In this note, a liquid layer above a thermoelastic halfspace is considered. The frequency equation for the waves generated at the interface is obtained and a few limiting cases are discussed

Harinath, K. S.

English

Defence Science Journal

A NOTE ON WAVES GENERATED AT A tIQUID-SOLID INTERYACE K. 8. HARINATEI Regional Engineering College* Ticuahirapalli faat- i3  ~ u g w t  1975) In thin note, a liquid layer above a therIria~lestio haI&gacer is coneidad. The hquenoy equation for the wavds gene- rated at the interfaae in obtained and a few limiting o m  an, diaouased. In a previoue paper' the author hadstudied the frequency equation for waves generated a t  the interface between two halfspacea one assumed elastic and the other thermalaatic. In this note, i t  is proposed to study the surface waves propagated at  the interface between a liquid layer and a thermalastic halfspace. Various limiting cases are discussedl. For the clasdcal case sf s liquid layer above an elastic halfspace see the lucid and also the discussion which followis, on geophysical studies of water-covered 8I'eaS. B A S I C  E Q U A T I O N S  We set up a cartesian coordinate ~yatem fs,p,%) in such a manner that the thermcelaatic halfspace occupies the region z 4 R, the liquid layet uia%npiesO < z < H and z = 0 is the free surface. So, the thickness of the liquid hper  i~ and z * @ is t-Be interface. For the sudace waves propagated in the $-direction the displacement components are gi,veh by; where the potential functions 4, # and the temperature perturbation T aatiefy the equations where p is t h ~  density of the thermalaatic medium, a is theisothermal (compressional) wave velocity, f i  ia the ehear wave velocity, k is the coefficient of thermal conductivity, s ib the specific heat a t  conatant strain, y is the ratio of the ccefficient of volume expansion to isothermal compreaaibility ~ n d  To is the  equilibrium temperature prior to the appearanoe of 8 disturbance throughout the region e 9 0. bsuming the aolutione of equations (2 ) in the form (T, 4, $1 = (T', 9 ' 9  #'I exP ( 8 ~  -4 3  (3) we ! obtain the solutions given by equation8 (4) below, where i t  ia clear that T, #, pt +O es z-+ cc, in , order that surface waves may be described. where y = @ - B', Re (&)a09 j = 1, 2, 8 and p,', qa2 are the roots of the equation h%4 - @ [kc' + iwpso" (1 + e)] + i ~ 8 ~ s  = o (5) where e = y2 Te/P'saais the coupl~ng conatant and , ,sj/p. The components of displacement in the liquid ia+f thickness H ir given by I .  - .  a = #*#, . r = o,, + = (6) where the potential function 4' eatis6;s where p is the velocity of tbe raveain the 1i;uid.j ' . .., .- - , - Let P' be the density of the liquid, p ita hpdrost.tic pmaaure aed r' ;= C - 3/@ ai th & (n) nbn-negati~e. The liquid layer is assumed to be maintained at the constant tamprature pQ. ~asuming again a aimple harmonic time dependenae factor exp (-+), we obtain q3' = [Ate-* f B'eT &*eel) i 1 k. 9) =iwp'+' (81 The constant. A, B, C in (4 ) and d: in (8) hw*: t; be d d n m i u d  by ming the boundary . * conditions. - -- * :  -- - A  H e  1 B O U  D A R Y " C O N D I T I O N ~  We impose the following conditiond: (a) At the free surface p = 0 / < - (b)  Across the interface r = R 3- - 1 uar=p, Q U = O S  T = o  W m W ,  i ,  w b m  $he stre8ses om, + a" are given by : ' S  a, = P~'(+W + 2P@'($CI-+,,r' - y2a Qu - pie8(%* + A#-- #a#) (9) I t  i8 to be n o w  that ecnditics on the tangential d i a ~ l c s p l e q t ~  pen be imposed a t  the interfwe, , I  HuwA~a :w&ea Generated at s, Liquid-Solid I n t d w  ~ R E Q U E N C Y  E Q U A T I O N  It is to  be noted that equation (10) id the thermalastic version of the equation (4 -154), given by Edng2, et.a2. As a remark, we can add that (10) cannot be deduced from the more general frequency equation for waves a t  the interface of a solid halfspace with a solid layer above by merely msuming one of the Lame constants to be zero. Hence, a separsrb study of the liquid-solid interface is necessary. . < - L I M I T I M U  C A S E S  - By the uae of the above five boundary conditions on a = 0 and z = H we obtain homogeneoua equations for the five constants A, B, C, A' B'. These have a nontrivial solution only when the matrix of the coefficients of A, B, C, A', B' is singular. This deterrnin.8 the frequency equation to  be There are various limiting caaeir and most of these are discussed'. Here we merely discuss the forms of the frequency equation for large and small frequencies (w) when the wove nvmber (6) ia fixed. The other cases for large and small wave lengths when the frequency is fixed and also for small valuea of the coupling constant may be obtained on lines similar to those in the author2 eedier paper". \ emH e4H 0 0 0 iw$ iw p' p (2g82-d)  p{2pV-~2) -2i$ap,9 0 0 2isX1$ 2i8X8fi2 2fia6a-oa 0 0 wE-aw wa--0(=gB2 0 rn --~n ,j c -XI 4 -4 is (i) For large frequelacies: Suppose o >> 1 and 8 is fixed. I n  case the liquid ltayer is also very thin (i.e. H << I), the frequency equation (10) takes the form = 0 where c = o/8 measures the phase-velocity ( = pe,(l/c)]-1) and the attenuation in the x-direction (= 0 F m  (l/c)I). Putting 6 = 0 in equation (11) we obtain ca [p' H - pa] = 4 p P  (12) which is nothing but equation (4-154) in Ewinga, et. al. for w >> 1 and H << 1. In  the general case when w >) 1, the frequency equation (10) takes the form : {p'wp tan (wHlP) - 4pS2/P] (pseas + 2ihco) - 2ipko(w3 = 0 (13) Various  case^ of intereat, vin: O<H< 1 Z I for real .dues of o of (13) may be diacusasd. (ie) For arc incompressible solid: For an inoompressible solid we h o w  that the complessiollaf wave velocity is infinite. Hence if the thermclekstio ha1fpe z<H ia incompressible equation (10) takes the form pm (2P8a - o2)s + p ' 6 ~ 3  tan (iml?) - 4pnsPpt/81-- 02[p = 0 (14) D m  8cn. 3.. VQL. 81, JUDABX 1977 .. For large or nmall vduea of the frequency o when S id fixed (14) may be evslusted. (iti) For . sal~Ufreqwwdes: . Suppose a << 1 and 8 is fixed. Then the frequency equation (10) takes the form \ p (9g8a - tdy + p 8 d  tan.(iSE) - 4 ~ 8 ~ / 3 4  = 0 (16) which shows that the phase velocity end the attenuation are independent of the compressional wave velocity a and hence for small valuea of the frequencies, (14) reduces to (15) simply by replacing m by 8 and neglecting aP/j9? C O N C L U B I O N S  In all the limiting oasea aonaidered above &o*p.be obaerved that the nave velocity delwnda on the frequency and hence there is dispersion. In other weds  in equations (11), (IS), (14), (15) co and O are always coupled whereas in. the elastic case (equakion (12)) w is proportional to 8. Thus the dispersion whichis absent in equation (12) but present in the other equations is due to the introduction of temperature perturbation. Further for small freqnenciea; the compressibility of the solid hw no in fluenae on the frequency equation as shown by equations (14) and (15). The above results hive direct bearing on defence studien of underground explosions, whether they are caused by nuclear deviaea or conventional oneb. These enable us to compute the and the frequency of the shock wave8 thus generated. The author is grateful to Prof. Dr. V. R. Thiruvenkataohar for many helpful discuaeiona. \ - R E F E R E N C E 8  1. R~IXATE,  K.S., w. Sci. J., 8% (1976), 67. 2. E m a ,  W.M.D. J C $ ~ K Y ,  %a. It PRSW, F., *El&o wawa iPi layered &B: 3. CEADWLU~~K, P., Prbg. X&, 1, (It%@) 865. 

A Note on Waves Generated at a Liquid Solid Interface

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A Note on Waves Generated at a Liquid Solid Interface

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