If probability functon of distribution of wave heights are assumed here to be as follows p(r)=-d/dre^<-(r/a)^n> Where r is wave height, a is n-th root mean of their values, then n<2 Swells, n>2 Seas, represent respectively. When sea is calm, or weather conditions do not change, and also n>2, and for in violent storm n<2. Rolling angle of a ship under these sea states, are expected as follows: [numerical formula] where θ0 is resonance angle in regular unique swells, θ0=√<πγ/2N0・180δ0e^<-βS^2>> (degrees), γ0≒08 coefficient of effective wave slope, N0≒0.02, damping coefficient, β_S=gT_S/2πu, resonance wave age, T_S: rolling period, u0: wind velocity. N: numbers of times of rolling, [numerical formula] β_n=maximum wave age which exists in the sea surface. Here steepness curve is used that of Neumann, ie. δ=δ0e-β^2,, δ0=√<2/10>. Frequently occurred rolling angles are about 2/3〜1/2 to those of regular swells, but when time passes moderately, maximum value approached gradually to that of perfect resonance.

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This paper is some analytical explanations on the probability of existence of wave period on sea surface, or terms "narrow banded" which is first assumed and introduced by Longuet-Higgins in their original studies of statistical distributions of heights of waves. The distributions are almost unchanged in the region of storm, and determined from Drs. Sverdrup-Munks' theory in the centre of storm, namely [numerical formula] where, β=gT/2πu; wave age, δ: wave steepness, u: wind velocity, u0: its maximum value, du/dt: wind velocity gradient per one hour, T: wave period. The maximum value of β=β_0 is, [numerical formula] usually 0≦β_0≦1. Then other general weather conditions after the storm (uβ)^5δ^2=(u_0β_0)^5δ0^2 The modulus, or here percentage of numbers of observed samples which lie between from (T-1) sec to (T+1) sec to that total numbers reached about more than 0.75%. where T is the period which is most frequently observed.

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When wave age exceeds 1.4, steepness varies especialy from 0.000 to 0.0219, The former expresses very calm and steady swells; on the other hand the latter corresponds to temporaly confused seas like in the eye of typhoon or in the centre of fronts. Statistical distribution of heights of waves shows that significant mean values are [numerical formula] where probability that wave height r lies between r and r+dr is defined a・P(r)=n(r/a)^<n-1>e^<-(r/a)^n> when n<2 extremely confused seas; n>2, calm swells. Then r/a=2.099(n=1), 1.415(n=2), 1.255(n=3), etc So that upper limit of steepness curves reach to straight gradually. In the decaying process of wave height is approximately H=uu_m/100,H(m),u(m/s);u_m(m/s) and relation between steepness and wave age is (β/1.40)^5(δ/0.022)^3=1 1.4≦β≦∞

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We defined the stability creterions for ship's rolling as [numerical formula] where Sd: dynamical stability arm, δ_r: stability range, m: metacentric height. θ0: resonance amplitude of rolling. u0: wind velocity. λu^2: statical couple lever due to wind pressure, θ0 can be calculated by the following formula. θ0≒1.86√δ, β_s=gT_S/2πu0 with the aid of Sverdrup-Munks' δ-β curve for ocean waves and the ship's natural period of rolling T_S. In the surrounding districts of region of storm, there occur the fully grown-up regular swells by the constant prevailling wind, which may cause the ship' perfectly resonamced rolling. However, in the centre of the storm waves are confused and very irregular only gales blowing violently. Then the above cited two creterions must be applicable to the outer and inside region of storm respectively. From the another point of view, ship is always more or less drifting by the various weather conditions. Then the character of resonance wave may be changed compared with the case when the ship's position is stationary, namely u=u0(1-ε0), [numerical formula] Where u is the relalive speed of wind to the ship and uo is absolute wind velocity on the sea surface, then the drifting speed of the ship is u0ε0. Here appear three resonance waves. The other β=β_s/2[-1-√<1+4ε/β_S>] is omitted for the reason of negative valued. The former corresponds to C_1, and the latter to C_2, which tend to each other when wind velocity is low or high respectively, but the former and the next vanished beyond u0≧gT_s/8πε0. Of course the three creterions which is calculated from above equations lie always between C_1 and C_2 for every wind speed. Thns we have proved that the C_1 and C_2 is the lower and upper limit of creterion of a ship for every weather conditions.

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[in Japanese]

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There are mutually many similar characters in heaving and pitching motion of a ship; namely, in her virtual mass, period, etc, but heaving's velocity, acceleration corresponds to pitching's amplitude, velocity, respectively. (1) For rate of increase of virtual inertia; Heaving: Δ_h=π/8B/h・2m/2m+1 Pitching: Δ_p=π/8B/h・2m/2m+3 where B: breadth, h: drought, C_w=m/(m+1); water-line coefficient. (2) For ships period, Heaving: T_h=2π√<h/g(1+Δ_h);> Pitching: T_p=2π√<h/g(1+Δ_p)> where g: gravitational acceleratioin. So that each value is almost equal, but for pitching are slightly less than those of for heaving. (3) For maximum resonanced amplitude: Heaving, [numerical formula] Pitching, [numerical formula] where, damping force is assumed to be proportional to square of velocily of motion, γ_h, γ_p: effective coefficient, its value is usually about 1/4〜1/3, L: ships length, β_h=gTh/2πu, β_p=gTp/2πu: resonance wave-age, u is wind velocity, δ0e-β_<h^2>: resonanced wave steepness. When a ship is rolling in confused seas, criical amplitude, velocity, acceleration are approximately [numerical formula] where [numerical formula] and n=1 forZ^^-; n=0, for Z and φ, n=-1 for Z and φ, n=-2, for φ respectivety. N is numbers of oscillations. Φ(-1) has maximum value o about 1/2, for β≒1/2 which implies that there may occur very heavy up-and-down motion.

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