1. Barenblatt, bodies) 2.

G.I., and Cherepanov,

Galin.

L.A.,

Smeshannye

(Mixed problems Doklady 3.

Galin,

Akad.

Radok,

5.

Nauk Vol.

khrnpkikh

tel (Wedging

of brittle

1953.

Sneddon,

I.N.,

Semi-Infinite 6.

Sneddon,

7.

Cole,

tcorii

On the solutions

Vol.

uprugosti with

s silami friction

treniia forces

dlia poluploakosti

for a half-plane)

39, No. 3, 1943. zadachi

1956,

teorii

of elasticity

Kontaktnye

I.R.M.,

Math.,

zadachi

of the theory

Gostekhizdat,

L.A.,

elasticity). 4.

G.P. 0 rasklinivanii

PMM Vol. 24, No. 4, 1960.

uprugosti

of problems

problems

(Contact

of dynamic

plane

of the theory

elasticity.

Quart.

of

Appl.

14, No. 3.

Stress

Produced Rend.

Solid.

by a Pulse

Circolo

I.N., Preobrazovaniia

J. and Huth, J., Stress

of Pressure

mat., Polermo,

Fur’e Prodncsd

(Fourier

Moving

Along

the Surface

of a

Vol. 2, 1952.

Transforms)

in a Half-Plane

Izd. inostr.

by Moving

lit.,

Loads

1955.

1. Appl.

Mech.,

1958, Vol. 25, No. 4. 8.

Craggs, Sot.

9.

I.W., Two-dimensional

Proc.,

Dang Dinh Ang. and Phys.,

10.

Vol. Elastic

1960,Vol.

Fillips,

Quart.

Appl.

waves

in an elastic

half-plane.

Cambridge

Philos.

56, p. 3. waves

generated

by a force moving along a crack.

I. Math.

38, No. 4.

Dang Dinh Ang. Transient plane.

11.

1960,

Math.,

O.M., Rezonansnye

motion 1960,

of a line Vol.

iavleniia

load on the surface

of an elastic

half-

18, No. 3. v gravitatsionnykh

mena in gravity waves) Sb. ‘Gidrodina-micherkaia instability’), Izd. ‘Mir’, 1964.

volnakh

neustoichivort’

(Resonance

pheno-

(Wydrodynamic

Tranrlated

by S.A.