BULLETIN of the Malaysan Mathematcal Scences Socety http://math.usm.my/bulletn Bull. Malays. Math. Sc. Soc. (2) 28(2) (2005), 163 172 Numercal Soluton of Boussnesq Equatons as a Model of Interfacal-wave Propagaton L.H. Wryanto Department of Mathematcs, Insttut Teknolog Bandung Jalan Ganesha 10 Bandung, Indonesa leo@dns.math.tb.ac.d Abstract. A model of nterfacal-wave propagaton s solved numercally based on one-dmensonal tme doman Boussnesq equatons, usng a predctor-corrector method. The numercal procedure s able to show the effect of non-lnearty of the model by observng the wave speed of soltary waves. The procedure s then used to smulate the wave propagaton generated, on the left boundary, by a snusodal functon. 2000 Mathematcs Subject Classfcaton: 76B07 (76B55) Key words and phrases: Boussnesq equatons, nterfacal wave, predctor-corrector method. 1. Introducton The exstence of a class of nonlnear waves n a densty-stratfed medum has been demonstrated by several nvestgators, such as Benney [1], Benjamn [2], Ono [6], and Kubota, Ko & Dobbs [5]. They derved a sngle equaton dfferently as the model of ther nternal waves. For two flud system Segur & Hammack [7] and Cho & Camassa [3] derved the model of the nterfacal waves n a KdV-type equaton. Another model descrbng the nterfacal waves s Boussnesq type. The dervaton of the model can be seen n Grmshaw & Pudjaprasetya [4] who use Hamltonan formulaton. Most of the equatons were then solved analytcally to descrbe soltary waves. Numercal solutons of the equatons are an alternatve way to observe the wave propagaton. Wryanto [8] solved the KdV equaton for the nterfacal wave by fnte dfference method. The procedure was able to smulate the wave propagaton n one drecton and ts deformaton at the nterface that can be explaned as a class of one-drecton waves from the model of Boussnesq type. In ths paper the propagaton of the nterfacal waves s observed numercally from the model of Boussnesq equatons. Ths numercal approach s able to show the propagaton for more general type of waves than just soltary, such as snusodal. The procedure s constructed by predctor-corrector method, and wll be used to Receved: October 11, 2004; Revsed: March 16, 2005.
164 L.H. Wryanto show the characterstc such as n analytcal works of soltary waves,.e. the wave breaks up nto two, and each wave propagates n dfferent drecton. Meanwhle, to preserve the soltary wave propagates n one drecton, t s requred a relaton for the ntal elevaton and velocty. In lnear case, the coeffcent q c s determned quanttatvely, and t s larger than the result from the analytcal formulaton as the nonlnear effect of Boussnesq equatons. The wave speed correspondng to the one-drecton wave s another quantty that can be determned followng q c. In presentng ths paper, we frst descrbe the Boussnesq equatons as the model of nterfacal wave n secton 2. To provde the condton of one-drecton wave, we derve the relaton between the elevaton and velocty based on depth average n secton 3. In the next secton, detals of the numercal procedure are gven, and followed by presentng the numercal computatons n secton 5. All descrbed n ths paper s fnally concluded n secton 6. 2. Boussnesq equatons The problem s the moton of gravty wave at the nterface of two-flud system havng dfferent densty ρ 1 for the upper flud and ρ 2 for the lower flud as shown n Fgure 1. Note that we then use varables wth subscrbe 1 for the upper flud and subscrbe 2 for the lower flud. In the undsturbed fluds, the depth of each layer s h 1 and h 2 measured from the nterface that s chosen as the horzontal x-axs. Therefore the nterfacal wave s presented as the elevaton of the nterfacal y(x, t) from the undsturbed level, where t s as the tme varable. Meanwhle, the flud system s bounded above and bellow by flat plane. Fgure 1. Sketch of the coordnates The fluds are assumed to be ncompressble and nvscd, and the flow s rrotatonal. Ths allows us to formulate the problem n potental functons φ 1 and φ 2 satsfyng Laplace equaton (2.1) 2 φ = 0 for = 1, 2 n each flud doman, the knematcs condtons (2.2) (2.3) (2.4) φ 1x = 0 on y = h 1 φ 2x = 0 on y = 2 y t + φ x y x = φ x on y = y(x, t) from upper and lower fluds, and dynamc condton on the nterface (2.5) ρ 1 {φ 1t + 1 } { 2 (φ2 1x + φ 2 1y) + gy = ρ 2 φ 2t + 1 } 2 (φ2 2x + φ 2 2y) + gy
Numercal Soluton of Boussnesq Equatons as a Model of Interfacal-wave Propagaton 165 Equaton (2.1) followed (2.2)-(2.5) s approxmated nto the Boussnesq equatons for the nterfacal wave as derved n [4], u t + g(ρ 1 ρ 2 )y x + 2νuu x = 0 (2.6) y t + h1h2 ρ 1h 2+ρ 2h 1 u x + 2ν(yu) x + 2βu xxx = 0 where g s acceleraton due to gravty, (2.7) ν = 1 ρ 2 h 2 1 ρ 1h 2 2 2 (ρ 1 h 2 + ρ 2 h 1 ) 2, β = h2 1 h2 2 ρ 1 h 1 + ρ 2 h 2 6 (ρ 1 h 2 + ρ 2 h 1 ) 2 and u s defned as (2.8) u = (ρ 2 φ 2 ρ 1 φ 1 ) x on y = y(x, t). For convenence, Equaton (2.6) s non-dmensonalzed by takng h 1 as the unty of length and gh 1 as the unty of speed. Therefore two parameters nvolved n the equatons are (2.9) h = h 2 h 1, ρ = ρ 2 ρ 1. The equatons are then wrtten n normalzed varables whch wll show the domnatng terms n the equatons, by defnng (2.10) η = y a, ξ = x a, τ = t a, ϑ = u a where a s a reference value of y such as the order of ampltude. These transform (2.6) to the form (2.11) ϑ τ + ( (ρ 1)η + a(ρ ) h2 ) 2(h + ρ) 2 ϑ = 0 ξ ( h (2.12) η τ + ρ + h ϑ + a(ρ h2 ) (h + ρ) 2 ηϑ + h2 a(ρh + 1) 3(ρ + h) 2 ϑ ξξ For small a, (2.11) and (2.12) can be reduced to a smple wave equaton of ϑ or η by elmnatng one of both unknowns, and can be solved analytcally by separatng the ndependent varable. On the other hand, the effect of the nonlnear and thrd dervatve terms wll be observed by solvng (2.11) and (2.12) numercally. 3. One-drecton wave Equatons (2.11) and (2.12) are able to explan wave propagaton n two drectons. In specal relaton of the ntal condtons ϑ and η, the equatons gve a wave travellng only n one drecton. We derve that relaton by consderng the depth-average horzontal velocty defned as (3.1) φ1x = 1 1 aη (3.2) φ2 x = 1 h + aη 1 aη φ 1x dy φ 2x dy n non-dmensonal varables correspondng to (2.10). ) for the upper flud for the lower flud ξ = 0
166 L.H. Wryanto The ntegraton n (3.1) and (3.2) can be expressed n η derved as followed. For the lower flud, we start from Laplace equaton (2.1) ntegrated on the thckness of the lower flud (3.3) aφ 2xx + φ 2yy dy = 0. From the knematcs condtons (2.3) and (2.4), and relaton (3.4) (3.3) becomes (3.5) x φ 2x dy = a φ 2x aη η x + x φ 2x dy + η t = 0. φ 2xx dy The next step, we suppose that the wave travels to the rght. Ths s expressed n η as (3.6) η t + cη x = 0 wth wave speed (3.7) c = h(ρ 1) h + ρ obtaned from the leadng order of (2.11) and (2.12). We then substtute (3.6) to (3.5), and ntegrate wth respect to x, gvng (3.8) φ 2x dy = cη The constant of ntegraton s zero as η, φ 2x 0 for x. Relaton (3.8) s then used to elmnate the ntegral n (3.2), so that we have (3.9) φ2x = cη h + aη. When the above procedure s followed for the upper flud, (13) becomes (3.10) φ1x = cη aη 1. Results (3.9) and (3.10) are used to approxmate u n Boussnesq equatons to obtan one-drecton travelng wave,.e. (3.11) u ( ρ h + aη 1 ) cη. aη 1
Numercal Soluton of Boussnesq Equatons as a Model of Interfacal-wave Propagaton 167 4. Numercal procedure In nvestgatng the propagaton of the nterfacal waves, the model n form of Boussnesq equatons s solved numercally. A numercal procedure s developed for (2.11) and (2.12). A predctor-corrector method s chosen as t can be appled to the equatons explctly. To smplfy n explanng the method, (2.11) and (2.12) are expressed n form ϑ τ = F(η, ϑ) where η τ = G(η, ϑ) (4.1) ( F(η, ϑ) = (ρ 1)η + a(ρ ) h2 ) 2(h + ρ) 2 ϑ2 ξ (4.2) ( h G(η, ϑ) = ρ + h ϑ + a(ρ h2 ) (h + ρ 2 ) ηϑ + h2 a(ρh + 1) 3(ρ + h) 2 ϑ ξξ The equatons are fnte-dfferenced on a grd n ξ and τ. The dscrete ndependent varables are expressed as ξ = ξ, τ = n τ wth level n referrng to nformaton at the present, known tme level. The predctor step s the thrd-order explct Adam-Bashforth scheme gven by (4.3) ϑ n+1 (4.4) η n+1 = ϑ n + τ 12 = η n + τ 12 [ 23F n 16F n 1 [ 23G n 16G n 1 + 5F n 2 ] + 5G n 2 ] The predcted values (4.3) and (4.4) are then used to evaluate F n+1 and G n+1 from (4.1) and (4.2) whch wll be used to correctng ϑ n+1 and η n+1 above. The scheme for t s the fourth-order Adams-Moulton method, gven by (4.5) ϑ n+1 (4.6) η n+1 = ϑ n + τ 24 = η n + τ 24 [ 9F n+1 [ 9G n+1 + 19F n 5F n 1 + 19G n 5G n 1 + F n 2 ] ) + G n 2 ]. Ths scheme s also calculated explctly. In runnng the numercal procedure, the predctor step requests two prevous tme levels at n 1 and n 2. Ths must be treated specally at n = 0 and 1 as we need values before the ntal ones. Ths problem can be solved by defnng the same value at the ntal condton. Another problem s boundary condtons and some values of ϑ n and η n outsde the observaton doman. We suppose to observe the wave propagaton n ξ [0, L] and we dscretze as ξ = ξ, ξ = L, = 0, 1, 2,, M. M. ξ
168 L.H. Wryanto Table 1. Data of the left-gong wave (a) and the rght-gong wave (b). t x a m 24.64147 57.24334 0.07345 26.87754 56.34891 0.07328 29.11360 55.45449 0.07311 31.34967 54.56006 0.07294 33.58574 53.66563 0.07278 35.82181 52.77120 0.07261 38.05788 51.87678 0.07244 40.29394 50.98235 0.07227 (a) t x a m 24.64147 76.92074 0.07345 26.87754 77.81517 0.07328 29.11360 78.70959 0.07311 31.34967 79.60402 0.07294 33.58574 80.49845 0.07278 35.82181 81.39288 0.07261 38.05788 82.28730 0.07244 40.29394 83.18173 0.07227 (b) The fnte dfference for the dervatve (up to thrd) of ϑ and η n the observaton doman needs values from = 2 to = M +2. As the values outsde the doman are provded every tme level n, we can overcome ths problem by lnearzng those values of both ϑ and η to the values n the doman. Ths can be done as the assumpton that the doman s relatvely long so that the boundary gves very small effect and can be neglected to the wave whch s propagatng far from the boundary. 5. Numercal results The numercal procedure descrbed n the prevous secton s used to observe the propagaton of nterfacal waves for varous ntal values ϑ(ξ, 0) and η(ξ, 0). Most results presented here are obtaned from calculaton usng ξ = 0.1, τ = 0.02, a = 0.1; and waves are plotted n non-dmensonal varables y versus x. The propagaton and deformaton of waves are shown by plottng the result of calculaton at some tme levels t n the same coordnates, but we shft the values y upper for hgher tme levels. Fgure 2 s typcal nterfacal waves when the ntal condtons are soltary n the form (5.1) y(x, 0) = A sech 2 [b(x xo)] for constants A, b, xo, and the horzontal velocty u = 0. The plot s the numercal result for quanttes h = 0.8, ρ = 1.5, and A = 0.15, b = 1.0, xo = 67.08. From the calculaton we found that the soltary wave breaks up nto two, and then each wave travels n dfferent drecton. The ampltude and the wave speed are calculated from Table 1a representng tme t, poston of wave peak x and the ampltude a m at that tme for the left-gong wave, and Table 1b for the other wave. We present the data for the last 8 curves as shown n Fgure 2 where the ntal wave has broken up nto two waves. The thrd column of both tables ndcates that the both waves have same ampltude, but t decreases by ncreasng tme. Our calculaton shows that up to t = 40.29394 the maxmum heght reaches 0.0722. Meanwhle, plot of the frst column versus second one produces a lne x = 0.4t + 67.1 for Table 1a, and x = 0.4t + 67.06 for Table 1b. These are the poston of the wave peak at tme t.
Numercal Soluton of Boussnesq Equatons as a Model of Interfacal-wave Propagaton 169 Fgure 2. Propagaton of nterfacal wave y(x, 0) = 0.15 sech 2 (x 67.08), u(x,0) = 0 for h = 0.8, ρ = 1.5 (a) (b) Fgure 3. Propagaton of wave for h = 0.8, ρ = 1.5 wth (a) u(x,0) = 0.7 y(x,0), (b) u(x, 0) = 1.5 y(x,0) The coeffcent of the lne represents the wave speed,.e. both waves have the same wave speed 0.4. The negatve sgn ndcates left drecton. Now when the above ntal soltary wave s followed by u(x, 0) = q y(x, 0), wth constant q, the wave wll break up nto two havng dfferent ampltude. Fgure 3a shows the wave deformaton for q = 0.7. The rght-gong wave has wave speed c R = 0.446 and ampltude 0.115. Meanwhle the other wave has wave speed c L = 0.400 and ampltude 0.029. For larger q the left-gong wave has smaller ampltude, and q c = 1.183 s the crtcal value where no left-gong wave. Ths crtcal value s dfferent wth the coeffcent of the lnearzed of (3.11),.e. 1.199, snce we solve the Boussnesq equatons nvolvng the nonlnear terms. Therefore, the dfference s the corrector factor of the lnearzed (3.11). For q > q c the left-gong wave appears wth negatve ampltude. We show ths n Fgure 3b for q = 1.5. In comparson wth the KdV-type equaton, we found that the numercal soluton of Boussnesq equatons can explan the character n KdV equaton by settng u(x, 0) = q c y(x, 0). For soltary waves deformng to some soltons, our procedure s able to show them usng the ntal elevaton as gven n (5.1) for a sutable value b wth respect to the ampltude. Fgure 4a shows a soltary wave deformng to two soltons. We obtan that result as the soluton of Bossnesq equatons usng ntal condton y(x, 0) = 0.4 sech 2 [0.5(x 10.0)] u(x, 0) = 1.183 y(x, 0)
170 L.H. Wryanto (a) (b) (c) Fgure 4. Wave deformaton from soltary to two soltons (a), three soltons (b), and appearng a tran of waves (c). Table 2. The crtcal value q c and wave speed c for varous h and ρ. ρ = 1.2 ρ = 1.5 h q c c 0.5 0.81 0.265 0.6 0.756 0.274 0.7 0.724 0.276 0.8 0.701 0.277 (a) h q c c 0.6 1.290 0.400 0.7 1.226 0.406 0.8 1.183 0.427 0.9 1.148 0.320 (b) and parameters h = 0.8, ρ = 1.5. For smaller values b the ntal soltary wave can deforms to more soltons, and larger values b a tran of small waves appears behnd the man soltary. We show these n Fgure 4b and 4c, correspondng to b = 0.2 and b = 1.7 respectvely. For other values h and ρ, the crtcal value q c was determned as descrbed above. The correspondng wave propagates to the rght wth wave speed c as gven n Table 2. The ampltude of the wave was observed durng propagatng and we found that t s relatvely constant around a m = 0.150. For larger values h, our numercal procedure s not able to gve smooth waves, snce ths ncreases the coeffcent of ϑ ξξξ n (12), and ths term domnates the equaton.
Numercal Soluton of Boussnesq Equatons as a Model of Interfacal-wave Propagaton 171 Fgure 5. Snusodal waves generated on the left as the boundary and propagatng along the nterface. Fgure 5 shows the propagaton of a typcal snusodal wave comng n the observaton doman havng ρ = 1.5 and h = 0.8. At the begnnng the nterface s undsturbed, and the left boundary s assumed to be a wave generator wth ( u(0, t) = y(0, t) = 0.15 sn(ct) ρ y(0, t) + h 1 y(0, t) 1 ) cy(0, t) where c s the ncomng wave speed defned n (3.7). From our calculaton, we obtaned that the elevaton of the nterface forms waves propagatng to the rght, and deformaton occurs as shown n Fgure 5. The crest of the snusodal deforms to soltary-lke wave and breaks to two waves. Ths s also obtaned from the KdV model, see [8]. 6. Conclusons Numercal solutons of Boussnesq equatons for nterfacal wave have been presented. The model was solved by a predctor-corrector method. We demonstrated the solutons as the wave propagaton for typcal soltary and snusodal. The ntal condton of the horzontal velocty plays an mportant role n observng the wave deformaton, manly break up to two waves travellng n dfferent drecton. In case where the ntal velocty u s lnear to the ntal elevaton, we found that the wave travels only n one drecton for a certan value of the coeffcent q c of that lnear relaton. The class of these one-drecton waves can explan the character of solutons n KdV model. The non-lnear effect of Boussnesq equatons s shown n values q c and wave speed c whch are dfferent from the lnearzed (3.11) and (3.7) for varous h and ρ. Acknowledgment. The research of ths report was supported by QUE (qualty undergraduate educaton) project for Mathematcs Insttut Teknolog Bandung. The author thanks Dr. Pudjaprasetya for stmulatng dscussons. References [1] D. J. Benney, Long nonlnear waves n flud flows, J. Math. Physcs. 45 (1966), 52 63. [2] T. B. Benjamn, Internal waves of permanent form n fluds of great depth, J. Flud Mech. 29 (1967), 559 592.
172 L.H. Wryanto [3] W. Cho and R. Camassa, Weakly nonlnear nternal waves n a two-flud system, J. Flud Mech. 313 (1996), 83 103. [4] R. Grmshaw and S.R. Pudjaprasetya, Hamltonan formulaton for the descrpton of nterfacal soltary waves, Nonlnear processes n geophyscs 5 (1998), 3 12. [5] T. Kubota, D. R. S. Ko, and L. D. Dobbs, Weakly-nonlnear, long nternal gravty waves n stratfed fluds of fnte depth, J. Hydronoutcs (1978), 157 165. [6] H. Ono, Algebrac soltary wave n stratfed fluds, J. Phys. Soc. Japan 39 (1975), 1082 1091. [7] H. Segur and J. L. Hammack, Solton models of long nternal waves, J. Flud Mech. 118 (1982), 285 304. [8] L. H. Wryanto, A fnte dfference method on KdV equaton of nternal waves, (n preparaton).