on waves in an elastic plate

On Waves in an Elastic PlateAuthor(s): Horace LambSource: Proceedings of the Royal Society of London. Series A, Containing Papers of aMathematical and Physical Character, Vol. 93, No. 648 (Mar. 1, 1917), pp. 114-128Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/93792 .

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114

On Waves ivn an Elastic Plate. By HORACE LAMB, F.R.S.

(Received July 10, 1916.)

The theory of waves in an infinitely long cylindrical rod was discussed by Pochhammer in 1876 in a well-known paper.* The somewhat simrpler problem of two-dimensional waves in a solid bounded by parallel planes was considered by Lord Ptayleigh+ and by the present writer: in 1889. The main object in these various investigations was to verify, or to ascertain small corrections to, the ordinary theory of the vibrations of thin rods or plates, and the wave-length was accordingly assumed in the end to be great in comparison with the thickness.

It occurred to me some time ago that a further exam-ination of the two- dimensional problem was desirable for more than one reason. In the first place, the number of cases in which the various types of vibration of a solid, none of whose dimensions is regarded as slmiall, have been studiedl is so restricted that any addition to it would have somne degree of interest, if mrnerely as a contribution to elastic theory. Again, modern seismiology has suggested various questions relating to waves and vibrations in an elastic stratum imagined as restinig on matter of a different elasticity and density.? These questions naturally present great mathematical difficulties, and it seemed unpromising to attempt any further discussion of themn unless the comparatively simiple problem which forms the subject of this paper shouild be found to adinit of a practical solution. In itself it has, of course, no bearing on the questions referred to.

Even in this case, however, the period-equation is at first sight rather intractable, and it is only recently that a nmethod of dealing with it (now pretty obvious) has suggested itself. The result is to give, I think, a fairly complete view of the more important modes of vibration of ani infinite plate, together with indications as to the character of the higher modes, which are of less interest. I may add that the numerical work has been greatly simplified by the help of the very full and convenient Tables of hyperbolic and circular functions issued by the Smithsonian Institution.j1

* 'Crelle,' vol. 81, p. 324; see also Love, 'Elasticity,' 1906, p. 275. t 'Proc. Lond. Math. Soc.,' vol. 20, p. 225; 'Scientific Papers,' vol. 3, p. 249. + See 'Proc. Lond. Math. Soc.,' vol. 21, p. 85. ? Love, 'Some Problems of Geodynainics,' 1911, Chap. XI.

'Hyperbolic Functions,' Washington, 1909. I have to thank Mr. J. E. Jonles for kindly verifying and, where necessary, correcting my calculations.



On Waves in an Elasttc Plute. 115

1. The motion is supposed to take place in two dimenlsions x, q, the origin being taken in the medial plane, and the axis of y normal to this. The thickness of the plate is delnoted by 2f. The stress-strain equations are, in the usual notation,

PXX= X (au + av/ay) + 2,u au/ax, 2),= k(3v/x + au/8ay), (1) Iyy= x(at/ax +avlay)+2,tavlay/

lt is kniown that the solution of the equations of mrotion is of the type

It = /ax/+xa*/ay', V = ap/ay_-a*/ax, (2) the funietions 0, -4 being subject to the equations

2*,~~~~~3 p +Dt =+2 I-)V , p 2=,2

where V2 = a22#X2 + =2/ay2 We now assume a time-factor eiot (olmiitted in the sequel), and write

1h2= Po- 7~2e-Pi- (4) X + 2,z P_ so that (v2+?h2)k = 0, (v2+k2)J = 0. (5)

We further assume, for the present purpose, periodicity with respect to X. This is most conveniently done by miieans of a factor ei$x, the wave-length being accordingly

V' = 27r/. (6) WVriting a2 = -27It2, 2 = 027U2 (7)

we have ay 20 62# = (8)

The equationis (1) now give

IlY - (02?+'2)0 2i?, ay

~~~~~~(9) PZ_Iy - i I(82+/32)f J

Symmre1riectl Vibrations.

2. When the illotion is synumetrical with respect to the plane y = 0 we assume, in accordance with (8),

= A cosh ayeitx>, - sinhi 3yeix. (10)

Tlhis gives, for the stresses on the faces y +f,

pyyl,t = {A(2 +?) cosh acf-B 2 iz/ coshl3f }Ied,$ x

_pxya = + {A 2ica sinh 0f + B (t2?/2) sinh /f }elf J

VOL. XCIII.-A L



11.6 Prof. EI. Lamb.

Equating these to zero, and eliminatilng the ratio A/B, we have the period- equation*

tanh/3f 4 V22$ .(12) tanh af (2? + 182)2 (

In t,he most; important type of long waves, U4, mf, are all smiall, anid the limBiting form of the equation is

(2h2)2.4t22a = 0, (13) whence Ic~~~2 -4i 2~ 4_ (X +, whe1lce ks/ ? (14)

in virLtue of (4). Hence if V be the wave-velocity

V\ 2 _ 4((?,u) (15) = 2X-+2fL p

This is in aareement with the ordinary theory, where the thickness is treated as infinitely small and the influence of lateral inertia is neglected.

For waves which are very short, on the- other hand, as compared with the thickness 2f, the quantities tf, af, /f are large, and the equation tends to the form

(2 t2 k 2)2_ 4 0 = 0. (16)

This is easily recogniised as the equation to deternmine the period of "R Ptayleigh waves" on the surface of an elastic solid.t It is kniown that if the substance be incompressible (h = 0, a = t) the wave-velocity is

V= 09554C(u/p)-, (17) whilst on Poisson's hypothesis of X = ,

ATV- 0 9194 (H/p)g (18)

These results will, in fact, present themselves later. 3. In virtue of the relations (4), (7), the equation (12) may be regarded as

an equation to find c- when e is given, i.e. to determuine the periods of the various modes correspondinig to any given wave-length; but from this point of view it is difficult to handle. It is easier to determine the values of t coiresponding, to given values of the ratio I3Ioc. This is equivalent to finding the wave-lengths corresponding to a giveln wave-velocity. Putting, in fact, ,8 = mnca, we find from (7),

= Ic2 - jii2h2 1r_M2 (19) 1-e

and therefore 02 -2 - (+2)-rn2) )

so 2 t X+2pL-i p,2nd ( so that V depends upon v?n only. * Ani equivalent equation is given by Rayleigh. t Rayleigh, 'Proc. Lond. Math. Soc.,' vol. 17, p. 3 (1887) ; 'Scientific Papers,' vol. 2,

i). 441.



On Waves in an Elastic Plate. 117

4. For simplicity we will suppose in the first ilnstanice that the substance is incompressible, so that X = Co h = 0, a-. Putting

/3= qmt, Uf =o, (21) the equation (12) beconmes

tanhnm,c = 4n (22) tanh co (1 + j)L2)2

whilst V2 = (1 -in2) p (23) It is evident that real values of m nmust be less than unity. Moreover we

have

d- tanhrnco = 2 ( Co, c, d1o tanh co co sinh 2 grco sinh 2 (24)

which latter expression is easily seen to be positive so long as m < 1. Hence as c, increases fronm 0, the first member of (22) increases from m to its asymptotic value unity. There is therefore one and only one value of w corresponding to any assigned value of em which makes the second meember of (22) less than unity. And since

(1 +T2)2 4i = (A-i) (rn3 +r2p 3rn-1), (25) the right-hand member of (22) is less than unity (for mn < 1) only so loiig as mn < 0-2956, which is the positive root of the second factor in (25). The; admissible real values of gm therefore range from 0-2956 to G. The former of these makes co = so, X' =0, and gives to V the value (1.7).

The values of co corresponding to a series of values of am between the above limits, together with the corresponding values (X'/2f) of the ratio of wave-length to thickness, and the corresponding wave-velocities, are given in Table I (p. 118).

So far f8 has been taken to be real, or t < k. In the opposite case we may write /l 2 = k2t2' (26)

and assume = A cosh ay ei r = B sin /3yei$x. (27) The period-equation is then found to be

tan ,83f - 4 21 .8 tanh af (-2 _122)2 (2)

b beinig still supposed to be real. In the case of incompressibility, writing

=w, = W.=) n, (29) tan nw 4n we have tanh c, -(1-nL2)2' (30)

whilst V2 = (1+nt2)Hp (31) L 2



118 Prof. H. Lamb.

If as n increases from 0 we take always the lowest root of (30), -we obtain a series of values of co continuous with the roots of (22) and dinminishing down to zero, when n = /3. This latter value makes V = 2V/ (plp), in agreement with the general formula (15) for long waves. Niumerical results for a series of values of nL ranging from 0 to \/3 are included in Table I.

Table I.-Syinmmetrical Type. X = o.

[The unit of V is '(,u/p).]

WI. n. a). '/2f. V.

0,2956 - _ 0-0 0 9554 0 29 - 867 0 362 0,957 0*28 7 09 0*442 0*960 0*27 6 38 0 *492 0 '963 0*26 5*93 0,530 0*966 0 .25 _ 561 0*560 0*968 0*20 - 457 0*662 0 979 015 4 35 0 1722 0-989 0 10 _ 414 0,759 0.995 0 05 - 4 -03 0 779 0-999 0 0 0 0 4-00 01785 1P0

0-1 3 872 0 -811 1P005 0-2 3 -570 0-880 1P020

-l _ 0 -3 3 '218 0 -976 1P044 0 - 0-4 2 883 1-090 1 P077

'I - 0-5 2,587 1 214 1-118 il - 00-6 2-331 1-348 1P166

l - 007 2-108 1,490 1P221 0-8 1 912 1-643 1P281 0-9 1P732 1-814 IP345 1-0 1r 2-0 V2

_ P1 Pl417 2-217 P 487 12 1269 2476 1562 1-3 1-121 2-803 1-640

__ 1-4 0-967 3-25 1P721 _ 1 -5 0,798 3-94 1 803 _ 1 -6 0-594 5*29 1P887

1 -7 0-292 10 -8 1i972 _ P *71 0-241 13-0 1P981 _ P1-72 0-175 15*0 1P989

1 -73 0-081 388 1-998 -V3 00 00 2 0

__________ ____~ ~ _ ,_ _ _ _ _ ____ _ __ _ _ _ _ _ _

The relation between wave-length and wave-velocity for the whole series ,of moides investigated in this and in the preceding section is shown by the curve A on the opposite page. The unit of the h-orizonital scale is X'/2f; that of the vertical scale is V/V/ (btp--I).

5. Oni the present hypothesis of incomnpressibility there is a displacement funrction P, analogous to the stream-function of hydrodynamics, viz., we hiave

U = aP/ay, V = -Nl/a. (32)




The lines T = const. give the directions in whiich the particles oscillate, whilst if they are drawn for small equidistant values of the constant their

2t0 _ .___ t _ .__ -.

1-6- -

1-4

1-2 'IO.

I*0

5 10o 1.S 2 0 25 30 3 S 4 0 4 5 5 0 5.5

FIG. 1.

greater or less degree of closeness indicates the relative amplitudes. In the case of standing waves the system of lines retains its position in space; in the case of a progressive wave-train it must be imagined to advance with the waves.

If in be real we find, omitting a constant factor, T = {2m cosh mwo sinb ty-(I +n2)cosh 6 sinh m.y}etx. (33)

In the opposite case we may write P = {2n cos not sinh ~y-(I -n2) cosh c sin nit}e. (34)

I have thought it worth while to make diagrams illustrating the configuration of the lines of displacement in the class of moodes of vibration which have so far been obtained.

The case of the Rayleigh waves, corresponding to a = cc, is, of course, of independelnt interest. In the present problem their wave-length is infinitely short; but if we transfer the origin to the surface y fg and then mnake f = oo, we get, omitting a numrerical factor,

T = (e6tY- 0 .5437 ebl) eitx, (35) where A = 0-2956. On this scale the wave-length may have any value. The diagram (fig. 2) shows the great difference in the character of the motion, and the slow diminution of amplitude with increasing clepth, as compared with the " surface waves " of hydrodynamics.* It follows, in fact,

* Lanmb, 'Hydrodynamics,' 4th ed., art. 228.



120 Prof. H. Lamb.

from (35) that the ratio of the vertical amplitude to that at the surface (in the same vertical) at depths of I 1, 1 wave-length, is 130, 0814,-0 340, respectively, whereas in the hydrodynamical problem the corresponding numbers(are 0-208, 0 043, 0-002.

FIG. 2.

In the case of t < 7l, the displacement function is given by (34). Ii have chosen for illustration, as giving a wave-length neither too great nor too small, the case of n =-6, which makes

= 0 594, a = 0 950, '/2f = 529, and,.accordingly,

T = (1010 sinh ysinnty) ei4X. (36)

The result is shown in fig. 3, which, like the former diag,ram, covers half a wave-length. The diagram-l may be taken to illustrate also, in a general

FIG. 3.



On Wcaves in an Elastic Plate. 121

way, the most importanit type of longitudinal vibrations in a cylindrical rod.

6. The modes of vibration so far obtained inielude all the more interesting types, from the physical point of view, of the symmetrical class; but there are, of course, an infinity of others. These correspond to the higher roots of the equation (30). Thus when n = 01 we easily find the approximate solutions

2 /-,-r 22-47, 42-47, 62-47, . .. (37) For i = 0-2

w/l-7r 12-28, 22-28, 32'48, .... (38) For i = 0-3

w/lI - 8-72, 15-39, 2205 ...; (39)

and so on. In these modes the plane xy is mapped out into rectangular compartments

whose boundaries are lines of displacement. This may be illustrated by the case of n = 1, wheni the internal compartments are squares. This happens to be particularly simple mathematically. The formula (34) for T is now indeterminate, but is easily evaluated. It is found from (30) that for small concomitant variations of w and X about n = I we have 83n) = 0. This leads to

T = sin (40) with =(2s+1)7r/2f, (41)

where s is an integer. The configuration is shown in fig. 4 for the case

IX - -F

FIG. 4.



122 Prof. H. Lamb.

s = 1; buit it is to be observed that the dotted lines in the diagram all represent planes which are free from stress, and that consequently any combination of themn may be taken to represent free boundaries. This particular solution is, moreover, independent of the hypothesis of incompressibility.* The surface- conditions (11) are, in fact, satisfied by

A = 0, 82 ,)192 = 0, cos,81f = 0, (42) which lead again to (40) anid (41).

As n is increased the compartments referred to become more elongated. For large values of i, and consequently small values of 0) or f, we have in the limit n;b = s7r, where s is integral. It is otherwise evident that the surface conditions are satisfied by

A-0, 0 0, sin,81f = 0, (43)

whence = 0, f = B sin (s7ry/f). (44)

The vibration now coInsists of a shearing motion parallel to x, with 2s nodal planes symmetrically situated on opposite sides of y = 0. The frequency is given by

2 S7r[tJ js 2 _

* S 7r H(45)

Asyntmetr iccl Miodes.

7. When the motion is anti-symmetrical with respect to the plane y= 0 we assume

= A sinh ay eitx, k = B cosh y etx, (46) where a, B are defined as before by (7). This gives for the stresses at the planies y = fn

pyy/b t = ? JA (S2 +j 32) sinh aSf- B 2 i t sinh ff } eil, (4)

p zJ/ = {A 2 i tc cosli af+ B (02 +?2) coshb,f } x. J These surfaces being free, we deduce

tanh/3f (- 2 +2)2 (48)t tanih af4 4~0'

4)

* It was noticed long ago by Lam6 as a possible miiode of transverse vibration (uniforn throughout the length) in a bar of square section, 'Theorie mathe6imatique de l"lasticit6,' 2nd ed., p. 170.

There is an analogous solution in the case of the synimmetrical vibrations of a cylindrical rod. The surface-conditions given on p. 277 of Love's 'Elasticity' (equation (54)) are satisfied by

A 07 Jl' (K'a) = 0, 2y' = p2p4p,.

In the notation of this paper the latter two conditions would be written

J1' (i3a) - 0, 292 -- .2 t (f. Rayleigh, loc. cit.



On Waves iM an Elastic Plate. 123

When the waves are infinitely short this reduces to the form (16) appropriate to Rayleigh waves.

In the case of long fiexural waves af, ff, tf are small. Writing

tanh aef = -af (1- cIa2f2), tanh8f = 8f (1- 2f'),

we find kS2 = _.4 2/ 12) 42 (49) on the supposition that k2/. 2, h2/ ,2 are small, which is seen to be verified. This makes

V 2 =4 02f2 X+A (50) X+2/jkp'(0

in agreement with the ordinary approximate theory. It may be pointed out in this connection that Fourier's well-known

calculation* of the effect of an arbitrary initial disturbance in an infinitely long bar is physically defective, in that it rests on the assumption that the formula analogous to (50) is valid for all wave-lengths. As a result, it makes the effect of a localised disturbance begin instantaneously at all distances, whereas there is a physical limit, viz. /{ (X+ 2,u) /p}, to the rate of propagation.

8. For the purpose of a further examination we assume the substance of the plate to be incompressible, so that =c _, and write 18 =T as before. The equation (48) becomes

tanh mno (1 + in2)2

tanhw 4im ' ~~~~~(51) tanh to 4?R m

where . = to f The wave-velocity is given by (23). Since in must be less than unity, whilst the second member of (51) exceeds

1 if ?n < 0-2956, it appears that for real solutions we are restricted to vallues of mn between 0-2956 and 1. A series of values of w corresponding to Values of in within this range is given on the next page.

The displacement-function is found to be

= {(1 + M2) cosh mco cosh y-2 cosh (o cosh m7yY} e?>. (52)

The fornms of the lines T = const. for the case of

09 = 0 9 w = 0 435 mo = 0-392, X/2f = 7'22,

are shown in fig. 5, for a range of half. a wave-length. Regarded as belol giDg to a staniding vibration, they indicate a rotation of the matter in the neig,h- bourhood of the nodes, about these points.

* See Todhuiiter, 'History of the Theory of Elasticity,' vol. 1, p. 112 ;# Rayleigh, Theory of Sound,' vol. 1, art. 192.



124 Prof. H. Lamb.

Table II.-Asymmetrical Type. X -oc. [The unit of V is 4(,kup).]

m}&. I w. x A'/ 2f. ) V.

0 2956 O 00 0) 9554 0'30 8'84 0 356 0 954 0'35 4'20 0,748 0'937 O '40 3 028 1 038 0-917 0'45 2'379 1-321 O'893 0 50 1 946 1 6414 0 866 0.55 1'627 1'931 0'835 0 60 1 377 2'282 0 800 0 65 1'171 2'683 0 760 0'70 0'995 3 157 0'714 075 0'841 3'736 0'661 0'80 0'700 4-45 0'600 0'85 0'563 5-58 0'527 0'90 0'435 7-22 0'436 0'95 0'300 10'5 0 312 1o0 00 xc 0 c0

* For small vatlues of ff the value of V is given by equation (50).

FiG. ). So far it has been supposed that k < ~, and consequently that m is real. In the opposite case we assume in place of (46)

= A sinh cy eitx, f = B cos ,fli eitx, (53)

and the period-equation is

tan /3f = - (1 _ 91 2)2 (54) tanih af - 4~23 ' (54

where 81 is defined by (26). In the case of incompressibility this becomies

tanue, __ (1-12)2

tanth o 4, '(55)

where w = n = i31/t. As the preceding investigation (summnarised in Table II) evidently covers all the modes in which v has the same signl throughout the thickness, the additional modes to which this equation relates may be dismissed with the remark that they are analogous to those referred



On 'Waves in an Elastic Plate. 125

to in ? 6 above. The particular case q = 1 is illustrated by fig. 4 if we imagine the lowest horizontal dotted line to form the lower boundary.

Influence of Cormpressibility. 9. Although the hypothesis of incompressibility has been adopted for

simplicity, the numerical calculations, so far as the mnore important modes are concerned, are not much more complicated if we abandon this restriction. It will be sufficient to consider the case of the symnmetrical types.

We have from (4) and (7)

2 kc2 -2 /2R2 (= + 2(X ) a22 R4 (56) * = Uk2-h 2 ?+

Hence if we write af = a), A = qlam , (57)

the equation (12) takes the formi

tanh qna = 4Mn (X + ?_) (X + 2 /-rn2O) (58) tanh a (X+2p.+?m2X)2

The relation of a to the wave-length is given by

52f2 = (X+2/_t-2a22u) (52

whilst the wave-velocity is given by (20). For numerical illustration, we may adopt Poisson's hypothesis as to the

relation between the elastic constants. Putting, then, X = ,w e have

tanh i - 8,n (3-_m2) tanh - (3 + 9n2)2 (

f = v/ (-(8-l2))) V2 - 3(1arn2) , (61) 3 - q2 p

PReal values of m nmust lie between 0 and 0'4641, this being the positive root of the equation

943+9rn12+15n-9 = 0 (62)

obtained by equating the second mnember of (60) to unity. The wave- velocity corresponding to this latter value of v? is

V _ 0 9194 v/ ( /P)) (63)

in accordance with the theory of Rayleigh waves, the wave-length being now infinitely small cQmpared with the thickness.



126 Prof. H. Lamb.

When m is imaginiary (= in), we have tanl n - 8n (3 + n 2)

tanh a (3-n92)2 '

{f= V/[1(3.In2)1o,V, =2 3(1?+n2)1 (65) 3?2 p The more important modes coming under these formulve are determined by the lowest root of (64) for values of n ranging fronm 0 to V,15, this latter number corresponding to an infinitesimal value of o, i.e. to waves of infinite length.

Numerical results are given in Table III, and the relation between wave-length and wave-velocity is shown by the cuLrve B in fig. 1 (p. 119). The unlit of the vertical scale is VI/V(p-1) as before.

Table III.-Symmetrical Type. X -. [The unit of V is 1/()t/p).]

m . .. _ _ __ __ /_2f. V.

0-4641 i 00 09194 0*45 5 220 0 '509 0 *925 0*40 3 *807 0 *692 0 -942 0 *35 - 3 343 0 *784 0 *956 0 *30 3 30855 0*844 0 *969 0 *25 2*918 0*888 0 978 0*20 - 2*805 0 921 0 986 0-15 2 727 0 944 0 992 0 110 - 2 678 0 -959 0 -997 0-0 010 2 640 0 972 1 0

- 0 1 2 604 0 983 1 003 0 -2 2 506 i o17 1-013

_ 04 2'214 1 129 1 049 - 0 6 1 911 1-268 1-102

0.8 1U649 1-412 1-163 I -0 1 432 1 -551 1-225 12- 1 253 1-683 1-284

_ 1-4 1-105 1 805 1-338 - 1 6 0i979 1 924 1 386

1/3 0 907 2 0 12 15-8 0-872 2 04 1,428 2-0 0-778 2-16 1-464

- 2'2 0 696 2-28 1A495 2-4 0 620 2 42 1 522

- 2 6 0 -551 2 58 1 544 _ 228 0-486 2-78 1P564

3-0 0-422 3 04 1P581 __ 3 -2 0-359 3.40 1 596 - 3-4 0O292 3.99 1`609

3 -6 0'216 5 -15 1 620 3 .8 0 109 9 8 1-630

l 1~/15 0 01-633

As it increases from 0, the modes corresponding to the higher roots of (64) have at first the same general character as in the case of. incompressibility




(? 6). When n = /3, we have the division into square colmpartmenits, to which fig. 4 refers. When n is infinite, whilst nw is finite, we have

c= 0, /31f =co = s7r, { f = s7r/I/V2. (66) A reference to (11) shows, in fact, that, independently of any special relation between the elastic constants, the boundary conditions are satisfied by a = 0, sin,3f= 0, and A (2_i312) + 2 B 0, cos sr = 0. (67)

This leads to = i(ossr++os~!L)i x,

It = C

(OS S7r+ 2 COS-fY e 1 - S \// (X)}sin/e.(68)

i= X }sin s There is here a transition to the case where a, as well as 3, is inmaginary.

Writing 2- h2-_2, 32 - 7,2 _2 (69)

and assuming (for the case of synmmetry)

= A cos aiy eix, 4 = B sin /3y eitx, (70) the period-equation is fouind to be

tan ,811 4 (271) tan a,f (fl1 2 - 2)2'(1

Since - h /1 2-1 2~2 -

_ _ _ _ _ _ _ _ _ t2=MW1 21 =H@12-(\+_K)MlS ~(72)

we find, writing 1f= n, flif= qw, (73)

tautqat 4(?t q(X +2IL- q2pk) (4 tan 4 (o2 +X + 2 I)2o (74)

Also

f2 _+ 2 2

2 _ (\~+ IL),(2 -2 (X?2ft)(q21 /L

(75)

q2/L-(X+ 2t) pp On Poisson's hypothesis these becomue

tan qw -8 (q2_ 3.) taii, (22+3))2 6)

,tf =V1 [ (q -3)]w, V2 - 3 21 - (77) qTh p

The value of q may range dlownwards from'' oc to v'/3.



128 On Waves in an Elastic Plate.

A mxinute examination of these modes would be laborious, and would hardly repay the trouble. In the extreme case where - = /3, the equation (76) is satisfied by either a zero value of tan q&, or an infinite value of tan a. The former alternative gives the shearing motions parallel to x, already re:ferred to at the eind of ? 6 (equations (43) and (44)). The other alternative gives a vibration at right angles to x. It is, in fact, obvious from (11) that the conditions are satisfied by

= 0, B = 0, cos 0, (78)

whence = cos(2s+1)WY, +r = 0, (79)

:r 2 = (S?+ )27 X+2 (80)

W hen q slightly exceeds \/3, we have modes of vibration resembling the above types, except for a gradual change of phase in the direction of x. The corresponding values of V, as given by (77), are very great, but it is to be remarked that the notion of " wave-velocity " is in reality hardly applicable (except in a purely geometrical or kinematical sense) to cases of this kind, and that resuLlts relating to moldes lying outside the limits of the numerical Tables are more appropriately expressed in terms of frequency.* As already remarked, there is a physical limit to the speed of propagation of an initially local disturbance.

* C(f. ' Hydrodynamics,' art. 261, where a similar point arises.



on waves in an elastic plate

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