We estimate the pressure ﬂuctuation inside fuselages. One of the most important point on the designers of fuselages for missiles, torpedous, etc... with very sensitive electronics and radioactive liquid explosives inside is to estimate the turbulent pressure component transmission to the fuselage interior ([1]). In this brief report, we propose a mathematical-pedagogical model to analyze this problem. Let us consider a inﬁnite beam backed on the lower side by a space of depth dwhich is ﬁlled with a ﬂuid of density ρ2and sound speed v2. On the upper side of the beam which is modelling the torpedous’ fuselage there is a supersonic boundary-layer turbulent pressure P. The ﬂuid on the upper side of the beam which is on the turbulence is supposed to have a free stream velocity U∞,d e n s i t y ρ1and sound speed v1. The eﬀective equation governing the “outside” pressure in our model is given by (d≤z<∞) /parenleftbigg∂ dt+U∞∂ ∂x/parenrightbigg2 p1(x, z, t)−v2 1/parenleftbigg∂2 ∂2x+∂2 ∂2z/parenrightbigg p1(x, z, t) = 0 (1.a) with the boundary condition (2ndNewton’s law) −ρ1/parenleftbigg∂ dt+U∞∂ ∂x/parenrightbigg2 W(x, t)=∂p 1(x, z, t) ∂z/vextendsingle/vextendsingle/vextendsingle z=d(1.b) CBPF-NF-044/01 2 Here the beam’s deﬂection W(x, t) is given by the beam small deﬂection equation B∂4W(x, t) ∂4x+m∂2W(x, t) ∂2t=P(x, t)+(p1−p2)(x, d, t)( 2 ) withBdenoting the bending rigidity, mthe mass per unit lenght of the beam and p(x, z, t) the (supersonic) boundary-layer pressure ﬂuctuation of the exterior medium. The searched induced pressure p2(x, z, t) in the ﬂuid interior medium (0 ≤z≤d)i s governed by ∂2p2 ∂2t(x, z, t)−(v2)2/parenleftbigg∂2 ∂2x+∂2 ∂2z/parenrightbigg p2(x, z, t) = 0 (3.a) ∂p 2(x, d, t) ∂z=−ρ2∂2 ∂2tW(x, t)( 3 . b ) The solution of eq. (1-a) and eq. (1-b) is straightforward obtained in the Fourier domain ˜p1(k,z,w )=−ρ1/braceleftBigg v1(w+U∞k)2 /radicalbig v2 1k2−(w+U∞k)2/bracerightBigg ˜W(k,w) exp(−1 v1−/radicalBig v2 1k2−(w+U∞k)2)(z−d)( 4 ) where the deﬂection beam ˜W(k,w) is explicitly given by ˜W(k,w)=(˜P(k,w)−˜p2(k,w,d )/braceleftBigg Bk4−mw2+ρ1(w+U∞k)2v1 /radicalbig v2 1k2−(w+U∞·k)2/bracerightBigg−1 (5) At this point, we solve our problem of determining the pressure ˜ p2(k,w,z )i nt h e interior domain eq. (3-a), eq. (3-b) if one knows the pressure ˜ p2(k,w,d ) on the “fuselage”. Let us, thus, consider the Taylor’s serie in the z-variable (0 ≤z≤d) around z=d,n a m e l y ˜p2(k,w,z )=˜p2(k,w,d )+∂p 2(k,w,z ) ∂z/vextendsingle/vextendsingle/vextendsingle z=d(z−d)+···+1 k!∂k˜p2(k,w,z ) ∂kz(z−d)k+···(6) From the boundary condition eq. (3-b), we have the explicitly expression for the second-derivative on the depth z ∂p 2(k,w,z ) ∂z/vextendsingle/vextendsingle/vextendsingle z=d=+ρ2w2˜W(k,w)= +ρ2w2[˜P(k,w)−˜p2(k,w,d )]×/braceleftBigg Bk4−mw2+ρ1(w+U∞k)2v1 /radicalbig v1k2−(w+U∞k)2/bracerightBigg−1 (7) CBPF-NF-044/01 3 The second z-derivative of the interior pressure (and the higher ones!) are easily obtained recursively from the wave equation (3-a) ( k≥0,k∈ZZ+) ∂2+k ∂2+kz˜p2(k,z,t)/vextendsingle/vextendsingle/vextendsingle z=d=/parenleftbigg1 (v2)2×(−w2+(v2)2k2)/parenrightbigg/parenleftbigg∂2+k−1 ∂2+k−1˜p2(k,z,t)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle z=d)(8) Let us ﬁnally comment that the general turbulent pressure is assumed to be expressible in a Stieltjes integral P(x, t)=/integraldisplay/integraldisplay+∞ −∞eikxei(w−ku)tdF(k,u)dG(u)( 9 ) where G(u) is a random function ([2]). Acknowledgements We are thankful to CBPF/MCT for warm hospitality. We herewith state our acknowl- edgements to Alexander Von Humboldt Stiftung-Deutschland for ﬁnancial support during 1991–1992 where my project research on waves physics has begun with Av.H Fellowship, besides all this research was carried out under the philosophy-“SLV’s-Vehicles for the third world”.