We model the eﬀect of carioca’s l´ ınguas negras on the coastal waves (beaches) by means of a ﬂuid with an eﬀective viscosity νsh≡νshear CBPF-NF-043/01 1 One of the most interesting problems in Fluid Waves Physics with a large spectrum for applications is that one related to the propagation of waves on surface ﬂuids ([1]). Although its intrinsic mathematical diﬃculty tosolve theassociated motionequations, it is very fortunate for hydrodynamicist to have few models exactly soluble ([1]). Anotherpoint worth remark is that among these fews exactly soluble models, one never considers the existence of internal friction on the ﬂuid motion of the form of usual mechanical rigid- body linear damping proportional to velocity; a experimental fact relevant for irrotational ﬂuid motions at very low frequencies, like water in reservoirs, bays, oceans, etc... Our aim in this note is to re-analyze the surface waves on ﬂuids in the presence of the above cited mechanical damping ([2]). A word for applications: It is well known that contaminents of biological origin, like the famous “l´ ınguas negras” (plumes) on the Carioca’s beaches, can be heavily spread in the ﬂuid medium, and, thus being an ecological candidate for our study ([3]). Letusstartouranalysisbyconsideringtheusual(irrotationalandincompressible) two- dimensional Navier-Stokes for very low ﬂuid velocities (the so called Bernoulli-equation!) in the presence of the gravity and a (small) viscosity parameter ν sh grad/parenleftbigg∂σ(x, z;t) ∂t+1 ρρ(x, z;t)+νshσ(x, z;t)+gz/parenrightbigg =0 ( 1 ) Here σ(x, z;t) is the hydrodynamical potential. The incompressibility condition on the ﬂuid is taken as a realistic assumption, since the carioca’s l´ ınguas negras do not have any sensible thermodynamical behaviour on the ﬂuid ∂2σ(x, g;t) ∂2x+∂2σ(x, z;t) ∂2z=0 ( 2 ) Finally, the elevation ﬂuid surface η(x, t)a tat i m e t>0 (for small ﬂuid oscillations) is given by the wave equation at z=0 0=−ρ∂σ(x, z;t) ∂t/vextendsingle/vextendsingle/vextendsingle z=0−pg η(x, t)( 3 ) As a last very important hypothesis, we consider our ﬂuid surface ilimited (−∞ <x< ∞) and the ﬂuid volume has a ﬁnite depth h(−h≤z≤0). In order to solve the Wave’s ﬂuid systems eq. (1)-eq. (3) we consider the “ansatz” σ(x, z;t)=exp(−νsh·t) σ(x, z;t)( 4 ) CBPF-NF-043/01 2 ItisstraightforwardtoseethattheFouriertransformedﬂuidvelocitypotentialsatisﬁes the following ordinary diﬀerential equation ˆΦ(k;z, t)=1 √ 2π/integraldisplay+∞ −∞dx eikx σ(x, z;t) (5.a) ˆΦ(k, z;t)= A(k, t)cosh[k(z+h)] (5.b) d2 A(k;t) d2t−νshd A(k;t) dt+w2(h) A(k, t)=0 ( 5 . c ) with the depth-dependent dispersion relation w2(h)=gt g h[k(z+h)] (6) At this point we point out that the modulation wave factor (see eq. (4)) A(k, t)= exp(−νsht) A(k, t) is given explicitly by the following formulae A(k;t)=e−(νsh 2)t/bracketleftbig α(k)eiΩ(h,kqνsh)t+β(k)e−iΩ(h,k,ν sh)t/bracketrightbig cosh(k(z+h)) (7) where Ω(h, k, ν sh)=(/radicalbig 4w2(h)−(νsh)2)/slashBig 2( 8 ) The coeﬃcients α(k)a n d β(k) are given by means of the initial conditions σ(x,0,0) = 0 ↔ α(k)+β(k)=0 ( 9) ∂ ∂tσ(x, z, t)/vextendsingle/vextendsingle/vextendsingle z=0;t=0=gf(x)↔α(k)=g 2iΩ(h, k, ν sh)˜f(k) cos(kh)(10) here ˜f(k)=1 √ 2π/integraldisplay+∞ −∞eikxf(x)dx (11) By grouping together all the results above obtained, we get our ﬁnal formulae for the hydrodynamical potential given as a Fourier Integral σ(x, z;t)=ge−(νsh 2)t √ 2π/integraldisplay+∞ 02˜f(k) /radicalbig 4Ω2(h, k, ν sh)−ν2 sh cosh[k(z+h)] cos(kh){sen(kx+Ω( h, k, ν sh)t)−sen(kx−Ω(h, k, ν sh)t)}(12) As an important application of eq. (12), let us take the initial proﬁle, a periodic pulse f(x)=/summationtext+∞ n=−∞fnexp{2πn Lx}. We have, thus, the following leading proﬁle for t>0a n d CBPF-NF-043/01 3 νshsmall n(x, t)∼=−1 √ 2π  +∞/summationdisplay n=−∞e−(νsh 2)t /radicalBig 4Ω2(νsh,h ,2πn L)−/parenleftbigνsh 2/parenrightbig2Ω(νsh,h ,2πn L) /parenleftbigg cos/bracketleftbigg (2πn L)x+Ω( νsh,h ,2πn L)t/bracketrightbigg −cos/bracketleftbigg (2πn L)x−Ω(νsh,h ,2πn L)t/bracketrightbigg/parenrightbigg/bracerightbigg (13) As a consequence of the above formulae, one can make the following sad conclusion for the surf in Rio de Janeiro: the heavy presence of contaminants of the kind νsh·t(l´ınguas negras) attenuates strongly the beaches’ waves. Acknowledgements We are thankful to CBPF/MCT for warm hospitality, specially to the “Akousmata-Pytagoric” physicist community of Rio de Janeiro city.