Dilatational and shear waves without energy dissipation in a rotating medium

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Coaammica~m, VoL 26, No. 2, I~. 22.$-227, 1999 ~ o 1999~ $ c ~ a c e !.~ Pdnaxlin the USA. All d61asresma~ 0093-641 ~

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D l l a t a t i o = a l a a d s h e a r maces r 2 t a o u t e n e r g y ddsslpattoa Jn a rota medium A. Khan and F. Ahmad Department of Mathem,.~ics, Quaid-i-Azam University, Islamabad, Pakistan.

(Received 13 July 1998; acceptedfor print 11 February 1999)

1.

Introduction Green and Naghdi [1] have formulated a theory of thermoelasficity without energy

dissipation. Chandrasekharaiah and Srinath [2] have considered this theory for a medium rotating uniformly. They have considered only dilatational and shear modes and have claimed that the phase velocity of both of these modes is reduced by a factor of I

(1 + q"sin:~0)-2

(1)

where q is the ratio of the frequency of rotation and the frequency of the wave and q~ is the angle between the polarisation vector of the displacement and the axis of rotation. Effects of rotation enter the theory through the factor (1) and disappear when cp = 0 i.e. when the displacement is in the direction of the axis of rotation. In this short note we shall show that for the pure dilatational or shear modes considered in [2] the angle • must vanish and the results of Chandrasekharaiah et al. [2] must reduce to earlier published results [3].

2.

Basic equations We follow the notation of [2]. The coupled thermoelastic equations of motion are C~ V 2 u + (C~ - C~ )Vdivu - C~VO =ii + p x (p×u)+2 p × u

(2)

C~ V:0 = 0 + edivii

(3)

Make the assumptions

and

u = a exp{i(~t - 11 n .x)}

(4)

0 = b exp{i(cot - 11 n .x)}

(5)

In the above n is a unit vector giving the propagation direction of the wave and p denotes the uniform angular velocity. Both n and p are real. However the vector a, indicating the direction 225

226

A.KHANandF.AHMAD

of the displacement,

the wave number n and the amplitude

complex a is an indication Substitution

of the fact that the components

of the displacement

are out of phase.

of the expressions (4) and (5) in Eqs. (2) and (3) gives

(w2 + p2 - Ci q2 )a - [(C?, -Cg )TJ’ ( a. n) - Ci i b$n and

b are assumed to be complex. A

- [(p.a)p + 2 io (px a)] = 0

(Cs q* - W2)b + imp2 (a.n) = 0

(6) (7)

The above equations are Eqs. (4) and (5) of [2]. Elimination (0’ + p2 - Cf n2 )a - [(C; -Ci ) - crcC$~~z T

of b from Eqs. (6) and (7) gives

In* (a. n) n - (p.a)p = 2 iw (px a)

(8)

Equation (8) is a vector equation and determines a in terms of n and p.

3. Dilatational and shear waves For a dilatational

wave a is parallel to n. Since n is real, a must also be real. Let

a=An Equation (8) can be written as

pxn=an

- ‘1 2iw

(p.n)p

(9)

where a=2.ow-+p 1 [ ’

3 - { l- c,$?}c;7~].

1

Let the angle between the vectors n and p be cp. Taking scalar products of Eq. (9) first with n, then with p we find a - $-p2

(10)

coscp = 0,

1 a p coscp - ~pj

coscp = 0.

If we let coscp = 0, it would imply p.n = 0 and a = 0, hence Eq. (9) will lead to p x n = 0, or sincp = 0, which contradicts the assumption 1 o-%p

2 =O

coscp = 0. Thus coscp f 0, and Eq. (11) gives

.

Now Eqs. (10) and (12) together imply cos2q = 1 or

(12)

WAVES W I T H O U T ENERGY DISSIPATION sinq) = 0

227 (13)

We have shown that it is impossible for a dilatational mode to propagate in a rotating medium within the framework of the Green and Naghdi theory [1] unless the wave is polarised in the direction of the axis of rotation i.e. q~ = 0 or n. Eq. (12) is the secular equation for the propagation of a dilatational mode and is equivalent to Eq. (13) of [2] if sinq) is taken equal to zero. For a shear wave a . n = 0 and Eq. (7) shows that this mode is not coupled with the thermal field. The governing equation, in this case is Eq. (2) with 0 = 0. This equation has been analysed by Schoenberg and Censor [4], who have shown that a transverse mode, with arbitrary frequency, will propagate with velocity C s only if the wave is polarised parallel to the axis of rotation. Thus in both cases considered in [2] sintp vanishes and the results derived there are devoid of any rotational effects. For a non trivial discussion of the rotational effects one must consider propagation in an arbitrary direction. Such modes are neither dilatational nor transverse in nature [4].

References t.

A. E. Green and P. M. Naghdi, J. Elasticity 31, 189 (1993).

2.

D. S. Chandrasekharaiah and K. S. Srinath, Mech. Res. Commn. 24, 551 (1997).

9.

D. S. Chandrasekharaiah, Mech. Res. Comnm. 23, 549 (1996).

4.

M. Schoenberg and D. Censor, Quart. Appl. Math. 31, 115 (1973).