. The collected papers of Sir Thomas Havelock on hydrodynamics. Ship resistance; Water waves; Hydrodynamics. THE FORCES ON A SUBMERGED BODY MOVING UNDER WAVES We also replace 13^2 in (41) in terms of J|/2 ^^'^ J5/2 and after some reduction we obtain, after putting cr = KT (V + c), the form M = (2 v-yi^ gpab^h e-"'' {k a e)"* / 1 + (|32_l) '^'J- 5/2 ' + (ft^)' i (1 + k{) (1 + k2) J5/2 + i (1 + kx) (1—2 ki) J,p M cos (cr f + a (44) where the argument of the J functions is k a e, and ;8 = length-beam ratio = ajb. 5. Summary of Notation and Results We may express these results more conve
. The collected papers of Sir Thomas Havelock on hydrodynamics. Ship resistance; Water waves; Hydrodynamics. THE FORCES ON A SUBMERGED BODY MOVING UNDER WAVES We also replace 13^2 in (41) in terms of J|/2 ^^'^ J5/2 and after some reduction we obtain, after putting cr = KT (V + c), the form M = (2 v-yi^ gpab^h e-"'' {k a e)"* / 1 + (|32_l) '^'J- 5/2 ' + (ft^)' i (1 + k{) (1 + k2) J5/2 + i (1 + kx) (1—2 ki) J,p M cos (cr f + a (44) where the argument of the J functions is k a e, and ;8 = length-beam ratio = ajb. 5. Summary of Notation and Results We may express these results more conveniently in the following notation, in which we also define suitable force and moment coefficients. L = Length of spheroid = 2 a. e = Eccentricity = (1 — b'^ja^)'^. ;S = Length-beam ratio = ajb. D = Displacement = f tt g p a b^. < ki, kz, k' = Axial, transverse, rotational virtual inertia coefficients, as defined and evaluated, for instance, in Ref. 3. V = Speed of advance (positive against the waves). /= Froude number = V/(^ L)-. d = Depth of axis below the surface. h = Amplitude of waves = half wave height. A = Wavelength = 2 ttJk. c = Wave velocity = (^/k:)*. 2 tt/ct = Period of encounter. CT = K (V + c). B = 1 TT {hjX) e~^"''l^ = Maximum effective wave slope at depth of axis. X, Z = Resultant forces in directions O x, O z. M = Resultant moment about O y. Cx = Surging force coefficient = X (max)/D d. Q = Heaving force coefficient = Z (max)/D 6. Cyy = Pitching moment coefficient = M (max)/D L d. In this notation, the results obtained are X = - D 0 C, cos ((7f + a); Z = - D e Q sin (a ? + a); M = D L 61C^^ cos ((T ? + a) . (45) l+^2+/(-^) (fc2-/c,)J '77 e L\ a = 2 77 e5/2 (l) J3/2(-^).(47) ^•^^ ~ 4 77 e"2 3V2 /An3/2 -i{\+k{){\+k2)^hli(^) + I (1 + fci) (1 - 2 k2) hn (^)} } • (48) 6. General Discussion The phase relations between the waves and the forces can be seen from a comparison of (4) with (45). It is of interest that these relations are unaltered by th
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