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PONAŠANJE BRODA NA TALASIMA____________________________Teorijska analiza ravanskih talasa na slobodnojpovršini idealne neograničene tečnosti...Koordinatni sistem x,zSlobodna površina z = ζ ( x, t)Potencijal brzine φ = φ( x, zt , )Važi Laplasova jednačinaGranični usloviUslov duboke vode∂ φ∂x∂ φ∂z2 2+ = 02 2φ = 0 , za z → −∞Poremećaj nestaje duboko pod slobodnompovršinom...Na slobodnoj površini veže dva uslova:dinamički i kinematički...Dinamički uslov: pritisak vode na slobodnojpovršini jednak atmosferskom pritisku...p = p , za z =ζ( x, t)atUprošćenje: zanemaren površinski napon...proučavamo tzv. gravitacione talaseZanemarene i fluktuacije atmosferskogpritiska usled vetra...Kinematički uslov: brzine vode i slobodnepovršine, u pravcu normale na slobodnupovršine, su jednakev = V , za z =ζ( x , t)N NζZamisli slučaj v N > V N ... ili v N < V NKinematički uslov definiče slobodnupovršinu...PREDAVANJA 2009. 37
PONAŠANJE BRODA NA TALASIMA____________________________Koristimo Koši-Lagranžev integral...2 2∂φ 1φ φpxzt ( , , ) pat ρgzρ ρ ⎡ ⎛ ∂ ⎞ ⎛ ∂ ⎞= − − − ⎤2⎢ + ⎥∂t ⎜∂x ⎟ ⎜∂z⎟⎢⎣⎝ ⎠ ⎝ ⎠ ⎥⎦i u njega smenjujemo dinamički uslov na slobodnoj površiniSladiOdnosno, nakon sređivanja2 2∂φ 1 ⎡⎛∂φ ⎞ ⎛∂φ⎞ ⎤pat−ρgζ−ρ− ⎢ + ⎥ = p∂t 2⎜∂x ⎟ ⎜∂z⎟⎢⎣⎝ ⎠ ⎝ ⎠ ⎥⎦2 21 ⎡∂φ 1⎛∂φ ⎞ 1⎛∂φ⎞ ⎤ζ =− ⎢ + + ⎥ , za z = ζ( x, t)g ∂t 2⎜∂x ⎟2⎜∂z⎟⎢⎣⎝ ⎠ ⎝ ⎠ ⎥⎦Izvođenje kinematičkog č uslova preskačemo...Dobija se, nakon obimnog izvođena i sređivanjaat∂φ ∂ζ ∂φ ∂ζ= + , za z = ζ ( x, t)∂z ∂t ∂x ∂xPREDAVANJA 2009. 38
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