Britain Hydromechanics of swimming propulsion. part - PDF document

J. Fluid Mech. (1971), vol. 46, part 2, pp. 337-355 printed ir& Great Britain Hydromechanics of swimming propulsion. part 1. two-dimensional flexible in an inviscid fluid By T. YAO-TSU WU California Institute of Technology, Pasadena, California (Received 21 July 1970) The most effective almost all sizes appear wave progressing along body from tail. The main the large Reynolds number, energy conservation. general problem unsteady forward speeds, is by applying inviscid initial-value harmonic motion shows For small-time solution is evaluated optimum shape is determined for under conditions Aquatic animals in water, liquid media, range in Large cetaceans, such and whales, have lengths from 2 to 30 m, and can swim at cruising speeds of from 6 to 10 m/s (Lang & Pryor organisms such spermatozoa, ranging 300pm down to 50pm in length length- speeds from 1000 to 80pmls. extremities there are many and aquatic Based on I of a body moving at velocity U kinematic viscosity w/v, the relative magnitude time average 108 for the rapid cetaceans, lo6 for migrating fishes, lo5-lo3 a great variety lo2 for tadpoles, 1 for Turbatrix, 10V less for 437), and to the extreme of 10-6 or less bacteria. Thus, Lighthill excellent survey propulsion employed a large number drastically different an a transverse wave propagating body from head great majority be singled as a pre-eminent propulsion. The remarkable cetaceans (dolphin, porpoises, whales, known game large aspect is only a from minute bacteria, primitive protozoa, tozoa, have employ either uniformly propagating transverse waves, or whip-like waves, helical waves along slender Reynolds number such a range. However, principles underlying different for large or small Reynolds number. For Reynolds number large, propulsion depends primarily effect, since outside a thin boundary layer is irrotational. is unimportant producing a a skin friction body surface. body performs an and attains a forward momentum, force pushes frictional resistance gives rise forward momentum by entraining wake due small thickness sheet; backward jet fluid expelled from body can, When a propelled body is backward momenta can nevertheless evaluated separately. large Reynolds numbers K&rm&n & a rigid plate in transverse byLighthill(1960); a two-dimensional pIa,te calculated by Wu (1961). small Reynolds numbers. propulsion in range depends almost entirely extremely small, except possibly for Oscillatory motions a by Stokes. Various organisms have 1952a, b), a propagating, monochromatic, transverse wave along a sheet immersed viscous fluid, and later the action of waving cylindrical tails of microscopic organisms. Further studies in contributed by (1953), Gray & Hancock (1955)~ Tuck (1968). ��Ohere are still other &#x/BBo;&#xx [1;.51;ˆ 6;.0;ࠃ ;@.7;ঈ ;إ.;䐇&#x ]00;&#x/BBo;&#xx [1;.51;ˆ 6;.0;ࠃ ;@.7;ঈ ;إ.;䐇&#x ]00;kinds of body motions, such as progressive waves along fringe belts &#x/BBo;&#xx [1;.51;ˆ 6;.0;ࠃ ;@.7;ঈ ;إ.;䐇&#x ]00;&#x/BBo;&#xx [1;.51;ˆ 6;.0;ࠃ ;@.7;ঈ ;إ.;䐇&#x ]00;flat fishes, and waving motion produced large number &#x/BBo;&#xx [1;.51;ˆ 6;.0;ࠃ ;@.7;ঈ ;إ.;䐇&#x ]00;&#x/BBo;&#xx [1;.51;ˆ 6;.0;ࠃ ;@.7;ঈ ;إ.;䐇&#x ]00;bassels underneath a fish, (c) squirming motion &#x/BBo;&#xx [1;.51;ˆ 6;.0;ࠃ ;@.7;ঈ ;إ.;䐇&#x ]00;&#x/BBo;&#xx [1;.51;ˆ 6;.0;ࠃ ;@.7;ঈ ;إ.;䐇&#x ]00;ofa &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;tail~less object in slow motion through a ciliated propulsion &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;of numerous micro-organisms by waving movements of a large number &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;attached to the body surface. Problem &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;(1963), and (c) has been analysed by Lighthill (1952). Close resemblance between &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;the movements of cilia and flagella has been contended by some investigators. The problem of self-propulsion of a deformable body in a perfect fluid, Saffman (1967). process begins swimming being, which converted, with &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;q,, into mechanical energy for maintaining &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;y,, into hydrodynamic energy for swimming. &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;y, say) of the hydrodynamic energy is spent useful work or dissipated, self-contained balance observations have reported. For example, Johannessen &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;& Harder (1960) porpoises, killer The boundary layer rigid, smooth surface Reynolds number range is skin friction such high speeds would can deliver only investigated care- migratory salmon careful investigation, two conclusions: these creatures have be achieved &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;ouOput per gram of muscle is much larger from physiological experiments on is known as the These puzzling explore various &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;the effect of compliant skin, mucous surface drag, studies &#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;&#x/BBo;&#xx [3;.55;† 5;g.8;Ѕ ;h.3;ই ;ն.;␆&#x ]00;thrust energy Energy lost ��& &#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;&#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;Daybell (1963) has given partial explanation to Gray's dilemma. The present study is devoted several hydromechanical problems a transverse wave along a or slender body energy conservation. The theory appears value in &#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;&#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;first approximation to the propulsion of lunate tail cetacean mammals high aspect If this type theory is to the &#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;&#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;it; is more realistic forward velocity up-and-down strokes, large backward-and-forward components a general artificial propulsive as the vertical-axis propeller and pitch), and for hydrofoils devices when &#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;&#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;be- haviour is of importance. With these a two-dimensional with a variable forward theory. The small-time solution a flexible plate starting with a acceleration from and the shape in determined for small body a rigid general optimum will be of Chis sheet shed a slender &#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;&#x/BBo;&#xx [2;0.1;֒ ;ذ ;ɥ.;䎖&#x 640;&#x.08 ;&#x]000;Cogether with the optimum movement. 2. Thrust; energy balance In order &#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];&#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];bo understand why the motion of a transverse wave progressing desirable for small amplitude, achieving t a rectilinear forward velocity &#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];&#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];U(t) through a fluid which is otherwise at rest. We choose Cartesian co-ordinate system (x, &#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];&#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];z) fixed at the mean position &#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];&#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];stretched plan form body lying &#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];&#x/BBo;&#xx [1;%.7;֔ ;ɕ.;ሁ&#x 134;&#x.639; 26;
.84; ];= &#x/BBo;&#xx [3;@.5;֗ ;Ź.;瘅&#x 345;&#x.119;&#x 186;&#x.720; ]0;&#x/BBo;&#xx [3;@.5;֗ ;Ź.;瘅&#x 345;&#x.119;&#x 186;&#x.720; ]0;0 plane and with the free-stream velocity &#x/BBo;&#xx [3;@.5;֗ ;Ź.;瘅&#x 345;&#x.119;&#x 186;&#x.720; ]0;&#x/BBo;&#xx [3;@.5;֗ ;Ź.;瘅&#x 345;&#x.119;&#x 186;&#x.720; ]0;U(t) pointing in the positive x direction. The body generally as &#x/BBo;&#xx [3;@.5;֗ ;Ź.;瘅&#x 345;&#x.119;&#x 186;&#x.720; ]0;&#x/BBo;&#xx [3;@.5;֗ ;Ź.;瘅&#x 345;&#x.119;&#x 186;&#x.720; ]0;Y = &#x/BBo;&#xx [2; .7;ঙ ;ķ.;⠈&#x 236;&#x.879; 14;.12;
];&#x/BBo;&#xx [2; .7;ঙ ;ķ.;⠈&#x 236;&#x.879; 14;.12;
];h(x, &#x/BBo;&#xx [2;@.4;ޗ ;ķ.;瘉&#x 246;&#x.72 ;ń.;␆&#x ]00;&#x/BBo;&#xx [2;@.4;ޗ ;ķ.;瘉&#x 246;&#x.72 ;ń.;␆&#x ]00;2, &#x/BBo;&#xx [2;P.5;֗ ;ķ.;⠈&#x 256;&#x.079; 14;.64;&#x ]00;&#x/BBo;&#xx [2;P.5;֗ ;ķ.;⠈&#x 256;&#x.079; 14;.64;&#x ]00;t) &#x/BBo;&#xx [2;g.8;9 1;7.2;ࠈ ;ɷ.;醉&#x 146;&#x.64 ;&#x]000;&#x/BBo;&#xx [2;g.8;9 1;7.2;ࠈ ;ɷ.;醉&#x 146;&#x.64 ;&#x]000;(x, &#x/BBo;&#xx [2;.5;ƒ ;Ĺ.; &#x 285;&#x.36 ;ń.;␆&#x ]00;&#x/BBo;&#xx [2;.5;ƒ ;Ĺ.; &#x 285;&#x.36 ;ń.;␆&#x ]00;2 &#x/BBo;&#xx [2;ˆ.7;Ɩ ;Ĺ.; &#x 293;&#x.519; 14;.24; ];&#x/BBo;&#xx [2;ˆ.7;Ɩ ;Ĺ.; &#x 293;&#x.519; 14;.24; ];E &#x/BBo;&#xx [2;–.6;Ζ ;ķ.;Ђ&#x 308;&#x.638; 14;.88; ];&#x/BBo;&#xx [2;–.6;Ζ ;ķ.;Ђ&#x 308;&#x.638; 14;.88; ];4, &#x/BBo;&#xx [4;".6;Ζ ;ĸ.;&#x 433;&#x.199; 14;.08; ];&#x/BBo;&#xx [4;".6;Ζ ;ĸ.;&#x 433;&#x.199; 14;.08; ];(1) where S is the stretched plan form body (when h vanishes identically), &#x/BBo;&#xx [4;".6;Ζ ;ĸ.;&#x 433;&#x.199; 14;.08; ];&#x/BBo;&#xx [4;".6;Ζ ;ĸ.;&#x 433;&#x.199; 14;.08; ];h is an arbitrary function of x, &#x/BBo;&#xx [2;.2;ޓ ;ă.;鈈&#x 221;&#x.759;&#x 110;&#x.400; ]0;&#x/BBo;&#xx [2;.2;ޓ ;ă.;鈈&#x 221;&#x.759;&#x 110;&#x.400; ]0;z, and t, with &#x/BBo;&#xx [2;ƒ.4;Ζ ;Ă.;⎙&#x 308;&#x.399; 11;.52; ];&#x/BBo;&#xx [2;ƒ.4;Ζ ;Ă.;⎙&#x 308;&#x.399; 11;.52; ];lah/atl and swimming velocity &#x/BBo;&#xx [4;&.7; 10;.56; 4;3.1;গ ;Ĕ.;爃&#x ]00;&#x/BBo;&#xx [4;&.7; 10;.56; 4;3.1;গ ;Ĕ.;爃&#x ]00;U assumed to be sufficiently small for regarded as &#x/BBo;&#xx [4;&.7; 10;.56; 4;3.1;গ ;Ĕ.;爃&#x ]00;&#x/BBo;&#xx [4;&.7; 10;.56; 4;3.1;গ ;Ĕ.;爃&#x ]00;incompressible, and with &#x/BBo;&#xx [1;#.3; 77;&#x.28 ;Ő.;閘&#x 88.;嘈&#x ]00;&#x/BBo;&#xx [1;#.3; 77;&#x.28 ;Ő.;閘&#x 88.;嘈&#x ]00;[ahlax/ and &#x/BBo;&#xx [1;y.5;ƙ ;w.2; 20;.91;‰ 8;.56; ];&#x/BBo;&#xx [1;y.5;ƙ ;w.2; 20;.91;‰ 8;.56; ];[ah/a~[ assumed also small enough Hydromechartics The Reynolds number ��Ullv, based on ��U and body length &#x/BBo;&#xx [3;u.1;Ɣ ;ة.;写&#x 377;&#x.759; 63;.96; ];&#x/BBo;&#xx [3;u.1;Ɣ ;ة.;写&#x 377;&#x.759; 63;.96; ];I &#x/BBo;&#xx [1;.95;‘ 6;.9;؂ ;).7;և ;آ.;耇&#x ]00;&#x/BBo;&#xx [1;.95;‘ 6;.9;؂ ;).7;և ;آ.;耇&#x ]00;(in the streamwise direction), is boundary layer is &#x/BBo;&#xx [1;.95;‘ 6;.9;؂ ;).7;և ;آ.;耇&#x ]00;&#x/BBo;&#xx [1;.95;‘ 6;.9;؂ ;).7;և ;آ.;耇&#x ]00;thin and the inertial effects can be evaluated under &#x/BBo;&#xx [2;F.9;֕ ;؅.;Ђ&#x 261;&#x.359; 61;.72; ];&#x/BBo;&#xx [2;F.9;֕ ;؅.;Ђ&#x 261;&#x.359; 61;.72; ];Dhe &#x/BBo;&#xx [2;c.9;ও ;؅.;Ђ&#x 298;&#x.799; 61;.72; ];&#x/BBo;&#xx [2;c.9;ও ;؅.;Ђ&#x 298;&#x.799; 61;.72; ];inviscid flow assumption. &#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;&#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;Then the boundary condition requiring normal component vanish prescribes &#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;&#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;velocity at the planar surface as The planar body may admit sharp leading and trailing edges. When the latter kind is present, as usual, the Kutta dependence on &#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;&#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;x simply drops out, and span in &#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;&#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;x direction. The thrust (positive when directed in the negative x direction) acting on body, based &#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;&#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;inviscid linear theory, is given by the forward direction, (Ap) denotes difference across &#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;&#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;&is thesingular force per unit arc length along to the last integral leading edge &#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;&#x/BBo;&#xx [1;.47; 59;�.40; 4;�.79;ˆ 5;˜.5;؄ ;&#x]000;x = &#x/BBo;&#xx [3;X.0;ޖ ;̉.;萈&#x 377;&#x.518; 31; .44; ];&#x/BBo;&#xx [3;X.0;ޖ ;̉.;萈&#x 377;&#x.518; 31; .44; ];b(x). The power required to maintain the motion is equal to the time rate of work done by the plate against the reaction of the fluid in the direction of the transverse plate motion, The third quantity of interest is mechanical energy to the &#x/BBo;&#xx [3;X.0;ޖ ;̉.;萈&#x 377;&#x.518; 31; .44; ];&#x/BBo;&#xx [3;X.0;ޖ ;̉.;萈&#x 377;&#x.518; 31; .44; ];inviscid flow is equal to the time rate of work done by the pressure over the body surface, or &#x/BBo;&#xx [3;X.0;ޖ ;̉.;萈&#x 377;&#x.518; 31; .44; ];&#x/BBo;&#xx [3;X.0;ޖ ;̉.;萈&#x 377;&#x.518; 31; .44; ];V(x,z,t)dS-T,U. &#x/BBo;&#xx [3;f.7;ƒ ;Ÿ.;嘈&#x 377;&#x.759; 18;.92;&#x ]00;&#x/BBo;&#xx [3;f.7;ƒ ;Ÿ.;嘈&#x 377;&#x.759; 18;.92;&#x ]00;(5) The above three quantities satisfy the principle of conservation of energy which is equal the rate the thrust, &#x/BBo;&#xx [3;f.7;ƒ ;Ÿ.;嘈&#x 377;&#x.759; 18;.92;&#x ]00;&#x/BBo;&#xx [3;f.7;ƒ ;Ÿ.;嘈&#x 377;&#x.759; 18;.92;&#x ]00;TU, plus the kinetic energy effects are skin friction must contain physical grounds periodic body forward velocity, U = const., h(x, z, t) = hl(x, z) exp (jut) (x, z E S), (7) where j = 4 1 is h,(x, z) generally be with respect j, h is interpreted by is over, kinetic energy vortex sheet U. average, cannot negative. (A this statement has given for plates in (1961).) from a U, h, and the components of the perturbation velocity (u, v, w) all vanish for t 0. generated in kinetic energy fluid (see 5 following discussion be based on presumption E 2 0. Under this condition we have, by (6), �PTU if E20. (9) be positive definite. is negative, energy is (like a 0 according Co (9), inertial drag acting on � 0, large enough power is is seen producing a positive ahlax same sign, for $ never negative. (9) and the ah/at cannot have qualitative picture, ahlax and ah/at everywhere opposite clearly h represents a transverse (see figure such periodic harmonic form (7), since, for dependence, all linear can be obtained by the Fourier synthesis; as for E, it can be in their time averages, with different multiple-frequencies two functions: - 1=' gh = lim So g(x, t) h(x, t) dt = +Re [C gn (x) h$ (x)] , T+m n ��where &#x/BBo;&#xx [4;.95;„ 6;'.3; 58;&#x.798; 63;.28;&#x ]00;&#x/BBo;&#xx [4;.95;„ 6;'.3; 58;&#x.798; 63;.28;&#x ]00;h* is the complex conjugate of h (with &#x/BBo;&#xx [4;.95;„ 6;'.3; 58;&#x.798; 63;.28;&#x ]00;&#x/BBo;&#xx [4;.95;„ 6;'.3; 58;&#x.798; 63;.28;&#x ]00;j). This result is readily &#x/BBo;&#xx [4;.95;„ 6;'.3; 58;&#x.798; 63;.28;&#x ]00;&#x/BBo;&#xx [4;.95;„ 6;'.3; 58;&#x.798; 63;.28;&#x ]00;,=tended to the integral form when g, h are integrals over to the &#x/BBo;&#xx [4;.95;„ 6;'.3; 58;&#x.798; 63;.28;&#x ]00;&#x/BBo;&#xx [4;.95;„ 6;'.3; 58;&#x.798; 63;.28;&#x ]00;[h, (x, z) exp &#x/BBo;&#xx [1;“.9;Ɔ ;ղ.;螙&#x 214;&#x.078; 58;.95;˜ ];&#x/BBo;&#xx [1;“.9;Ɔ ;ղ.;螙&#x 214;&#x.078; 58;.95;˜ ];(j(wt - &#x/BBo;&#xx [2;$.8;ށ ;ճ.;6 2;E.2;ކ ;ւ.;閘&#x ]00;&#x/BBo;&#xx [2;$.8;ށ ;ճ.;6 2;E.2;ކ ;ւ.;閘&#x ]00;kx))] (x, &#x/BBo;&#xx [2;i.9;উ ;յ.;疔&#x 273;&#x.838; 58;�.56; ];&#x/BBo;&#xx [2;i.9;উ ;յ.;疔&#x 273;&#x.838; 58;�.56; ];z &#x/BBo;&#xx [2;v.4;ކ ;յ.;疔&#x 281;&#x.278; 58;�.79;™ ];&#x/BBo;&#xx [2;v.4;ކ ;յ.;疔&#x 281;&#x.278; 58;�.79;™ ];E S), &#x/BBo;&#xx [3;`.9;֑ ;ղ.;螙&#x 377;&#x.518; 58;.24; ];&#x/BBo;&#xx [3;`.9;֑ ;ղ.;螙&#x 377;&#x.518; 58;.24; ];(12) &#x/BBo;&#xx [8;.23;ˆ 4;B.7;খ ;„.9;ք ;ђ.;㦔&#x ]00;&#x/BBo;&#xx [8;.23;ˆ 4;B.7;খ ;„.9;ք ;ђ.;㦔&#x ]00;I &#x/BBo;&#xx [1;.9;Ɖ ;й.;枔&#x 135;&#x.118; 44;.84;
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]0;I &#x/BBo;&#xx [2; .7;ঈ ;й.;枔&#x 239;&#x.998; 44;.84;
];&#x/BBo;&#xx [2; .7;ঈ ;й.;枔&#x 239;&#x.998; 44;.84;
];i'hlux- &#x/BBo;&#xx [2;A.4;Έ ;с.;萅&#x 246;&#x.238; 44;.16; ];&#x/BBo;&#xx [2;A.4;Έ ;с.;萅&#x 246;&#x.238; 44;.16; ];&#x/BBo;&#xx [2;A.4;Έ ;с.;萅&#x 246;&#x.238; 44;.16; ]; &#x/BBo;&#xx [2;H.3;ই ;с.;萅&#x 251;&#x.998; 44;.59;• ];&#x/BBo;&#xx [2;H.3;ই ;с.;萅&#x 251;&#x.998; 44;.59;• ];0 I I I &#x/BBo;&#xx [8;.99;‚ 4;.1;গ ;„.9;ք ;Х.;刃&#x ]00;&#x/BBo;&#xx [8;.99;‚ 4;.1;গ ;„.9;ք ;Х.;刃&#x ]00;I &#x/BBo;&#xx [1;$.3;Ƒ ;В.;ޖ&#x 146;&#x.638; 41; .99;– ];&#x/BBo;&#xx [1;$.3;Ƒ ;В.;ޖ&#x 146;&#x.638; 41; .99;– ];&#x/BBo;&#xx [1;$.3;Ƒ ;В.;ޖ&#x 146;&#x.638; 41; .99;– ];4pO &#x/BBo;&#xx [1;‚.8;ބ ;В.;ޖ&#x 183;&#x.598;&#x 423;&#x.840; ]0;&#x/BBo;&#xx [1;‚.8;ބ ;В.;ޖ&#x 183;&#x.598;&#x 423;&#x.840; ]0;I &#x/BBo;&#xx [2;(.4;ރ ;В.;考&#x 251;&#x.518; 42;�.47;— ];&#x/BBo;&#xx [2;(.4;ރ ;В.;考&#x 251;&#x.518; 42;�.47;— ];4piO I I I I I &#x/BBo;&#xx [8;.23;ˆ 3;ˆ.5;؁ ;….1;™ 3;˜.6; ]0;&#x/BBo;&#xx [8;.23;ˆ 3;ˆ.5;؁ ;….1;™ 3;˜.6; ]0;I &#x/BBo;&#xx [1;.8;΃ ;΃.;( 1;A.5;঄ ;Α.; &#x ]00;&#x/BBo;&#xx [1;.8;΃ ;΃.;( 1;A.5;঄ ;Α.; &#x ]00;&#x/BBo;&#xx [1;.8;΃ ;΃.;( 1;A.5;঄ ;Α.; &#x ]00;a11p, 0 I I &#x/BBo;&#xx [2;".7;Ɓ ;΃.;冕&#x 239;&#x.759;&#x 391;&#x.439; ]0;&#x/BBo;&#xx [2;".7;Ɓ ;΃.;冕&#x 239;&#x.759;&#x 391;&#x.439; ]0;ahlat &#x/BBo;&#xx [2;A.4;Έ ;΅.;䎙&#x 246;&#x.478; 38; .52; ];&#x/BBo;&#xx [2;A.4;Έ ;΅.;䎙&#x 246;&#x.478; 38; .52; ];o I I I &#x/BBo;&#xx [2;‘.8;Ά ;ͩ.;ᆘ&#x 292;&#x.798; 37; .92; ];&#x/BBo;&#xx [2;‘.8;Ά ;ͩ.;ᆘ&#x 292;&#x.798; 37; .92; ];I I I I FIGURE 1. Consideration of energy conservation indicates forward swimming, transverse movements body wave also backward relative fluid, since &#x/BBo;&#xx [2;‘.8;Ά ;ͩ.;ᆘ&#x 292;&#x.798; 37; .92; ];&#x/BBo;&#xx [2;‘.8;Ά ;ͩ.;ᆘ&#x 292;&#x.798; 37; .92; ];2 &#x/BBo;&#xx [2;….1;ƃ ;̣.;Θ&#x 293;&#x.518; 32; .28;&#x ]00;&#x/BBo;&#xx [2;….1;ƃ ;̣.;Θ&#x 293;&#x.518; 32; .28;&#x ]00;U. which represents a simple wave planar body direction with phase velocity &#x/BBo;&#xx [2;….1;ƃ ;̣.;Θ&#x 293;&#x.518; 32; .28;&#x ]00;&#x/BBo;&#xx [2;….1;ƃ ;̣.;Θ&#x 293;&#x.518; 32; .28;&#x ]00;c = &#x/BBo;&#xx [2;9.9;অ ;ʃ.;醗&#x 254;&#x.878; 29;.99;– ];&#x/BBo;&#xx [2;9.9;অ ;ʃ.;醗&#x 254;&#x.878; 29;.99;– ];w/k and amplitude &#x/BBo;&#xx [3;7.9;Ɔ ;ʂ.;閕&#x 338;&#x.638; 29;.24; ];&#x/BBo;&#xx [3;7.9;Ɔ ;ʂ.;閕&#x 338;&#x.638; 29;.24; ];1 &#x/BBo;&#xx [3;@.3;Ƒ ;ʃ.;栂&#x 348;&#x.718;
29;.51;• ];&#x/BBo;&#xx [3;@.3;Ƒ ;ʃ.;栂&#x 348;&#x.718;
29;.51;• ];h, (x, &#x/BBo;&#xx [3;c.5;˜ 2;ƒ.4;Ζ ;ͱ.;Α&#x 293;&#x.039; ]0;&#x/BBo;&#xx [3;c.5;˜ 2;ƒ.4;Ζ ;ͱ.;Α&#x 293;&#x.039; ]0;x) &#x/BBo;&#xx [3;r.7;Ɖ ;ʂ.;r 3;w.2;ނ ;ʓ.;瘁&#x ]00;&#x/BBo;&#xx [3;r.7;Ɖ ;ʂ.;r 3;w.2;ނ ;ʓ.;瘁&#x ]00;1. Substituting (12) (3) and &#x/BBo;&#xx [3;r.7;Ɖ ;ʂ.;r 3;w.2;ނ ;ʓ.;瘁&#x ]00;&#x/BBo;&#xx [3;r.7;Ɖ ;ʂ.;r 3;w.2;ނ ;ʓ.;瘁&#x ]00;(4), and taking the time average, we obtain &#x/BBo;&#xx [1;).3;օ ;ɘ.;␂&#x 134;&#x.399;&#x 265;&#x.680; ]0;&#x/BBo;&#xx [1;).3;օ ;ɘ.;␂&#x 134;&#x.399;&#x 265;&#x.680; ]0;k 1 &#x/BBo;&#xx [2;%.5;ঈ ;ɖ.;㆘&#x 239;&#x.759;&#x 266;&#x.640; ]0;&#x/BBo;&#xx [2;%.5;ঈ ;ɖ.;㆘&#x 239;&#x.759;&#x 266;&#x.640; ]0;ah; &#x/BBo;&#xx [1;.1;ƃ ;Ɉ.;㦘&#x 114;&#x.958; 26;�.88; ];&#x/BBo;&#xx [1;.1;ƃ ;Ɉ.;㦘&#x 114;&#x.958; 26;�.88; ];Fp = -Re (Ap,) &#x/BBo;&#xx [1;‘.5;Ɓ ;ɉ.;6 2;.9;Ɖ ;ɠ.;ᖕ&#x ]00;&#x/BBo;&#xx [1;‘.5;Ɓ ;ɉ.;6 2;.9;Ɖ ;ɠ.;ᖕ&#x ]00;jht &#x/BBo;&#xx [2;.5;Y 2;P.8; ;ȕ.;·&#x 257;&#x.519; ]0;&#x/BBo;&#xx [2;.5;Y 2;P.8; ;ȕ.;·&#x 257;&#x.519; ]0;+ - - exp &#x/BBo;&#xx [2;g.3;։ ;ɉ.;6 2;‡.9;উ ;ə.;枔&#x ]00;&#x/BBo;&#xx [2;g.3;։ ;ɉ.;6 2;‡.9;উ ;ə.;枔&#x ]00;(jkx) &#x/BBo;&#xx [2;.1;ֈ ;ɐ.;ޖ&#x 304;&#x.078; 25; .20; ];&#x/BBo;&#xx [2;.1;ֈ ;ɐ.;ޖ&#x 304;&#x.078; 25; .20; ];dS, &#x/BBo;&#xx [1;).5;˜ 2;E.2;ޗ ;ij.;醉&#x 252;&#x ]00;&#x/BBo;&#xx [1;).5;˜ 2;E.2;ޗ ;ij.;醉&#x 252;&#x ]00;2 &#x/BBo;&#xx [1;I.2;މ ;Ƀ.;ᆘ&#x 160;&#x.558; 26;.91;— ];&#x/BBo;&#xx [1;I.2;މ ;Ƀ.;ᆘ&#x 160;&#x.558; 26;.91;— ];1s &#x/BBo;&#xx [1;ˆ.3;™ 2;C.1;Ƙ ;ƒ.;⎈&#x 265;&#x.680; ]0;&#x/BBo;&#xx [1;ˆ.3;™ 2;C.1;Ƙ ;ƒ.;⎈&#x 265;&#x.680; ]0;( &#x/BBo;&#xx [2;.4;΁ ;Ƀ.;㘄&#x 244;&#x.318; 26;.91;— ];&#x/BBo;&#xx [2;.4;΁ ;Ƀ.;㘄&#x 244;&#x.318; 26;.91;— ];kax) &#x/BBo;&#xx [1;.7;ƅ ;ȣ.; &#x 116;&#x.158; 23;.07;– ];&#x/BBo;&#xx [1;.7;ƅ ;ȣ.; &#x 116;&#x.158; 23;.07;– ];P = - Re &#x/BBo;&#xx [1;e.1; 2; .7;ঙ ;Ɔ.;閇&#x 230;&#x.879; ]0;&#x/BBo;&#xx [1;e.1; 2; .7;ঙ ;Ɔ.;閇&#x 230;&#x.879; ]0;(Ap,) &#x/BBo;&#xx [1;‰.3;ւ ;Ƞ.;禙&#x 209;&#x.758; 23;
.59;• ];&#x/BBo;&#xx [1;‰.3;ւ ;Ƞ.;禙&#x 209;&#x.758; 23;
.59;• ];(jh?) exp &#x/BBo;&#xx [2;3.0;· ;Ƞ.;禙&#x 253;&#x.678; 23;�.87;™ ];&#x/BBo;&#xx [2;3.0;· ;Ƞ.;禙&#x 253;&#x.678; 23;�.87;™ ];(jkx) &#x/BBo;&#xx [2;U.8;Ά ;ȡ.;瘁&#x 269;&#x.758; 23;�.87;™ ];&#x/BBo;&#xx [2;U.8;Ά ;ȡ.;瘁&#x 269;&#x.758; 23;�.87;™ ];dS, &#x/BBo;&#xx [1;S.5;ঈ ;ȕ.;Θ&#x 164;&#x.398; 23;.35;– ];&#x/BBo;&#xx [1;S.5;ঈ ;ȕ.;Θ&#x 164;&#x.398; 23;.35;– ];Js where (Ap,) &#x/BBo;&#xx [7;.83;† 2;.5;؄ ;‚.5;Y 2;.9;֘ ;&#x]000;&#x/BBo;&#xx [7;.83;† 2;.5;؄ ;‚.5;Y 2;.9;֘ ;&#x]000;= (Ap) exp &#x/BBo;&#xx [1;).1; 1;˜.4;ࠁ ;Š.;禁&#x 208;&#x.560;
]0;&#x/BBo;&#xx [1;).1; 1;˜.4;ࠁ ;Š.;禁&#x 208;&#x.560;
]0;(-jut), is independent a result &#x/BBo;&#xx [1;).1; 1;˜.4;ࠁ ;Š.;禁&#x 208;&#x.560;
]0;&#x/BBo;&#xx [1;).1; 1;˜.4;ࠁ ;Š.;禁&#x 208;&#x.560;
]0;%due to the leading-edge suction is from inequality &#x/BBo;&#xx [1;).1; 1;˜.4;ࠁ ;Š.;禁&#x 208;&#x.560;
]0;&#x/BBo;&#xx [1;).1; 1;˜.4;ࠁ ;Š.;禁&#x 208;&#x.560;
]0;P &#x/BBo;&#xx [1;s.7;և ;ř.;萁&#x 179;&#x.518; 16;.79;™ ];&#x/BBo;&#xx [1;s.7;և ;ř.;萁&#x 179;&#x.518; 16;.79;™ ];2 &#x/BBo;&#xx [1;„.7;ঈ ;Š.;考&#x 199;&#x.438;
16; .92;&#x ]00;&#x/BBo;&#xx [1;„.7;ঈ ;Š.;考&#x 199;&#x.438;
16; .92;&#x ]00;UF &#x/BBo;&#xx [2;.5;Ƅ ;ř.;妕&#x 209;&#x.518;
16;.79;™ ];&#x/BBo;&#xx [2;.5;Ƅ ;ř.;妕&#x 209;&#x.518;
16;.79;™ ];2 &#x/BBo;&#xx [2;.9;উ ;Ř.;㦘&#x 238;&#x.558; 16; .67;” ];&#x/BBo;&#xx [2;.9;উ ;Ř.;㦘&#x 238;&#x.558; 16; .67;” ];uTP, &#x/BBo;&#xx [3;`.9;֑ ;ŗ.;栂&#x 377;&#x.518; 16;.51;• ];&#x/BBo;&#xx [3;`.9;֑ ;ŗ.;栂&#x 377;&#x.518; 16;.51;• ];(14) provided E &#x/BBo;&#xx [7;.87;„ 1;D.2;Ε ;€.6;΅ ;ő.; &#x ]00;&#x/BBo;&#xx [7;.87;„ 1;D.2;Ε ;€.6;΅ ;ő.; &#x ]00;2 &#x/BBo;&#xx [8;.19; 14;.19;— 9;.15;ˆ 1;R.1;֕ ;&#x]000;&#x/BBo;&#xx [8;.19; 14;.19;— 9;.15;ˆ 1;R.1;֕ ;&#x]000;0. Consequently, if &#x/BBo;&#xx [1;w.1;ƃ ;Ń.;Θ&#x 203;&#x.038; 15;.11;˜ ];&#x/BBo;&#xx [1;w.1;ƃ ;Ń.;Θ&#x 203;&#x.038; 15;.11;˜ ];ah,/ax = 0, or if &#x/BBo;&#xx [2;U.8;Ά ;ł.;㈂&#x 256;&#x.798; 15;.59;™ ];&#x/BBo;&#xx [2;U.8;Ά ;ł.;㈂&#x 256;&#x.798; 15;.59;™ ];1 &#x/BBo;&#xx [2;X.7;Ɓ ;ł.;㈂&#x 273;&#x.598;&#x 153;&#x.599; ]0;&#x/BBo;&#xx [2;X.7;Ɓ ;ł.;㈂&#x 273;&#x.598;&#x 153;&#x.599; ]0;ahll &#x/BBo;&#xx [2;u.5;Ƅ ;ł.;㈂&#x 286;&#x.798;
15;.59;™ ];&#x/BBo;&#xx [2;u.5;Ƅ ;ł.;㈂&#x 286;&#x.798;
15;.59;™ ];8x1 &#x/BBo;&#xx [2;‘.8;Ά ;Ń.;疔&#x 298;&#x.798; 15;
.67;” ];&#x/BBo;&#xx [2;‘.8;Ά ;Ń.;疔&#x 298;&#x.798; 15;
.67;” ];4 &#x/BBo;&#xx [3;.5;঄ ;Ł.;萁&#x 325;&#x.438;
15;.36; ];&#x/BBo;&#xx [3;.5;঄ ;Ł.;萁&#x 325;&#x.438;
15;.36; ];I1chll, then from (13) and (14) we immediately have &#x/BBo;&#xx [3;.5;঄ ;Ł.;萁&#x 325;&#x.438;
15;.36; ];&#x/BBo;&#xx [3;.5;঄ ;Ł.;萁&#x 325;&#x.438;
15;.36; ];wllc &#x/BBo;&#xx [2;.1;ং ;Ė.;ᖕ&#x 211;&#x.199;&#x 123;&#x.36 ;&#x]000;&#x/BBo;&#xx [2;.1;ং ;Ė.;ᖕ&#x 211;&#x.199;&#x 123;&#x.36 ;&#x]000;2 &#x/BBo;&#xx [2;.7;ƅ ;ė.;ᆘ&#x 226;&#x.318; 12;.80; ];&#x/BBo;&#xx [2;.7;ƅ ;ė.;ᆘ&#x 226;&#x.318; 12;.80; ];U. &#x/BBo;&#xx [3;`.7;ƅ ;Ĕ.;␂&#x 377;&#x.278; 12;.84;
];&#x/BBo;&#xx [3;`.7;ƅ ;Ĕ.;␂&#x 377;&#x.278; 12;.84;
];(15) This result shows that not only is a &#x/BBo;&#xx [3;`.7;ƅ ;Ĕ.;␂&#x 377;&#x.278; 12;.84;
];&#x/BBo;&#xx [3;`.7;ƅ ;Ĕ.;␂&#x 377;&#x.278; 12;.84;
];also its phase velocity be greater &#x/BBo;&#xx [3;`.7;ƅ ;Ĕ.;␂&#x 377;&#x.278; 12;.84;
];&#x/BBo;&#xx [3;`.7;ƅ ;Ĕ.;␂&#x 377;&#x.278; 12;.84;
];U (under the stated conditions), in order to achieve a given swimming velocity &#x/BBo;&#xx [2;.3;ঃ ;v.5;؄ ;ȧ.;馂&#x 84.;⎙&#x ]00;&#x/BBo;&#xx [2;.3;ঃ ;v.5;؄ ;ȧ.;馂&#x 84.;⎙&#x ]00;U. This qualitative feature remains true for amplitude function with variable forward ��in plane flow Though the flow around swimming fish is certainly three-dimensional, ��the theory of two-dimensional swimming motion is still of considerable interest, large aspect ratio some species ��the faster sharks, or even the propulsion of wing flappings of migrating birds. ��We derive in the following the main features of swimming with arbitrary forward velocity in plane flows. Here we consider the incompressible plane flow of an &#x/BBo;&#xx [3;7.1;™ 5;.5;ƙ ;ͱ.;馓&#x 510;&#x.959; ]0;&#x/BBo;&#xx [3;7.1;™ 5;.5;ƙ ;ͱ.;馓&#x 510;&#x.959; ]0;inviscid fluid past &#x/BBo;&#xx [4;$.0; 50;.56; 4;(.3;ঘ ;ԇ.;怂&#x ]00;&#x/BBo;&#xx [4;$.0; 50;.56; 4;(.3;ঘ ;ԇ.;怂&#x ]00;a flexible plate of zero thickness, spanning a waving motion general form continuous function x, t, always small. (The small thickness a planar body can be estimated The motion &#x/BBo;&#xx [4;$.0; 50;.56; 4;(.3;ঘ ;ԇ.;怂&#x ]00;&#x/BBo;&#xx [4;$.0; 50;.56; 4;(.3;ঘ ;ԇ.;怂&#x ]00;= &#x/BBo;&#xx [8;.55;— 4;.4;Ι ;“.1; 4;.3;ঘ ;&#x]000;&#x/BBo;&#xx [8;.55;— 4;.4;Ι ;“.1; 4;.3;ঘ ;&#x]000;0 from a uniform free-stream velocity &#x/BBo;&#xx [8;.55;— 4;.4;Ι ;“.1; 4;.3;ঘ ;&#x]000;&#x/BBo;&#xx [8;.55;— 4;.4;Ι ;“.1; 4;.3;ঘ ;&#x]000;U(t) may depend on t. Let &#x/BBo;&#xx [8;.55;— 4;.4;Ι ;“.1; 4;.3;ঘ ;&#x]000;&#x/BBo;&#xx [8;.55;— 4;.4;Ι ;“.1; 4;.3;ঘ ;&#x]000;v again denote respectively y component &#x/BBo;&#xx [8;.55;— 4;.4;Ι ;“.1; 4;.3;ঘ ;&#x]000;&#x/BBo;&#xx [8;.55;— 4;.4;Ι ;“.1; 4;.3;ঘ ;&#x]000;pa is the pressure at infinity and &#x/BBo;&#xx [2;b.5;Y 3;9.6; ;ɨ.;ޖ&#x 346;&#x.799; ]0;&#x/BBo;&#xx [2;b.5;Y 3;9.6; ;ɨ.;ޖ&#x 346;&#x.799; ]0;p is the fluid density. In the linear theory of this incompressible irrotational flow, &#x/BBo;&#xx [2;€.0; 32;.12;
2;ˆ.7;Ɩ ;̴.;㈆&#x ]00;&#x/BBo;&#xx [2;€.0; 32;.12;
2;ˆ.7;Ɩ ;̴.;㈆&#x ]00;p, and hence also &#x/BBo;&#xx [2;€.0; 32;.12;
2;ˆ.7;Ɩ ;̴.;㈆&#x ]00;&#x/BBo;&#xx [2;€.0; 32;.12;
2;ˆ.7;Ɩ ;̴.;㈆&#x ]00;$, is a harmonic function of x, y for all harmonic function &#x/BBo;&#xx [2;€.0; 32;.12;
2;ˆ.7;Ɩ ;̴.;㈆&#x ]00;&#x/BBo;&#xx [2;€.0; 32;.12;
2;ˆ.7;Ɩ ;̴.;㈆&#x ]00;$(x, y, t) conjugate to &#x/BBo;&#xx [3;†.6;Ζ ;̔.;搇&#x 392;&#x.159; 32;.48;
];&#x/BBo;&#xx [3;†.6;Ζ ;̔.;搇&#x 392;&#x.159; 32;.48;
];$ may be defined by &#x/BBo;&#xx [1;!.9;Ɨ ;́.;䐇&#x 131;&#x.999; 31;
.52; ];&#x/BBo;&#xx [1;!.9;Ɨ ;́.;䐇&#x 131;&#x.999; 31;
.52; ];$, = &#x/BBo;&#xx [1;F.1;֙ ;̀.;r 1;Y.8;9 3;.7;؁ ;&#x]000;&#x/BBo;&#xx [1;F.1;֙ ;̀.;r 1;Y.8;9 3;.7;؁ ;&#x]000;$,, &#x/BBo;&#xx [1;c.6;ޘ ;̀.;r 1;s.7;֘ ;̑.;分&#x ]00;&#x/BBo;&#xx [1;c.6;ޘ ;̀.;r 1;s.7;֘ ;̑.;分&#x ]00;$, = - &#x/BBo;&#xx [1;˜.7;Ɩ ;́.;栂&#x 212;&#x.399; 31;.00; ];&#x/BBo;&#xx [1;˜.7;Ɩ ;́.;栂&#x 212;&#x.399; 31;.00; ];$x, where the subscripts x y denote &#x/BBo;&#xx [1;˜.7;Ɩ ;́.;栂&#x 212;&#x.399; 31;.00; ];&#x/BBo;&#xx [1;˜.7;Ɩ ;́.;栂&#x 212;&#x.399; 31;.00; ];$+i$ and the complex velocity w = u- &#x/BBo;&#xx [1;H.0;ޒ ;ɸ.;蠂&#x 155;&#x.999; 28;.32; ];&#x/BBo;&#xx [1;H.0;ޒ ;ɸ.;蠂&#x 155;&#x.999; 28;.32; ];iv are analytic &#x/BBo;&#xx [1;H.0;ޒ ;ɸ.;蠂&#x 155;&#x.999; 28;.32; ];&#x/BBo;&#xx [1;H.0;ޒ ;ɸ.;蠂&#x 155;&#x.999; 28;.32; ];x = &#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];&#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];x+iy for all real t. (We borrow &#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];&#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];x for this different purpose in this related by &#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];&#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];Euler's equation of motion, The linearized boundary conditions are Here, condition (19) &#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];&#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];(18); condition (20) &#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];&#x/BBo;&#xx [3;‰.5;ƒ ;ɷ.;栅&#x 413;&#x.519; 28;.28; ];$ is even, and hence &#x/BBo;&#xx [3;P.3;™ 6;.08;&#x 355;&#x.919; 74;&#x.159; ]0;&#x/BBo;&#xx [3;P.3;™ 6;.08;&#x 355;&#x.919; 74;&#x.159; ]0;$ is odd in propulsion. Part ��is &#x/BBo;&#xx [2; .03;‘ 6;&.6;Є ;B.9;֘ ;ش.;㆘&#x ]00;&#x/BBo;&#xx [2; .03;‘ 6;&.6;Є ;B.9;֘ ;ش.;㆘&#x ]00;the Kutta condition fo'r the flow at the trailing edge &#x/BBo;&#xx [2;c.9;ও ;ا.;萁&#x 267;&#x.839;&#x 632;&#x.880; ]0;&#x/BBo;&#xx [2;c.9;ও ;ا.;萁&#x 267;&#x.839;&#x 632;&#x.880; ]0;x = 1. Condition (22) for &#x/BBo;&#xx [1; .43;’ 6;.4;Ѓ ;%.9;Ɖ ;ؘ.;爃&#x ]00;&#x/BBo;&#xx [1; .43;’ 6;.4;Ѓ ;%.9;Ɖ ;ؘ.;爃&#x ]00;w &#x/BBo;&#xx [2;.79;– 6;.7;ȇ ;G.7;֘ ;ؠ.;ᘆ&#x ]00;&#x/BBo;&#xx [2;.79;– 6;.7;ȇ ;G.7;֘ ;ؠ.;ᘆ&#x ]00;may also be specified as &#x/BBo;&#xx [1;8.9;֕ ;ؒ.;爇&#x 147;&#x.359; 62;.76; ];&#x/BBo;&#xx [1;8.9;֕ ;ؒ.;爇&#x 147;&#x.359; 62;.76; ];1x1 &#x/BBo;&#xx [1;P.4;ޗ ;ؓ.;鈄&#x 171;&#x.839; 62;�.64; ];&#x/BBo;&#xx [1;P.4;ޗ ;ؓ.;鈄&#x 171;&#x.839; 62;�.64; ];+a, &#x/BBo;&#xx [1;v.6;Γ ;ؒ.;䠁&#x 193;&#x.199;&#x 623;&#x.520; ]0;&#x/BBo;&#xx [1;v.6;Γ ;ؒ.;䠁&#x 193;&#x.199;&#x 623;&#x.520; ]0;larg &#x/BBo;&#xx [1;–.3;Ƒ ;ؒ.;䠁&#x 202;&#x.319; 62;.52; ];&#x/BBo;&#xx [1;–.3;Ƒ ;ؒ.;䠁&#x 202;&#x.319; 62;.52; ];zl &#x/BBo;&#xx [2;.8;Δ ;ؕ.;㘇&#x 213;&#x.599; 62;�.40;
];&#x/BBo;&#xx [2;.8;Δ ;ؕ.;㘇&#x 213;&#x.599; 62;�.40;
];&#x/BBo;&#xx [2;.8;Δ ;ؕ.;㘇&#x 213;&#x.599; 62;�.40;
]; &#x/BBo;&#xx [2;.1;֙ ;ؓ.;枘&#x 225;&#x.119; 62;.32; ];&#x/BBo;&#xx [2;.1;֙ ;ؓ.;枘&#x 225;&#x.119; 62;.32; ];0, &#x/BBo;&#xx [2;).1;™ 6;.3;؇ ;Ɂ.;枔&#x 622;&#x.800; ]0;&#x/BBo;&#xx [2;).1;™ 6;.3;؇ ;Ɂ.;枔&#x 622;&#x.800; ]0;i.e. as &#x/BBo;&#xx [2;W.9;খ ;ؕ.;㘇&#x 261;&#x.839; 62;�.40;
];&#x/BBo;&#xx [2;W.9;খ ;ؕ.;㘇&#x 261;&#x.839; 62;�.40;
];a &#x/BBo;&#xx [2;c.9;ও ;ؕ.;㘇&#x 282;&#x.72 ;ؠ.;搇&#x ]00;&#x/BBo;&#xx [2;c.9;ও ;ؕ.;㘇&#x 282;&#x.72 ;ؠ.;搇&#x ]00;+cc in the region exclud- ing the trailing vortex sheet. Integration of (17) to obtain the boundary value of &#x/BBo;&#xx [2;t.0;ޒ ;ֈ.;&#x 281;&#x.279; 59;.08; ];&#x/BBo;&#xx [2;t.0;ޒ ;ֈ.;&#x 281;&#x.279; 59;.08; ];@ on the plate can be &#x/BBo;&#xx [1;.95;‘ 5;v.2;І ;9.1; 5;ƒ.9; ]0;&#x/BBo;&#xx [1;.95;‘ 5;v.2;І ;9.1; 5;ƒ.9; ]0;done by using the method of characteristics. However, with variable &#x/BBo;&#xx [1;.95;‘ 5;v.2;І ;9.1; 5;ƒ.9; ]0;&#x/BBo;&#xx [1;.95;‘ 5;v.2;І ;9.1; 5;ƒ.9; ]0;U(t), it is &#x/BBo;&#xx [1;.95;‘ 5;c.7;؁ ;@.3;Ƙ ;հ.;&#x ]00;&#x/BBo;&#xx [1;.95;‘ 5;c.7;؁ ;@.3;Ƙ ;հ.;&#x ]00;more convenient to make use of the &#x/BBo;&#xx [1;s.9;ও ;գ.;Ѕ&#x 209;&#x.039;
57;.87;™ ];&#x/BBo;&#xx [1;s.9;ও ;գ.;Ѕ&#x 209;&#x.039;
57;.87;™ ];Laplace transform method. We first introduce &#x/BBo;&#xx [1;.95;‘ 5;Q.5;ȃ ;2.8;ޙ ;ՙ.;䐃&#x ]00;&#x/BBo;&#xx [1;.95;‘ 5;Q.5;ȃ ;2.8;ޙ ;ՙ.;䐃&#x ]00;the variable &#x/BBo;&#xx [1;‚.6;Ή ;Ը.;嘈&#x 209;&#x.039;
54;.16; ];&#x/BBo;&#xx [1;‚.6;Ή ;Ը.;嘈&#x 209;&#x.039;
54;.16; ];U(t)dt (t &#x/BBo;&#xx [2;0.1;֒ ;Հ.;䠁&#x 235;&#x.919; 54;.76;
];&#x/BBo;&#xx [2;0.1;֒ ;Հ.;䠁&#x 235;&#x.919; 54;.76;
];&#x/BBo;&#xx [2;0.1;֒ ;Հ.;䠁&#x 235;&#x.919; 54;.76;
]; &#x/BBo;&#xx [2;@.4;ޗ ;Ը.;㈂&#x 251;&#x.039; 54;.68; ];&#x/BBo;&#xx [2;@.4;ޗ ;Ը.;㈂&#x 251;&#x.039; 54;.68; ];O), &#x/BBo;&#xx [3;` 5;7.8;Ё ;Ͷ.;㆑&#x 547;&#x.200; ]0;&#x/BBo;&#xx [3;` 5;7.8;Ё ;Ͷ.;㆑&#x 547;&#x.200; ]0;(23) &#x/BBo;&#xx [1;.71;– 5;.5;ঙ ;4.5;֗ ;Ԣ ;&#x]000;&#x/BBo;&#xx [1;.71;– 5;.5;ঙ ;4.5;֗ ;Ԣ ;&#x]000;and assume its inverse function &#x/BBo;&#xx [1;.71;– 5;.5;ঙ ;4.5;֗ ;Ԣ ;&#x]000;&#x/BBo;&#xx [1;.71;– 5;.5;ঙ ;4.5;֗ ;Ԣ ;&#x]000;t(r) &#x/BBo;&#xx [1;™.6;ޘ ;ԕ.;Ђ&#x 206;&#x.399;&#x 522;&#x.480;
]0;&#x/BBo;&#xx [1;™.6;ޘ ;ԕ.;Ђ&#x 206;&#x.399;&#x 522;&#x.480;
]0;is unique so that &#x/BBo;&#xx [2;.5;ƒ ;ԕ.;Ђ&#x 288;&#x.479;&#x 522;&#x.480;
]0;&#x/BBo;&#xx [2;.5;ƒ ;ԕ.;Ђ&#x 288;&#x.479;&#x 522;&#x.480;
]0;U = U(t &#x/BBo;&#xx [3;.9;Ɨ ;Ԓ.;搇&#x 333;&#x.599; 52;.24; ];&#x/BBo;&#xx [3;.9;Ɨ ;Ԓ.;搇&#x 333;&#x.599; 52;.24; ];(7)) is a &#x/BBo;&#xx [3;W.5;ক ;Ԕ.;㈆&#x 376;&#x.799; 51; .60; ];&#x/BBo;&#xx [3;W.5;ক ;Ԕ.;㈆&#x 376;&#x.799; 51; .60; ];one- &#x/BBo;&#xx [1;.47; 50;
.36;&#x 47.;疘&#x 510;&#x.000; ]0;&#x/BBo;&#xx [1;.47; 50;
.36;&#x 47.;疘&#x 510;&#x.000; ]0;valued function of r, this being the case so long as the swimming proceeds in &#x/BBo;&#xx [1;.47; 50;
.36;&#x 47.;疘&#x 510;&#x.000; ]0;&#x/BBo;&#xx [1;.47; 50;
.36;&#x 47.;疘&#x 510;&#x.000; ]0;one direction. Regarding &#x/BBo;&#xx [1;.47; 50;
.36;&#x 47.;疘&#x 510;&#x.000; ]0;&#x/BBo;&#xx [1;.47; 50;
.36;&#x 47.;疘&#x 510;&#x.000; ]0;x and &#x/BBo;&#xx [2;e.4;Ζ ;҈.;䀅&#x 272;&#x.399; 49;.64; ];&#x/BBo;&#xx [2;e.4;Ζ ;҈.;䀅&#x 272;&#x.399; 49;.64; ];T, (17) becomes &#x/BBo;&#xx [9;.67;” 3;”.8; ;Ĕ.;閘&#x 406;&#x.08 ;&#x]000;&#x/BBo;&#xx [9;.67;” 3;”.8; ;Ĕ.;閘&#x 406;&#x.08 ;&#x]000;I(,, &#x/BBo;&#xx [1;.5;ঙ ;Δ.;考&#x 124;&#x.319;
40;.16; ];&#x/BBo;&#xx [1;.5;ঙ ;Δ.;考&#x 124;&#x.319;
40;.16; ];s) = &#x/BBo;&#xx [1;8.2;Ι ;Έ.;嘁&#x 168;&#x.239;
41;
.12; ];&#x/BBo;&#xx [1;8.2;Ι ;Έ.;嘁&#x 168;&#x.239;
41;
.12; ];/:exP ( &#x/BBo;&#xx [1;w.1;Ɣ ;Δ.;考&#x 196;&#x.799; 40;.16; ];&#x/BBo;&#xx [1;w.1;Ɣ ;Δ.;考&#x 196;&#x.799; 40;.16; ];-ST) &#x/BBo;&#xx [1;™.4;Β ;Δ.;嘈&#x 216;&#x.959;
40;.16; ];&#x/BBo;&#xx [1;™.4;Β ;Δ.;嘈&#x 216;&#x.959;
40;.16; ];B(x, T) &#x/BBo;&#xx [2;(.9;֕ ;Ζ.;爇&#x 238;&#x.559; 40;.16; ];&#x/BBo;&#xx [2;(.9;֕ ;Ζ.;爇&#x 238;&#x.559; 40;.16; ];d~ (Re &#x/BBo;&#xx [2;g.5;ক ;Ζ.;爇&#x 271;&#x.199; 40;
.52; ];&#x/BBo;&#xx [2;g.5;ক ;Ζ.;爇&#x 271;&#x.199; 40;
.52; ];s &#x/BBo;&#xx [2;u.5;ƕ ;Ζ.;爇&#x 281;&#x.279; 40;
.76;
];&#x/BBo;&#xx [2;u.5;ƕ ;Ζ.;爇&#x 281;&#x.279; 40;
.76;
];&#x/BBo;&#xx [2;u.5;ƕ ;Ζ.;爇&#x 281;&#x.279; 40;
.76;
]; &#x/BBo;&#xx [2;….8;9 3;”.3;Ȃ ;ʓ.;冕&#x 403;&#x.680; ]0;&#x/BBo;&#xx [2;….8;9 3;”.3;Ȃ ;ʓ.;冕&#x 403;&#x.680; ]0;0) &#x/BBo;&#xx [3;Y.2;ޓ ;Δ.;ࠇ&#x 375;&#x.839;&#x 403;&#x.200; ]0;&#x/BBo;&#xx [3;Y.2;ޓ ;Δ.;ࠇ&#x 375;&#x.839;&#x 403;&#x.200; ]0;(26) to &#x/BBo;&#xx [3;�.47;” 3;i.3;؄ ;H.9;֕ ;͹.; &#x ]00;&#x/BBo;&#xx [3;�.47;” 3;i.3;؄ ;H.9;֕ ;͹.; &#x ]00;(24), under zero initial conditions, yields &#x/BBo;&#xx [3;�.47;” 3;i.3;؄ ;H.9;֕ ;͹.; &#x ]00;&#x/BBo;&#xx [3;�.47;” 3;i.3;؄ ;H.9;֕ ;͹.; &#x ]00;x = - &#x/BBo;&#xx [1;.0; 31;.23;™ 2;�.8;ޕ ;̥.; &#x ]00;&#x/BBo;&#xx [1;.0; 31;.23;™ 2;�.8;ޕ ;̥.; &#x ]00;co, using conditions &#x/BBo;&#xx [2;„.3;ঘ ;̗.;刃&#x 303;&#x.36 ;̧.;ሁ&#x ]00;&#x/BBo;&#xx [2;„.3;ঘ ;̗.;刃&#x 303;&#x.36 ;̧.;ሁ&#x ]00;(22), and expressing in terms of &#x/BBo;&#xx [7;.71;’ 3; 8;.83; 31;.64; ];&#x/BBo;&#xx [7;.71;’ 3; 8;.83; 31;.64; ];@, and vice versa, we obtain its imaginary part as On the plate, with &#x/BBo;&#xx [7;.71;’ 3; 8;.83; 31;.64; ];&#x/BBo;&#xx [7;.71;’ 3; 8;.83; 31;.64; ];+ and &#x/BBo;&#xx [1;€ 2;#.2; ;Ɖ.;㖓&#x 234;&#x.480;
]0;&#x/BBo;&#xx [1;€ 2;#.2; ;Ɖ.;㖓&#x 234;&#x.480;
]0;1x1 &#x/BBo;&#xx [1;”.8;ޙ ;Ȧ.; 2;�.3;ঔ ;ȱ.;6 ];&#x/BBo;&#xx [1;”.8;ޙ ;Ȧ.; 2;�.3;ঔ ;ȱ.;6 ];1, &#x/BBo;&#xx [2;.8;ޕ ;ȣ.;’ 2;4.4;y 2;3.2;ࠄ ;&#x]000;&#x/BBo;&#xx [2;.8;ޕ ;ȣ.;’ 2;4.4;y 2;3.2;ࠄ ;&#x]000;G(x, &#x/BBo;&#xx [2;7.3;֖ ;Ȧ.; 2;A.9;Ɖ ;ȳ.;Θ&#x ]00;&#x/BBo;&#xx [2;7.3;֖ ;Ȧ.; 2;A.9;Ɖ ;ȳ.;Θ&#x ]00;0 &#x/BBo;&#xx [2;D.5;Y 2;$.8;ࠂ ;ɐ.;ޖ&#x 232;&#x.080; ]0;&#x/BBo;&#xx [2;D.5;Y 2;$.8;ࠂ ;ɐ.;ޖ&#x 232;&#x.080; ]0;+ , s) = &#x/BBo;&#xx [2;x.8;ޒ ;ȣ.;栅&#x 296;&#x.159; 23;.44; ];&#x/BBo;&#xx [2;x.8;ޒ ;ȣ.;栅&#x 296;&#x.159; 23;.44; ];v(x, s), which is the &#x/BBo;&#xx [1; 21;
.44; 5;
.83; 22;
.52; ];&#x/BBo;&#xx [1; 21;
.44; 5;
.83; 22;
.52; ];Laplace transform of (18). Application of this condition to &#x/BBo;&#xx [2;†.3;Ƒ ;Ȑ.;阆&#x 308;&#x.879; 22;�.56; ];&#x/BBo;&#xx [2;†.3;Ƒ ;Ȑ.;阆&#x 308;&#x.879; 22;�.56; ];(28a) yields &#x/BBo;&#xx [9;
.43;’ 1;v.1;؃ ;•.7;Y 1;w.8;Ё ;&#x]000;&#x/BBo;&#xx [9;
.43;’ 1;v.1;؃ ;•.7;Y 1;w.8;Ё ;&#x]000;* where &#x/BBo;&#xx [8; .75;” 1;f.0;ࠃ ;Ĵ.;ᖕ&#x 175;&#x.920; ]0;&#x/BBo;&#xx [8; .75;” 1;f.0;ࠃ ;Ĵ.;ᖕ&#x 175;&#x.920; ]0;KC&.)= &#x/BBo;&#xx [1;9.9;Ɨ ;ř.;萁&#x 196;&#x.319;
18;.40; ];&#x/BBo;&#xx [1;9.9;Ɨ ;ř.;萁&#x 196;&#x.319;
18;.40; ];-(&+s)/~ &#x/BBo;&#xx [2;.0;ޒ ;ť.;萈&#x 248;&#x.399; 17;.60; ];&#x/BBo;&#xx [2;.0;ޒ ;ť.;萈&#x 248;&#x.399; 17;.60; ];Y(x1,s)dxl &#x/BBo;&#xx [2;Y.9;Ɖ ;ť.;ሁ&#x 273;&#x.599;
17;.16; ];&#x/BBo;&#xx [2;Y.9;Ɖ ;ť.;ሁ&#x 273;&#x.599;
17;.16; ];(1x1 &#x/BBo;&#xx [2;y.1;Ƙ ;ŧ.;瘁&#x 284;&#x.639; 17;.04; ];&#x/BBo;&#xx [2;y.1;Ƙ ;ŧ.;瘁&#x 284;&#x.639; 17;.04; ];1), - 1 &#x/BBo;&#xx [3;R.7;খ ;ť.;㘇&#x 365;&#x.999; 17;.72;&#x ]00;&#x/BBo;&#xx [3;R.7;খ ;ť.;㘇&#x 365;&#x.999; 17;.72;&#x ]00;(29 &#x/BBo;&#xx [3;g.9;Ɖ ;ť.;怂&#x 375;&#x.36 ;ŵ.; &#x ]00;&#x/BBo;&#xx [3;g.9;Ɖ ;ť.;怂&#x 375;&#x.36 ;ŵ.; &#x ]00;b) - 1 &#x/BBo;&#xx [1; 13;.80; 3;.59;• 1;D.2;І ;&#x]000;&#x/BBo;&#xx [1; 13;.80; 3;.59;• 1;D.2;І ;&#x]000;and A, &#x/BBo;&#xx [6;.15;’ 1;4.1;؆ ;w.9;খ ;Ń.;瘅&#x ]00;&#x/BBo;&#xx [6;.15;’ 1;4.1;؆ ;w.9;খ ;Ń.;瘅&#x ]00;(s) = - &#x/BBo;&#xx [1;
.2;ޗ ;Ķ.;嘁&#x 105;&#x.119; 14;
.36;&#x ]00;&#x/BBo;&#xx [1;
.2;ޗ ;Ķ.;嘁&#x 105;&#x.119; 14;
.36;&#x ]00;s &#x/BBo;&#xx [1;&.2;Ε ;Ĵ.;ᘆ&#x 141;&#x.599; 14;.76; ];&#x/BBo;&#xx [1;&.2;Ε ;Ĵ.;ᘆ&#x 141;&#x.599; 14;.76; ];qx, &#x/BBo;&#xx [1;D.4;y 1;6.5;؁ ;ʼn.;Δ&#x 143;&#x.519; ]0;&#x/BBo;&#xx [1;D.4;y 1;6.5;؁ ;ʼn.;Δ&#x 143;&#x.519; ]0;o &#x/BBo;&#xx [1;Q.6;ޔ ;ĵ.;㘄&#x 157;&#x.199;&#x 142;&#x.320; ]0;&#x/BBo;&#xx [1;Q.6;ޔ ;ĵ.;㘄&#x 157;&#x.199;&#x 142;&#x.320; ]0;+ , &#x/BBo;&#xx [1;d.3;ঔ ;Ĵ.;ᘆ&#x 171;&#x.119; 14;.76; ];&#x/BBo;&#xx [1;d.3;ঔ ;Ĵ.;ᘆ&#x 171;&#x.119; 14;.76; ];s) dx = &#x/BBo;&#xx [1;˜ 1;(.1;֙ ;Ȑ.;⎙&#x 150;&#x.720; ]0;&#x/BBo;&#xx [1;˜ 1;(.1;֙ ;Ȑ.;⎙&#x 150;&#x.720; ]0;s/ &#x/BBo;&#xx [2;#.1;ও ;ij.;’ 2;9.0;Δ ;Ł.;6 ];&#x/BBo;&#xx [2;#.1;ও ;ij.;’ 2;9.0;Δ ;Ł.;6 ];exp &#x/BBo;&#xx [2;A.9;Ɖ ;Ĵ.;ᘆ&#x 257;&#x.278; 14;.51;™ ];&#x/BBo;&#xx [2;A.9;Ɖ ;Ĵ.;ᘆ&#x 257;&#x.278; 14;.51;™ ];(s(x + &#x/BBo;&#xx [2;i.0;Θ ;Ĵ.;ᘆ&#x 279;&#x.119; 14;.76; ];&#x/BBo;&#xx [2;i.0;Θ ;Ĵ.;ᘆ&#x 279;&#x.119; 14;.76; ];1)) &#x/BBo;&#xx [2;‚.2;Ι ;ij.;’ 3;
.6;ޑ ;Ņ.;枘&#x ]00;&#x/BBo;&#xx [2;‚.2;Ι ;ij.;’ 3;
.6;ޑ ;Ņ.;枘&#x ]00;P(x, 0, &#x/BBo;&#xx [3;.9;Ɖ ;ij.;’ 3; .8;ޙ ;Ń.;⠄&#x ]00;&#x/BBo;&#xx [3;.9;Ɖ ;ij.;’ 3; .8;ޙ ;Ń.;⠄&#x ]00;s) &#x/BBo;&#xx [3;#.2;ޓ ;ĵ.;萅&#x 335;&#x.759; 14;.51;™ ];&#x/BBo;&#xx [3;#.2;ޓ ;ĵ.;萅&#x 335;&#x.759; 14;.51;™ ];dx. &#x/BBo;&#xx [3;S.2;ޗ ;ij.;栅&#x 375;&#x.119; 14;.03;˜ ];&#x/BBo;&#xx [3;S.2;ޗ ;ij.;栅&#x 375;&#x.119; 14;.03;˜ ];(29c) &#x/BBo;&#xx [2;.9;֕ ;Ĩ.;蠆&#x 221;&#x.999; 13;.00; ];&#x/BBo;&#xx [2;.9;֕ ;Ĩ.;蠆&#x 221;&#x.999; 13;.00; ];-m The last equality &#x/BBo;&#xx [2;.9;֕ ;Ĩ.;蠆&#x 221;&#x.999; 13;.00; ];&#x/BBo;&#xx [2;.9;֕ ;Ĩ.;蠆&#x 221;&#x.999; 13;.00; ];(29c) is obtained readily &#x/BBo;&#xx [2;.9;֕ ;Ĩ.;蠆&#x 221;&#x.999; 13;.00; ];&#x/BBo;&#xx [2;.9;֕ ;Ĩ.;蠆&#x 221;&#x.999; 13;.00; ];(28b) with &#x/BBo;&#xx [3;9.8;9 1;.8; ;͢.;掉&#x 116;&#x.160; ]0;&#x/BBo;&#xx [3;9.8;9 1;.8; ;͢.;掉&#x 116;&#x.160; ]0;(28a) at &#x/BBo;&#xx [1;.51;™ 9;.80; 2;.07;’ 1;.8;Ј ;&#x]000;&#x/BBo;&#xx [1;.51;™ 9;.80; 2;.07;’ 1;.8;Ј ;&#x]000;x = - 1, &#x/BBo;&#xx [5;.07;’ 9;.68; 6;.11;˜ 1;
.6; ]0;&#x/BBo;&#xx [5;.07;’ 9;.68; 6;.11;˜ 1;
.6; ]0;y = &#x/BBo;&#xx [7;.51;• 9;.84;
8;.47;” 1;.7;ঙ ;&#x]000;&#x/BBo;&#xx [7;.51;• 9;.84;
8;.47;” 1;.7;ঙ ;&#x]000;0. Thus is known except for &#x/BBo;&#xx [7;.51;• 9;.84;
8;.47;” 1;.7;ঙ ;&#x]000;&#x/BBo;&#xx [7;.51;• 9;.84;
8;.47;” 1;.7;ঙ ;&#x]000;constank term &#x/BBo;&#xx [3;H.4;ޗ ;“.9;؆ ;͵.;6 1;.4;ࠈ ;&#x]000;&#x/BBo;&#xx [3;H.4;ޗ ;“.9;؆ ;͵.;6 1;.4;ࠈ ;&#x]000;AO(s). Furthermore, from 346 T. Y. Wzc This Riemann-Hilbert problem, (29), (30), and conditions (21), (22), can be readily solved (for the general method, see, e.g., Muskhelishvili 1953, pp. 235-8), giving in which the function (z - l)* (x + 1)-* with a branch z = - 1 to x 1 as 1x1 +co. The leading-edge singularity can be separated above solution i P(x, S) = iA, (s) - -6, (s) 2 dE, (31a) Now, substituting the value of T(x, 0,s) for x deduced from (31a), integral representation (29c), appropriate integrations by (29 (s), and hence also A,(s) from (31 B(s) = K1(s) K, (s) + K, 6)' KO, Kl being the modified Beseel inverse transform (31), the solution off becomes ao (7) = - J o [bo (7') + bl (r')] ~(r - r') dr' + b1 (T), (34) and A, (7) inverse transform $ body surface is $+(x, t) = $(x, 0 + , t) = - $(x, 0 - , t) = - $-(x, t), ��in &#x/BBo;&#xx [3;.59;• 6;0.4;ࠁ ;`.2;Α ;ظ.;蠂&#x ]00;&#x/BBo;&#xx [3;.59;• 6;0.4;ࠁ ;`.2;Α ;ظ.;蠂&#x ]00;wllich C over the integral sign denotes its Cauchy principal value. &#x/BBo;&#xx [3;.59;• 6;0.4;ࠁ ;`.2;Α ;ظ.;蠂&#x ]00;&#x/BBo;&#xx [3;.59;• 6;0.4;ࠁ ;`.2;Α ;ظ.;蠂&#x ]00;term in (38) gives &#x/BBo;&#xx [3;.59;• 6;0.4;ࠁ ;`.2;Α ;ظ.;蠂&#x ]00;&#x/BBo;&#xx [3;.59;• 6;0.4;ࠁ ;`.2;Α ;ظ.;蠂&#x ]00;regular wherever is continuous. &#x/BBo;&#xx [3;.59;• 6;0.4;ࠁ ;`.2;Α ;ظ.;蠂&#x ]00;&#x/BBo;&#xx [3;.59;• 6;0.4;ࠁ ;`.2;Α ;ظ.;蠂&#x ]00;a, &#x/BBo;&#xx [2;Q.0;· ;؄.;㈆&#x 261;&#x.598; 61;.92; ];&#x/BBo;&#xx [2;Q.0;· ;؄.;㈆&#x 261;&#x.598; 61;.92; ];(T) is the only quantity in the &#x/BBo;&#xx [2;
.59;‘ 5;’.0;ࠇ ;V.8;ޕ ;؁.; &#x ]00;&#x/BBo;&#xx [2;
.59;‘ 5;’.0;ࠇ ;V.8;ޕ ;؁.; &#x ]00;solution that is influenced by the history (see (34) and (37)) and requires expres- sion in terms &#x/BBo;&#xx [2;
.59;‘ 5;’.0;ࠇ ;V.8;ޕ ;؁.; &#x ]00;&#x/BBo;&#xx [2;
.59;‘ 5;’.0;ࠇ ;V.8;ޕ ;؁.; &#x ]00;T. The pressure difference across &#x/BBo;&#xx [2;
.59;‘ 5;’.0;ࠇ ;V.8;ޕ ;؁.; &#x ]00;&#x/BBo;&#xx [2;
.59;‘ 5;’.0;ࠇ ;V.8;ޕ ;؁.; &#x ]00;(16), Ap &#x/BBo;&#xx [1;.4;ކ ;գ.;首&#x 120;&#x.959; 56;.84; ];&#x/BBo;&#xx [1;.4;ކ ;գ.;首&#x 120;&#x.959; 56;.84; ];E &#x/BBo;&#xx [1;#.8;9 5;a.1;ȁ ;ņ.;掉&#x 570;&#x.480; ]0;&#x/BBo;&#xx [1;#.8;9 5;a.1;ȁ ;ņ.;掉&#x 570;&#x.480; ]0;p-(x, t) &#x/BBo;&#xx [1;W.4;Ζ ;ՠ.;蠆&#x 187;&#x.679; 57;�.48; ];&#x/BBo;&#xx [1;W.4;Ζ ;ՠ.;蠆&#x 187;&#x.679; 57;�.48; ];-p+(x, t) = &#x/BBo;&#xx [2;.7;Ɖ ;ՠ.;蠆&#x 243;&#x.839; 57;�.96; ];&#x/BBo;&#xx [2;.7;Ɖ ;ՠ.;蠆&#x 243;&#x.839; 57;�.96; ];2pq5+(x, t) &#x/BBo;&#xx [2;c.7;և ;ՠ.;ᖙ&#x 277;&#x.438; 57;
.44; ];&#x/BBo;&#xx [2;c.7;և ;ՠ.;ᖙ&#x 277;&#x.438; 57;
.44; ];(1x1 &#x/BBo;&#xx [2;‚.9;֕ ;գ.;( 2;ˆ.7;Ɩ ;ը.;嘁&#x ]00;&#x/BBo;&#xx [2;‚.9;֕ ;գ.;( 2;ˆ.7;Ɩ ;ը.;嘁&#x ]00;1). &#x/BBo;&#xx [3;d.5;֓ ;ՠ.;蠆&#x 381;&#x.119;&#x 570;&#x.239; ]0;&#x/BBo;&#xx [3;d.5;֓ ;ՠ.;蠆&#x 381;&#x.119;&#x 570;&#x.239; ]0;(39) The lift L acting on &#x/BBo;&#xx [3;d.5;֓ ;ՠ.;蠆&#x 381;&#x.119;&#x 570;&#x.239; ]0;&#x/BBo;&#xx [3;d.5;֓ ;ՠ.;蠆&#x 381;&#x.119;&#x 570;&#x.239; ]0;plabe and the moment of force, M, about the &#x/BBo;&#xx [3;Y.9;উ ;Մ.;嘄&#x 381;&#x.119;&#x 552;&#x.239; ]0;&#x/BBo;&#xx [3;Y.9;উ ;Մ.;嘄&#x 381;&#x.119;&#x 552;&#x.239; ]0;mid- cord (positive in nose-up sense) readily obtained by straightforward can be deduced by integration same singularity &#x/BBo;&#xx [3;Y.9;উ ;Մ.;嘄&#x 381;&#x.119;&#x 552;&#x.239; ]0;&#x/BBo;&#xx [3;Y.9;উ ;Մ.;嘄&#x 381;&#x.119;&#x 552;&#x.239; ]0;F, namely This singularity of w is known in the aerodynamic theory to give rise &#x/BBo;&#xx [3;Y.9;উ ;Մ.;嘄&#x 381;&#x.119;&#x 552;&#x.239; ]0;&#x/BBo;&#xx [3;Y.9;উ ;Մ.;嘄&#x 381;&#x.119;&#x 552;&#x.239; ]0;leading- edge suction (directed upstream), in which &#x/BBo;&#xx [6;.87;’ 2;x.1;؆ ;r.7;Ɩ ;ʈ.;阂&#x ]00;&#x/BBo;&#xx [6;.87;’ 2;x.1;؆ ;r.7;Ɩ ;ʈ.;阂&#x ]00;a$ stands for the complex conjugate of &#x/BBo;&#xx [2;@.9;և ;ɷ.;’ 2;Q.9;উ ;ʅ.;6 ];&#x/BBo;&#xx [2;@.9;և ;ɷ.;’ 2;Q.9;উ ;ʅ.;6 ];a,. Finally, the thrust T, power P and energy loss E can be determined in &#x/BBo;&#xx [2;@.9;և ;ɷ.;’ 2;Q.9;উ ;ʅ.;6 ];&#x/BBo;&#xx [2;@.9;և ;ɷ.;’ 2;Q.9;উ ;ʅ.;6 ];h(x, t) by substituting &#x/BBo;&#xx [2;@.9;և ;ɷ.;’ 2;Q.9;উ ;ʅ.;6 ];&#x/BBo;&#xx [2;@.9;և ;ɷ.;’ 2;Q.9;উ ;ʅ.;6 ];(38), (39) and (43) in &#x/BBo;&#xx [2;&.3;ƕ ;ɓ.;ᦗ&#x 257;&#x.519; 26;.80; ];&#x/BBo;&#xx [2;&.3;ƕ ;ɓ.;ᦗ&#x 257;&#x.519; 26;.80; ];(3)-(5), with &#x/BBo;&#xx [2;„.1;֒ ;ɓ.;ᦗ&#x 292;&#x.559; 26;.27;— ];&#x/BBo;&#xx [2;„.1;֒ ;ɓ.;ᦗ&#x 292;&#x.559; 26;.27;— ];q5, h,, and &#x/BBo;&#xx [3;(.5;֓ ;ɒ.;阂&#x 337;&#x.918; 26;.80; ];&#x/BBo;&#xx [3;(.5;֓ ;ɒ.;阂&#x 337;&#x.918; 26;.80; ];hz in (3)-(5) all assuming their real pulation, is simply from ��(6), &#x/BBo;&#xx [2;E.9;ও ;ر.;醓&#x 253;&#x.199; 63; .35;“ ];&#x/BBo;&#xx [2;E.9;ও ;ر.;醓&#x 253;&#x.199; 63; .35;“ ];P = &#x/BBo;&#xx [2;g.5;ক ;ز.;ᖙ&#x 283;&#x.199;&#x 639;&#x.599; ]0;&#x/BBo;&#xx [2;g.5;ক ;ز.;ᖙ&#x 283;&#x.199;&#x 639;&#x.599; ]0;TU + E. The manipulation involved above result can considerably facilitated &#x/BBo;&#xx [2;g.5;ক ;ز.;ᖙ&#x 283;&#x.199;&#x 639;&#x.599; ]0;&#x/BBo;&#xx [2;g.5;ক ;ز.;ᖙ&#x 283;&#x.199;&#x 639;&#x.599; ]0;two arbitrary functions &#x/BBo;&#xx [2;g.5;ক ;ز.;ᖙ&#x 283;&#x.199;&#x 639;&#x.599; ]0;&#x/BBo;&#xx [2;g.5;ক ;ز.;ᖙ&#x 283;&#x.199;&#x 639;&#x.599; ]0;g(x) and their derivatives &#x/BBo;&#xx [2;g.5;ক ;ز.;ᖙ&#x 283;&#x.199;&#x 639;&#x.599; ]0;&#x/BBo;&#xx [2;g.5;ক ;ز.;ᖙ&#x 283;&#x.199;&#x 639;&#x.599; ]0;g'(x) are continuous in - 1 &#x/BBo;&#xx [1;p.6;Ζ ;յ.;➓&#x 176;&#x.639; 58;.24; ];&#x/BBo;&#xx [1;p.6;Ζ ;յ.;➓&#x 176;&#x.639; 58;.24; ];x &#x/BBo;&#xx [1;‰.8;Δ ;յ.;➓&#x 195;&#x.839;&#x 582;&#x.479; ]0;&#x/BBo;&#xx [1;‰.8;Δ ;յ.;➓&#x 195;&#x.839;&#x 582;&#x.479; ]0;6 1, &#x/BBo;&#xx [2;.9;֕ ;ն ;Ȩ.;閕&#x 583;&#x.679; ]0;&#x/BBo;&#xx [2;.9;֕ ;ն ;Ȩ.;閕&#x 583;&#x.679; ]0;the% This theorem can readily proved parts and contributions from Cauchy principal limits &#x/BBo;&#xx [2;.9;֕ ;ն ;Ȩ.;閕&#x 583;&#x.679; ]0;&#x/BBo;&#xx [2;.9;֕ ;ն ;Ȩ.;閕&#x 583;&#x.679; ]0;6 = x - &#x/BBo;&#xx [3;` 4;d.8;ޙ ;ͤ.;ޒ&#x 469;&#x.679; ]0;&#x/BBo;&#xx [3;` 4;d.8;ޙ ;ͤ.;ޒ&#x 469;&#x.679; ]0;E and &#x/BBo;&#xx [3;“.1;Ɣ ;Ѣ.;閕&#x 397;&#x.679; 47;.31;˜ ];&#x/BBo;&#xx [3;“.1;Ɣ ;Ѣ.;閕&#x 397;&#x.679; 47;.31;˜ ];6 = x &#x/BBo;&#xx [4;.7;ঙ ;ѣ.;醗&#x 425;&#x.039;
47;�.64;&#x ]00;&#x/BBo;&#xx [4;.7;ঙ ;ѣ.;醗&#x 425;&#x.039;
47;�.64;&#x ]00;+ &#x/BBo;&#xx [4;&.9;֕ ;Ѥ.;掓&#x 430;&#x.799; 46; .67;˜ ];&#x/BBo;&#xx [4;&.9;֕ ;Ѥ.;掓&#x 430;&#x.799; 46; .67;˜ ];s cancel out as &#x/BBo;&#xx [1;3.6;ޔ ;ђ.;d 1;H.3;Ƙ ;ї.;枔&#x ]00;&#x/BBo;&#xx [1;3.6;ޔ ;ђ.;d 1;H.3;Ƙ ;ї.;枔&#x ]00;s+ &#x/BBo;&#xx [1;P.4;ޗ ;ђ.;㦔&#x 157;&#x.199;&#x 459;&#x.599; ]0;&#x/BBo;&#xx [1;P.4;ޗ ;ђ.;㦔&#x 157;&#x.199;&#x 459;&#x.599; ]0;0. The integrals in (44) and (45) can into a &#x/BBo;&#xx [1;P.4;ޗ ;ђ.;㦔&#x 157;&#x.199;&#x 459;&#x.599; ]0;&#x/BBo;&#xx [1;P.4;ޗ ;ђ.;㦔&#x 157;&#x.199;&#x 459;&#x.599; ]0;series representation upon substituting respective Fourier coefficients given &#x/BBo;&#xx [1;P.4;ޗ ;ђ.;㦔&#x 157;&#x.199;&#x 459;&#x.599; ]0;&#x/BBo;&#xx [1;P.4;ޗ ;ђ.;㦔&#x 157;&#x.199;&#x 459;&#x.599; ]0;(46), into the integrand and carrying out the &#x/BBo;&#xx [1;P.4;ޗ ;ђ.;㦔&#x 157;&#x.199;&#x 459;&#x.599; ]0;&#x/BBo;&#xx [1;P.4;ޗ ;ђ.;㦔&#x 157;&#x.199;&#x 459;&#x.599; ]0;b,, &#x/BBo;&#xx [2;T.3;ঔ ;̕.;ᆘ&#x 264;&#x.959; 32;.19;— ];&#x/BBo;&#xx [2;T.3;ঔ ;̕.;ᆘ&#x 264;&#x.959; 32;.19;— ];,8, is implied. Other flow quantities of related interest are vorticity distribution trailing vortex vorticity in point (x, &#x/BBo;&#xx [2;T.3;ঔ ;̕.;ᆘ&#x 264;&#x.959; 32;.19;— ];&#x/BBo;&#xx [2;T.3;ঔ ;̕.;ᆘ&#x 264;&#x.959; 32;.19;— ];0) of the wake (x &#x/BBo;&#xx [4;
.9;খ ;ɹ.;莔&#x 407;&#x.759; 28;.11;” ];&#x/BBo;&#xx [4;
.9;খ ;ɹ.;莔&#x 407;&#x.759; 28;.11;” ];&#x/BBo;&#xx [4;
.9;খ ;ɹ.;莔&#x 407;&#x.759; 28;.11;” ]; 1) is &#x/BBo;&#xx [7; 26;.43;– 8; .27;“ 2;u.0;Δ ;&#x]000;&#x/BBo;&#xx [7; 26;.43;– 8; .27;“ 2;u.0;Δ ;&#x]000;y(x, &#x/BBo;&#xx [9;.87;• 2;e.6;ࠂ ;ē.;➉&#x 275;&#x.039; ]0;&#x/BBo;&#xx [9;.87;• 2;e.6;ࠂ ;ē.;➉&#x 275;&#x.039; ]0;t)dx, positive in counterclockwise sense, &#x/BBo;&#xx [9;.87;• 2;e.6;ࠂ ;ē.;➉&#x 275;&#x.039; ]0;&#x/BBo;&#xx [9;.87;• 2;e.6;ࠂ ;ē.;➉&#x 275;&#x.039; ]0;y(x, t) = - &#x/BBo;&#xx [2;X.2;Α ;Ʌ.;➗&#x 284;&#x.879; 25;.64;&#x ]00;&#x/BBo;&#xx [2;X.2;Α ;Ʌ.;➗&#x 284;&#x.879; 25;.64;&#x ]00;2u+(x, t), &#x/BBo;&#xx [4;.7;Ɩ ;Ʌ.;刃&#x 430;&#x.799; 25;.35;– ];&#x/BBo;&#xx [4;.7;Ɩ ;Ʌ.;刃&#x 430;&#x.799; 25;.35;– ];(49) which, by virtue &#x/BBo;&#xx [4;.7;Ɩ ;Ʌ.;刃&#x 430;&#x.799; 25;.35;– ];&#x/BBo;&#xx [4;.7;Ɩ ;Ʌ.;刃&#x 430;&#x.799; 25;.35;– ];(20), satisfies the equation It therefore follows that By Kelvin's circulation theorem, circulation (positive &#x/BBo;&#xx [4;.7;Ɩ ;Ʌ.;刃&#x 430;&#x.799; 25;.35;– ];&#x/BBo;&#xx [4;.7;Ɩ ;Ʌ.;刃&#x 430;&#x.799; 25;.35;– ];F(t), varies at the rate which gives, upon Hydromechanics of swimming propulsion. Part 1 349 Thus, both y(x,r) and r(r) are determined once u+(1,7) is found. This can be best by evaluating Laplace transform G+(l, s). From the real part of condition (20) ~+(l, s) = - s e+ exp (8x1 6+(x, s) dx, 11, where 6+(x, s) is the real part of P(x, s) given by (31 a) evaluated at y = 0 + , 1.1 1. Substituting this expression for 6+(x,s) in the above integral, using again (47), we obtain ~+(l, s) = &ns e-S [(a, + 9,) I, (s) - (6, - 6,) I, (s)], where In first kind. (32a, b) and the Wronskian I, (s) K, (s) +I, (s) KO (s) = l/s, the above expression reduces to G+( 1, s) = $7 e-s [6", (s) + 6, (s)]/[K, (s) + K, (s)]. (53) u+(l, r) particularly significant general solution is bow +b,(t) = 0, (54a) or equivalently, if V(x, Fourier expansion (35)), 00 V(x, t) = b, (t) (4 - cos 8) + C b, (t) cos nB (x = cos B), n=2 (54b) then, according to (53), u+(l, t) = 0, and hence the circulation I' remains constant (see (51)), and the plabe sheds no wake regardless b, (t), b, . . there are infinite number such modes unsteady motion trailing vortex Furthermore, by (34), also implies b, (t) = - b, (t). It therefore follows from (44), terms in expression for the total shed under condition (54), those values arise only from average values under condition (54), implying no energy loss, any net required over each last property was (1961), to play a particularly significant role h(x, swimming efficiency. an aquatic animal propels the total monient of force must balance the time rate of change of their corresponding momentum. Considering the typical case of a three-dimensional (or slender) secondary details such is reasonable ��power will come only from pure moment can be produced muscular contractions relaxations. This moment is applied bending moment ��is possible for an aquatic animal in reality be represented a linear open question, living soft tissues ��Fung (1967) to be strongly non-linear. We shall, however, assume to the bending moment linear elastic relation- elementary beam moments acting a longitudinal transverse movements. The bending moment ��M includes the active moment &#x/BBo;&#xx [1;5.5;ঈ ;ɥ.; &#x 147;&#x.838; 27;.12;
];&#x/BBo;&#xx [1;5.5;ঈ ;ɥ.; &#x 147;&#x.838; 27;.12;
];Ma due to &#x/BBo;&#xx [1;y.2;ޓ ;ɥ.; &#x 232;&#x.799; 27;.60; ];&#x/BBo;&#xx [1;y.2;ޓ ;ɥ.; &#x 232;&#x.799; 27;.60; ];asymmetrical muscular contractions and relaxations. Taking the free-body diagram a longitudinal section, &#x/BBo;&#xx [1;y.2;ޓ ;ɥ.; &#x 232;&#x.799; 27;.60; ];&#x/BBo;&#xx [1;y.2;ޓ ;ɥ.; &#x 232;&#x.799; 27;.60; ];length dx, of the body (see figure &#x/BBo;&#xx [1;y.2;ޓ ;ɥ.; &#x 232;&#x.799; 27;.60; ];&#x/BBo;&#xx [1;y.2;ޓ ;ɥ.; &#x 232;&#x.799; 27;.60; ];aqpx + 4 &#x/BBo;&#xx [2;f.3;ই ;ȑ.;䐇&#x 272;&#x.879; 21;.08; ];&#x/BBo;&#xx [2;f.3;ই ;ȑ.;䐇&#x 272;&#x.879; 21;.08; ];= o, &#x/BBo;&#xx [4;.2;Έ ;ȇ.;ለ&#x 430;&#x.558; 21;.20; ];&#x/BBo;&#xx [4;.2;Έ ;ȇ.;ለ&#x 430;&#x.558; 21;.20; ];(554 where &#x/BBo;&#xx [1;.1;֒ ;Ħ.;阂&#x 123;&#x.599; 13;.80; ];&#x/BBo;&#xx [1;.1;֒ ;Ħ.;阂&#x 123;&#x.599; 13;.80; ];q(x, t) is the longitudinal tension induced &#x/BBo;&#xx [1;.1;֒ ;Ħ.;阂&#x 123;&#x.599; 13;.80; ];&#x/BBo;&#xx [1;.1;֒ ;Ħ.;阂&#x 123;&#x.599; 13;.80; ];4, which represents the longitudinal component of hydrodynamic shear lift per arising from unit length, elastic shear cross-sectional plane, &#x/BBo;&#xx [1;.1;֒ ;Ħ.;阂&#x 123;&#x.599; 13;.80; ];&#x/BBo;&#xx [1;.1;֒ ;Ħ.;阂&#x 123;&#x.599; 13;.80; ];Ma represents the applied bending moment muscular contractions. Hydromechanics of swimming propulsion. Part 1 35 1 quantity (E,I) stands for the bending rigidity, E, being the effective young's inertia about bending axis. (55a-c) applied moment Ma that must be required for the prescribed body motion h, versa, with (e.g. Q = Ma = 0 and the ends). ~ualjtatively thrust and drag are about distributed along a self-propelling body, I$ everywhere small. Furthermore, Fn generally small compared L, large Reynolds if the cross-flow does not separate. Under these presumptions, we integrate (55a-c) slender or body, giving for [llm(x) h&, t) dx, (56a) M(t) = - xm(x) htt (x, t) dx. S', (56b) may be remarked here after integration, (56b) contains a term E,Ihzz vanish with bending moments body. Although (56a, b) were given by Lighthill (1960), the set of differential equations (55a-c) are still thought useful for biological studies of the activating couple Ma. We further note that, although (56a, observed for all they are always satisfied motions. However, h of (12), may exclude the possibility of this motion being realized without extra 'rigid-body ' yaw being present two-dimensional lunate tail large aspect ratio, or (56a, settled together with Harmonic time U = const. special case simple harmonic motion U = const., of a two-dimensional flexible plate which starts impulsively from h = 0 at t = 0. evaluated by Wu (1961) method (compared with function- be simpler). shall supplement previous work = 0 for t 0. U = const., hence T = Ut. Then (4 = 0 (& +jg) hl (x) ��= &#x/BBo;&#xx [1;.7;অ ;ز.;ᖙ&#x 123;&#x.358; 63; .11;˜ ];&#x/BBo;&#xx [1;.7;অ ;ز.;ᖙ&#x 123;&#x.358; 63; .11;˜ ];0 for t &#x/BBo;&#xx [1;G.8;Ά ;ز.;ᖙ&#x 153;&#x.598; 63;.43;™ ];&#x/BBo;&#xx [1;G.8;Ά ;ز.;ᖙ&#x 153;&#x.598; 63;.43;™ ];&#x/BBo;&#xx [1;X.1;֒ ;ر.;醓&#x 164;&#x.878; 63; .11;˜ ];&#x/BBo;&#xx [1;X.1;֒ ;ر.;醓&#x 164;&#x.878; 63; .11;˜ ];0. The asymptotic solution involves primarily for large small t, only history-dependent &#x/BBo;&#xx [1;X.1;֒ ;ر.;醓&#x 164;&#x.878; 63; .11;˜ ];&#x/BBo;&#xx [1;X.1;֒ ;ر.;醓&#x 164;&#x.878; 63; .11;˜ ];Laplace transform of &#x/BBo;&#xx [2;€.5;Y 6;.5;؁ ;ʘ.;㆕&#x 614;&#x.399; ]0;&#x/BBo;&#xx [2;€.5;Y 6;.5;؁ ;ʘ.;㆕&#x 614;&#x.399; ]0;V(x, t), &#x/BBo;&#xx [2;.6;ޔ ;ֆ.; 2;#.6;ޔ ;֗.;莗&#x ]00;&#x/BBo;&#xx [2;.6;ޔ ;ֆ.; 2;#.6;ޔ ;֗.;莗&#x ]00;qx, &#x/BBo;&#xx [2;&.0;މ ;ֆ.;㆕&#x 233;&#x.038; 59;.67;˜ ];&#x/BBo;&#xx [2;&.0;މ ;ֆ.;㆕&#x 233;&#x.038; 59;.67;˜ ];8) = &#x/BBo;&#xx [2;G.6;ޑ ;օ.;莔&#x 255;&#x.119;&#x 596;&#x.159; ]0;&#x/BBo;&#xx [2;G.6;ޑ ;օ.;莔&#x 255;&#x.119;&#x 596;&#x.159; ]0;q &#x/BBo;&#xx [2;W.0;Δ ;օ.;莔&#x 279;&#x.359; 59;.15;™ ];&#x/BBo;&#xx [2;W.0;Δ ;օ.;莔&#x 279;&#x.359; 59;.15;™ ];(x)/(s &#x/BBo;&#xx [2;.5;ƒ ;օ.;莔&#x 305;&#x.518; 59;.67;˜ ];&#x/BBo;&#xx [2;.5;ƒ ;օ.;莔&#x 305;&#x.518; 59;.67;˜ ];-j4, in &#x/BBo;&#xx [8; .51;ˆ 5;g.3;؄ ;ĩ.;ę ;ո.;㦔&#x ]00;&#x/BBo;&#xx [8; .51;ˆ 5;g.3;؄ ;ĩ.;ę ;ո.;㦔&#x ]00;(32),[and applying the inversion theorem &#x/BBo;&#xx [8; .51;ˆ 5;g.3;؄ ;ĩ.;ę ;ո.;㦔&#x ]00;&#x/BBo;&#xx [8; .51;ˆ 5;g.3;؄ ;ĩ.;ę ;ո.;㦔&#x ]00;6, (s), we obtain &#x/BBo;&#xx [2; .5;֓ ;Ց.;疘&#x 223;&#x.679; 55;.48;
];&#x/BBo;&#xx [2; .5;֓ ;Ց.;疘&#x 223;&#x.679; 55;.48;
];1 &#x/BBo;&#xx [2;7.5;঑ ;Փ.;醗&#x 254;&#x.158; 55; .19;— ];&#x/BBo;&#xx [2;7.5;঑ ;Փ.;醗&#x 254;&#x.158; 55; .19;— ];e+im &#x/BBo;&#xx [3;H.2;Α ;Ց.;刃&#x 357;&#x.599; 55;.96; ];&#x/BBo;&#xx [3;H.2;Α ;Ց.;刃&#x 357;&#x.599; 55;.96; ];ds &#x/BBo;&#xx [9;.59;‘ 5;B.6; 10;.23;ˆ 5;P.0; ]0;&#x/BBo;&#xx [9;.59;‘ 5;B.6; 10;.23;ˆ 5;P.0; ]0;a, &#x/BBo;&#xx [1;.3;ই ;Ղ.;螕&#x 112;&#x.798; 55;.47;” ];&#x/BBo;&#xx [1;.3;ই ;Ղ.;螕&#x 112;&#x.798; 55;.47;” ];(t) = &#x/BBo;&#xx [1;'.6;އ ;Ղ.;d 1;5.1;Ƈ ;Ւ.;䞔&#x ]00;&#x/BBo;&#xx [1;'.6;އ ;Ղ.;d 1;5.1;Ƈ ;Ւ.;䞔&#x ]00;b1 &#x/BBo;&#xx [1;7.2;ކ ;Ղ.;螕&#x 145;&#x.919; 55;.23;™ ];&#x/BBo;&#xx [1;7.2;ކ ;Ղ.;螕&#x 145;&#x.919; 55;.23;™ ];(t) &#x/BBo;&#xx [1;H.5;֓ ;Շ.;ᦓ&#x 154;&#x.798; 54;.15;• ];&#x/BBo;&#xx [1;H.5;֓ ;Շ.;ᦓ&#x 154;&#x.798; 54;.15;• ];- &#x/BBo;&#xx [1;W.6;ޑ ;Ղ.;㦔&#x 168;&#x.478; 55;.47;” ];&#x/BBo;&#xx [1;W.6;ޑ ;Ղ.;㦔&#x 168;&#x.478; 55;.47;” ];[b, &#x/BBo;&#xx [1;p.3;™ 5;B.6; 17; .03;‡ 5;R.2;Ι ;&#x]000;&#x/BBo;&#xx [1;p.3;™ 5;B.6; 17; .03;‡ 5;R.2;Ι ;&#x]000;(t) + &#x/BBo;&#xx [1;.3;ƕ ;Ղ.;d 1;—.9;উ ;Ւ.;䞔&#x ]00;&#x/BBo;&#xx [1;.3;ƕ ;Ղ.;d 1;—.9;উ ;Ւ.;䞔&#x ]00;b1 &#x/BBo;&#xx [2;�.1;ֈ ;Ղ.;d 2;.3;ই ;Ւ.;⎙&#x ]00;&#x/BBo;&#xx [2;�.1;ֈ ;Ղ.;d 2;.3;ই ;Ւ.;⎙&#x ]00;(t)] &#x/BBo;&#xx [2;.7;঒ ;Զ.;搄&#x 236;&#x.878; 55; .19;— ];&#x/BBo;&#xx [2;.7;঒ ;Զ.;搄&#x 236;&#x.878; 55; .19;— ];-1 &#x/BBo;&#xx [2;V.0;ޒ ;Ղ.;d 2;q.9;Ɠ ;Չ.;莔&#x ]00;&#x/BBo;&#xx [2;V.0;ޒ ;Ղ.;d 2;q.9;Ɠ ;Չ.;莔&#x ]00;exp &#x/BBo;&#xx [2;t.7;ঈ ;Ղ.;㦔&#x 289;&#x.439; 55;.23;™ ];&#x/BBo;&#xx [2;t.7;ঈ ;Ղ.;㦔&#x 289;&#x.439; 55;.23;™ ];[r(s - &#x/BBo;&#xx [2;™.0;Α ;Ղ.;㦔&#x 316;&#x.078; 55;.23;™ ];&#x/BBo;&#xx [2;™.0;Α ;Ղ.;㦔&#x 316;&#x.078; 55;.23;™ ];jcr)] &#x/BBo;&#xx [3;.9;֕ ;Ղ.;d 3;7.6;ޑ ;Ք.;㦘&#x ]00;&#x/BBo;&#xx [3;.9;֕ ;Ղ.;d 3;7.6;ޑ ;Ք.;㦘&#x ]00;a(s) &#x/BBo;&#xx [3;9.8;9 5;D.5;֓ ;ͣ.;莆&#x 548;&#x.159; ]0;&#x/BBo;&#xx [3;9.8;9 5;D.5;֓ ;ͣ.;莆&#x 548;&#x.159; ]0;- &#x/BBo;&#xx [3;u.3;։ ;Ղ.;d 3;‚.0;ޒ ;Ւ.;&#x ]00;&#x/BBo;&#xx [3;u.3;։ ;Ղ.;d 3;‚.0;ޒ ;Ւ.;&#x ]00;(6 &#x/BBo;&#xx [3;†.3;™ 5;D.7;ঙ ;Β.;ᖒ&#x 549;&#x.839; ]0;&#x/BBo;&#xx [3;†.3;™ 5;D.7;ঙ ;Β.;ᖒ&#x 549;&#x.839; ]0;&#x/BBo;&#xx [3;†.3;™ 5;D.7;ঙ ;Β.;ᖒ&#x 549;&#x.839; ]0; &#x/BBo;&#xx [3;–.4;y 5;B.6; 40;.03;‘ 5;R.2;Ι ;&#x]000;&#x/BBo;&#xx [3;–.4;y 5;B.6; 40;.03;‘ 5;R.2;Ι ;&#x]000;O), (59) &#x/BBo;&#xx [2;.0;· ;Ը.;㈂&#x 228;&#x.718; 54;.99;– ];&#x/BBo;&#xx [2;.0;· ;Ը.;㈂&#x 228;&#x.718; 54;.99;– ];27~i &#x/BBo;&#xx [3;9.8;9 5;8.5;֗ ;̓.;枇&#x 543;&#x.359; ]0;&#x/BBo;&#xx [3;9.8;9 5;8.5;֗ ;̓.;枇&#x 543;&#x.359; ]0;S &#x/BBo;&#xx [3;E.8;Ά ;Զ.;ᘃ&#x 364;&#x.318; 54;.35;– ];&#x/BBo;&#xx [3;E.8;Ά ;Զ.;ᘃ&#x 364;&#x.318; 54;.35;– ];-J6 where &#x/BBo;&#xx [1;.5;ƈ ;ԗ.;醗&#x 117;&#x.839; 52;.76;
];&#x/BBo;&#xx [1;.5;ƈ ;ԗ.;醗&#x 117;&#x.839; 52;.76;
];b,, &#x/BBo;&#xx [1;!.6;ޑ ;ԕ.;刃&#x 129;&#x.838; 52;.76;
];&#x/BBo;&#xx [1;!.6;ޑ ;ԕ.;刃&#x 129;&#x.838; 52;.76;
];b, are given by (35) (the time factor &#x/BBo;&#xx [1;!.6;ޑ ;ԕ.;刃&#x 129;&#x.838; 52;.76;
];&#x/BBo;&#xx [1;!.6;ޑ ;ԕ.;刃&#x 129;&#x.838; 52;.76;
];(jwt) of &#x/BBo;&#xx [3;A.0;· ;ԗ.;栂&#x 351;&#x.118; 52;.51;• ];&#x/BBo;&#xx [3;A.0;· ;ԗ.;栂&#x 351;&#x.118; 52;.51;• ];b,, &#x/BBo;&#xx [3;T.9;֕ ;ԗ.;䎖&#x 362;&#x.638; 52;.76;
];&#x/BBo;&#xx [3;T.9;֕ ;ԗ.;䎖&#x 362;&#x.638; 52;.76;
];b, being recovered here) and &#x/BBo;&#xx [1;".1;֒ ;ԅ.;䐃&#x 140;&#x.398; 51;.27;— ];&#x/BBo;&#xx [1;".1;֒ ;ԅ.;䐃&#x 140;&#x.398; 51;.27;— ];H(s) is given by &#x/BBo;&#xx [1;”.6;Γ ;ԅ.;䐃&#x 207;&#x.598; 51;.04; ];&#x/BBo;&#xx [1;”.6;Γ ;ԅ.;䐃&#x 207;&#x.598; 51;.04; ];(32 b). The imaginary &#x/BBo;&#xx [1;”.6;Γ ;ԅ.;䐃&#x 207;&#x.598; 51;.04; ];&#x/BBo;&#xx [1;”.6;Γ ;ԅ.;䐃&#x 207;&#x.598; 51;.04; ];j in the above integrand can clearly be taken to be the same as &#x/BBo;&#xx [2;5.4;Β ;ҕ.;ᆘ&#x 238;&#x.798; 50;.55;— ];&#x/BBo;&#xx [2;5.4;Β ;ҕ.;ᆘ&#x 238;&#x.798; 50;.55;— ];i = &#x/BBo;&#xx [2;R.9;֑ ;ґ.;首&#x 258;&#x.958; 50;.27;“ ];&#x/BBo;&#xx [2;R.9;֑ ;ґ.;首&#x 258;&#x.958; 50;.27;“ ];,/ - 1. The integrand has a &#x/BBo;&#xx [2;R.9;֑ ;ґ.;首&#x 258;&#x.958; 50;.27;“ ];&#x/BBo;&#xx [2;R.9;֑ ;ґ.;首&#x 258;&#x.958; 50;.27;“ ];s &#x/BBo;&#xx [8;.71;‰ 4;ƒ.8;Ё ;‘.9;Ɠ ;҆.;䠁&#x ]00;&#x/BBo;&#xx [8;.71;‰ 4;ƒ.8;Ё ;‘.9;Ɠ ;҆.;䠁&#x ]00;= &#x/BBo;&#xx [9;.51;„ 4;‚.6;Є ;ą.;㖉&#x 490;&#x.080; ]0;&#x/BBo;&#xx [9;.51;„ 4;‚.6;Є ;ą.;㖉&#x 490;&#x.080; ]0;ia and a logarithmic branch point at &#x/BBo;&#xx [2;h.0;ޅ ;҂.;搄&#x 271;&#x.919; 48;.44; ];&#x/BBo;&#xx [2;h.0;ޅ ;҂.;搄&#x 271;&#x.919; 48;.44; ];s = 0, and is regular elsewhere in the finites plane with a branch cut negative real s-axis. &#x/BBo;&#xx [2;h.0;ޅ ;҂.;搄&#x 271;&#x.919; 48;.44; ];&#x/BBo;&#xx [2;h.0;ޅ ;҂.;搄&#x 271;&#x.919; 48;.44; ];l?or large &#x/BBo;&#xx [1;3.9;Ɖ ;ї.;枔&#x 138;&#x.479; 46;.47;” ];&#x/BBo;&#xx [1;3.9;Ɖ ;ї.;枔&#x 138;&#x.479; 46;.47;” ];r (actually large &#x/BBo;&#xx [2;.0;ޅ ;ѕ.;Δ&#x 233;&#x.038; 46;.11;” ];&#x/BBo;&#xx [2;.0;ޅ ;ѕ.;Δ&#x 233;&#x.038; 46;.11;” ];Utll, &#x/BBo;&#xx [2;7.8;Ά ;ї.;枔&#x 240;&#x.478; 46;.11;” ];&#x/BBo;&#xx [2;7.8;Ά ;ї.;枔&#x 240;&#x.478; 46;.11;” ];I being the half-chord which is being taken to be unity here), the above path of integration can be into a &#x/BBo;&#xx [2;7.8;Ά ;ї.;枔&#x 240;&#x.478; 46;.11;” ];&#x/BBo;&#xx [2;7.8;Ά ;ї.;枔&#x 240;&#x.478; 46;.11;” ];s = &#x/BBo;&#xx [2;B.1;ք ;в.;䠁&#x 251;&#x.998; 43; .92;&#x ]00;&#x/BBo;&#xx [2;B.1;ք ;в.;䠁&#x 251;&#x.998; 43; .92;&#x ]00;icr and a contour &#x/BBo;&#xx [3;!.3;։ ;в.;䠁&#x 327;&#x.358; 43; .92;&#x ]00;&#x/BBo;&#xx [3;!.3;։ ;в.;䠁&#x 327;&#x.358; 43; .92;&#x ]00;I? circumventing counter- contour integral &#x/BBo;&#xx [3;!.3;։ ;в.;䠁&#x 327;&#x.358; 43; .92;&#x ]00;&#x/BBo;&#xx [3;!.3;։ ;в.;䠁&#x 327;&#x.358; 43; .92;&#x ]00;s = &#x/BBo;&#xx [1;2.2;Α ;Ї.;刃&#x 142;&#x.078; 41;.96; ];&#x/BBo;&#xx [1;2.2;Α ;Ї.;刃&#x 142;&#x.078; 41;.96; ];i6 is given immediately residue theorem, whereas &#x/BBo;&#xx [1;2.2;Α ;Ї.;刃&#x 142;&#x.078; 41;.96; ];&#x/BBo;&#xx [1;2.2;Α ;Ї.;刃&#x 142;&#x.078; 41;.96; ];I'- &#x/BBo;&#xx [7;.27;‰ 3;’.3;ঘ ;Ő.;⎑&#x 403;&#x.439; ]0;&#x/BBo;&#xx [7;.27;‰ 3;’.3;ঘ ;Ő.;⎑&#x 403;&#x.439; ]0;contourlintegral can be evaluated for large &#x/BBo;&#xx [7;.27;‰ 3;’.3;ঘ ;Ő.;⎑&#x 403;&#x.439; ]0;&#x/BBo;&#xx [7;.27;‰ 3;’.3;ঘ ;Ő.;⎑&#x 403;&#x.439; ]0;r, according to Watson's lemma, &#x/BBo;&#xx [7;.27;‰ 3;’.3;ঘ ;Ő.;⎑&#x 403;&#x.439; ]0;&#x/BBo;&#xx [7;.27;‰ 3;’.3;ঘ ;Ő.;⎑&#x 403;&#x.439; ]0;JsJ. The &#x/BBo;&#xx [3;.3;֓ ;΂.;㆘&#x 334;&#x.798; 39;�.23;™ ];&#x/BBo;&#xx [3;.3;֓ ;΂.;㆘&#x 334;&#x.798; 39;�.23;™ ];find result is &#x/BBo;&#xx [7;.75; 29;
.12;
9;.91;‰ 3;
.2;
;&#x]000;&#x/BBo;&#xx [7;.75; 29;
.12;
9;.91;‰ 3;
.2;
;&#x]000;@(cr) is the Theodorsen function, &#x/BBo;&#xx [2;7.1; 2;’.7;ঙ ;ɦ.;掓&#x 300;&#x.479; ]0;&#x/BBo;&#xx [2;7.1; 2;’.7;ঙ ;ɦ.;掓&#x 300;&#x.479; ]0;%and &#x/BBo;&#xx [2;q.1;আ ;ʓ.;Δ&#x 278;&#x.638; 30;�.47;” ];&#x/BBo;&#xx [2;q.1;আ ;ʓ.;Δ&#x 278;&#x.638; 30;�.47;” ];9 being its &#x/BBo;&#xx [3;%.4;Β ;ʒ.;禙&#x 342;&#x.479;&#x 300;&#x.479; ]0;&#x/BBo;&#xx [3;%.4;Β ;ʒ.;禙&#x 342;&#x.479;&#x 300;&#x.479; ]0;real and imaginary part respectively, and &#x/BBo;&#xx [1;Y.8;9 2;€.5;؁ ;ť.;莆&#x 285;&#x.599; ]0;&#x/BBo;&#xx [1;Y.8;9 2;€.5;؁ ;ť.;莆&#x 285;&#x.599; ]0;cr is the reduced frequency based The last (60) diminishes monotonically like &#x/BBo;&#xx [1;Y.8;9 2;€.5;؁ ;ť.;莆&#x 285;&#x.599; ]0;&#x/BBo;&#xx [1;Y.8;9 2;€.5;؁ ;ť.;莆&#x 285;&#x.599; ]0;t-2 (noting that the harmonic time &#x/BBo;&#xx [1;Y.8;9 2;€.5;؁ ;ť.;莆&#x 285;&#x.599; ]0;&#x/BBo;&#xx [1;Y.8;9 2;€.5;؁ ;ť.;莆&#x 285;&#x.599; ]0;-+ &#x/BBo;&#xx [1;‘.9;ও ;ɓ.;醓&#x 202;&#x.559; 26;�.88; ];&#x/BBo;&#xx [1;‘.9;ও ;ɓ.;醓&#x 202;&#x.559; 26;�.88; ];co, yielding the steady-state solution of &#x/BBo;&#xx [3;q.7;և ;ɒ.;阂&#x 382;&#x.559; 26;�.15;• ];&#x/BBo;&#xx [3;q.7;և ;ɒ.;阂&#x 382;&#x.559; 26;�.15;• ];a,, a, (t) = &#x/BBo;&#xx [2;5.1;আ ;ȴ.;䠁&#x 243;&#x.118; 24;.56;
];&#x/BBo;&#xx [2;5.1;আ ;ȴ.;䠁&#x 243;&#x.118; 24;.56;
];b, - &#x/BBo;&#xx [2;T.3;ঔ ;ȴ.;䠁&#x 264;&#x.959; 24;.56;
];&#x/BBo;&#xx [2;T.3;ঔ ;ȴ.;䠁&#x 264;&#x.959; 24;.56;
];(b, + &#x/BBo;&#xx [2;u.7;Y 2;4.2;Ε ;ʆ.;喆&#x 244;&#x.799; ]0;&#x/BBo;&#xx [2;u.7;Y 2;4.2;Ε ;ʆ.;喆&#x 244;&#x.799; ]0;b,) &#x/BBo;&#xx [2;‰.4;Β ;ȴ.;熖&#x 311;&#x.759;&#x 244;&#x.319; ]0;&#x/BBo;&#xx [2;‰.4;Β ;ȴ.;熖&#x 311;&#x.759;&#x 244;&#x.319; ]0;@(cr). &#x/BBo;&#xx [4;.2;މ ;ȴ.;阂&#x 435;&#x.119;&#x 245;&#x.040; ]0;&#x/BBo;&#xx [4;.2;މ ;ȴ.;阂&#x 435;&#x.119;&#x 245;&#x.040; ]0;(62) For small &#x/BBo;&#xx [1;7.2;ކ ;Ȗ.;阂&#x 144;&#x.239; 22;.43;™ ];&#x/BBo;&#xx [1;7.2;ކ ;Ȗ.;阂&#x 144;&#x.239; 22;.43;™ ];r, the asymptotic value of the integral in (59) &#x/BBo;&#xx [1;7.2;ކ ;Ȗ.;阂&#x 144;&#x.239; 22;.43;™ ];&#x/BBo;&#xx [1;7.2;ކ ;Ȗ.;阂&#x 144;&#x.239; 22;.43;™ ];Is[, giving This result starts, the &#x/BBo;&#xx [1;7.2;ކ ;Ȗ.;阂&#x 144;&#x.239; 22;.43;™ ];&#x/BBo;&#xx [1;7.2;ކ ;Ȗ.;阂&#x 144;&#x.239; 22;.43;™ ];(b, + b,) is &#x/BBo;&#xx [1;f.0;ޒ ;Ł.;6 1;s.0;Α ;ő.;ᦓ&#x ]00;&#x/BBo;&#xx [1;f.0;ޒ ;Ł.;6 1;s.0;Α ;ő.;ᦓ&#x ]00;9, which changes over to &#x/BBo;&#xx [2;€.3;ƕ ;Ł.;6 2;™.9;ও ;Ő.;閘&#x ]00;&#x/BBo;&#xx [2;€.3;ƕ ;Ł.;6 2;™.9;ও ;Ő.;閘&#x ]00;@(a) as t &#x/BBo;&#xx [3; .3;ই ;Ń.;写&#x 341;&#x.518; 14;.79;™ ];&#x/BBo;&#xx [3; .3;ই ;Ń.;写&#x 341;&#x.518; 14;.79;™ ];+a. This feature is Wagner function for expressions for &#x/BBo;&#xx [3; .3;ই ;Ń.;写&#x 341;&#x.518; 14;.79;™ ];&#x/BBo;&#xx [3; .3;ই ;Ń.;写&#x 341;&#x.518; 14;.79;™ ];(GO), &#x/BBo;&#xx [2;P.0;ޅ ;Ė.;䀁&#x 268;&#x.799; 12; ]0;&#x/BBo;&#xx [2;P.0;ޅ ;Ė.;䀁&#x 268;&#x.799; 12; ]0;(63), can be directly small values is obvious &#x/BBo;&#xx [2;P.0;ޅ ;Ė.;䀁&#x 268;&#x.799; 12; ]0;&#x/BBo;&#xx [2;P.0;ޅ ;Ė.;䀁&#x 268;&#x.799; 12; ]0;T, E and &#x/BBo;&#xx [4;'.9;Ɔ ;ć.;➓&#x 434;&#x.878; 11;.19;“ ];&#x/BBo;&#xx [4;'.9;Ɔ ;ć.;➓&#x 434;&#x.878; 11;.19;“ ];P differ from their respective steady-state value by a &#x/BBo;&#xx [4;'.9;Ɔ ;ć.;➓&#x 434;&#x.878; 11;.19;“ ];&#x/BBo;&#xx [4;'.9;Ɔ ;ć.;➓&#x 434;&#x.878; 11;.19;“ ];0(r-2), which becomes negligible for &#x/BBo;&#xx [1;7.2;ކ ;.3;؄ ;ł.;ޅ&#x 86.;ᘃ&#x ]00;&#x/BBo;&#xx [1;7.2;ކ ;.3;؄ ;ł.;ޅ&#x 86.;ᘃ&#x ]00;r &#x/BBo;&#xx [1;F.1;ֈ ;.3;؄ ;Œ.;ᖄ&#x 86.;搄&#x ]00;&#x/BBo;&#xx [1;F.1;ֈ ;.3;؄ ;Œ.;ᖄ&#x 86.;搄&#x ]00;&#x/BBo;&#xx [1;F.1;ֈ ;.3;؄ ;Œ.;ᖄ&#x 86.;搄&#x ]00; 10, or after the body travelled chord lengths, motion falls ��T, E and P follow immediately from ��(44), (45) by applying the averaging formula (11). ��F &#x/BBo;&#xx [2;.95;‡ 5;™.2; 31;&#x.438; 60;.16; ];&#x/BBo;&#xx [2;.95;‡ 5;™.2; 31;&#x.438; 60;.16; ];= &#x/BBo;&#xx [3;.51;ˆ 5;–.4; ;P.8;ވ ;؆.;⎙&#x ]00;&#x/BBo;&#xx [3;.51;ˆ 5;–.4; ;P.8;ވ ;؆.;⎙&#x ]00;frp Re [(a, + &#x/BBo;&#xx [9;.59;‘ 5;–.8;ࠆ ;ā.;➆&#x 606;&#x.960; ]0;&#x/BBo;&#xx [9;.59;‘ 5;–.8;ࠆ ;ā.;➆&#x 606;&#x.960; ]0;b, &#x/BBo;&#xx [1;.4;΅ ;֖.;蠆&#x 124;&#x.078; 60;.39;˜ ];&#x/BBo;&#xx [1;.4;΅ ;֖.;蠆&#x 124;&#x.078; 60;.39;˜ ];-A) &#x/BBo;&#xx [1;&.7;ƅ ;֗.;ሁ&#x 139;&#x.199;&#x 607;&#x.920; ]0;&#x/BBo;&#xx [1;&.7;ƅ ;֗.;ሁ&#x 139;&#x.199;&#x 607;&#x.920; ]0;(a: - &#x/BBo;&#xx [1;P.2;Α ;֗.;ሁ&#x 160;&#x.078; 60;.68; ];&#x/BBo;&#xx [1;P.2;Α ;֗.;ሁ&#x 160;&#x.078; 60;.68; ];b: &#x/BBo;&#xx [1;b.4;y 5;–.8;ࠆ ;ƅ.;冄&#x 608;&#x.160; ]0;&#x/BBo;&#xx [1;b.4;y 5;–.8;ࠆ ;ƅ.;冄&#x 608;&#x.160; ]0;+I?) &#x/BBo;&#xx [1;ˆ.1;ք ;֖.;蠆&#x 221;&#x.518; 60;.39;˜ ];&#x/BBo;&#xx [1;ˆ.1;ք ;֖.;蠆&#x 221;&#x.518; 60;.39;˜ ];+,8,,8:] = &#x/BBo;&#xx [3;.75;” 5;.0;Ѕ ;Q.3;։ ;֐.;螙&#x ]00;&#x/BBo;&#xx [3;.75;” 5;.0;Ѕ ;Q.3;։ ;֐.;螙&#x ]00;in-p &#x/BBo;&#xx [5;.51;ˆ 5;€.5;؄ ;X.7;ঈ ;֑.;萁&#x ]00;&#x/BBo;&#xx [5;.51;ˆ 5;€.5;؄ ;X.7;ঈ ;֑.;萁&#x ]00;{I &#x/BBo;&#xx [6;�.47;† 5;.5;Ȇ ;h.3;ই ;֑.;ህ&#x ]00;&#x/BBo;&#xx [6;�.47;† 5;.5;Ȇ ;h.3;ই ;֑.;ህ&#x ]00;b, &#x/BBo;&#xx [7;�.55;† 5;ƒ.2; ;w.0;Δ ;։.;栂&#x ]00;&#x/BBo;&#xx [7;�.55;† 5;ƒ.2; ;w.0;Δ ;։.;栂&#x ]00;+ &#x/BBo;&#xx [7; .19;“ 5;.0;Ѕ ;‰.0;· ;֒.;ࠇ&#x ]00;&#x/BBo;&#xx [7; .19;“ 5;.0;Ѕ ;‰.0;· ;֒.;ࠇ&#x ]00;b,l &#x/BBo;&#xx [9;.71;’ 5;.7;؁ ;ē.;➉&#x 591;&#x.36 ;&#x]000;&#x/BBo;&#xx [9;.71;’ 5;.7;؁ ;ē.;➉&#x 591;&#x.36 ;&#x]000;(P2 + &#x/BBo;&#xx [1;$.5;ֆ ;փ.;’ 1;5.1;Ƈ ;֑.;怆&#x ]00;&#x/BBo;&#xx [1;$.5;ֆ ;փ.;’ 1;5.1;Ƈ ;֑.;怆&#x ]00;g2 - &#x/BBo;&#xx [1;E.6;އ ;ց.;瘁&#x 159;&#x.119; 59;
.36;&#x ]00;&#x/BBo;&#xx [1;E.6;އ ;ց.;瘁&#x 159;&#x.119; 59;
.36;&#x ]00;9) - Re &#x/BBo;&#xx [1;„.7;ঈ ;ց.;( 1;‘.5;ƒ ;֑.;6 ];&#x/BBo;&#xx [1;„.7;ঈ ;ց.;( 1;‘.5;ƒ ;֑.;6 ];(j &#x/BBo;&#xx [1;“.1;™ 5;.2; 21;.63;‰ 5;‘.8;Ё ;&#x]000;&#x/BBo;&#xx [1;“.1;™ 5;.2; 21;.63;‰ 5;‘.8;Ё ;&#x]000;Ua(b, + &#x/BBo;&#xx [2;).4;΅ ;ց.;( 2;@.2;Α ;֑.;6 ];&#x/BBo;&#xx [2;).4;΅ ;ց.;( 2;@.2;Α ;֑.;6 ];b,) &#x/BBo;&#xx [2;B.6;΅ ;ց.;( 2;`.1;ք ;֒.;㈂&#x ]00;&#x/BBo;&#xx [2;B.6;΅ ;ց.;( 2;`.1;ք ;֒.;㈂&#x ]00;[(P: - &#x/BBo;&#xx [2;p.9;֑ ;ց.;分&#x 285;&#x.358; 59;.32; ];&#x/BBo;&#xx [2;p.9;֑ ;ց.;分&#x 285;&#x.358; 59;.32; ];P:) &#x/BBo;&#xx [2;‡.9;উ ;փ.;栅&#x 294;&#x.478; 59;
.36;&#x ]00;&#x/BBo;&#xx [2;‡.9;উ ;փ.;栅&#x 294;&#x.478; 59;
.36;&#x ]00;O (a) + &#x/BBo;&#xx [3;.7;Ɖ ;ց.;( 3;C.1;আ ;֒.;喗&#x ]00;&#x/BBo;&#xx [3;.7;Ɖ ;ց.;( 3;C.1;আ ;֒.;喗&#x ]00;P:])), (64) E = &#x/BBo;&#xx [6;.75;‡ 5;c.0;Ѕ ;.3;֓ ;ղ.;螙&#x ]00;&#x/BBo;&#xx [6;.75;‡ 5;c.0;Ѕ ;.3;֓ ;ղ.;螙&#x ]00;&rp &#x/BBo;&#xx [8;.03;‘ 5;e.6;ࠅ ;.2;Ε ;ճ.;ህ&#x ]00;&#x/BBo;&#xx [8;.03;‘ 5;e.6;ࠅ ;.2;Ε ;ճ.;ህ&#x ]00;U Re &#x/BBo;&#xx [1;.2;ޓ ;բ.;禙&#x 122;&#x.159; 57;.64; ];&#x/BBo;&#xx [1;.2;ޓ ;բ.;禙&#x 122;&#x.159; 57;.64; ];[(a, + &#x/BBo;&#xx [1;2.9;և ;գ.;Ѕ&#x 143;&#x.759; 57;.12; ];&#x/BBo;&#xx [1;2.9;և ;գ.;Ѕ&#x 143;&#x.759; 57;.12; ];bO) &#x/BBo;&#xx [1;F.3;ঔ ;գ.;( 1;X.6;Γ ;ճ.;怆&#x ]00;&#x/BBo;&#xx [1;F.3;ঔ ;գ.;( 1;X.6;Γ ;ճ.;怆&#x ]00;(b: - &#x/BBo;&#xx [1;p.3;™ 5;c.0;Ѕ ;Ɔ.;閇&#x 573;&#x.600; ]0;&#x/BBo;&#xx [1;p.3;™ 5;c.0;Ѕ ;Ɔ.;閇&#x 573;&#x.600; ]0;a$)] = &#x/BBo;&#xx [2;
.8;Ά ;գ.;Ѕ&#x 217;&#x.439; 57;.87;™ ];&#x/BBo;&#xx [2;
.8;Ά ;գ.;Ѕ&#x 217;&#x.439; 57;.87;™ ];$n-p &#x/BBo;&#xx [2;.1; 5;e.2; ;Ȧ.;㆕&#x 572;&#x.879; ]0;&#x/BBo;&#xx [2;.1; 5;e.2; ;Ȧ.;㆕&#x 572;&#x.879; ]0;U &#x/BBo;&#xx [2;'.7;և ;բ.;㆘&#x 228;&#x.479; 57;.36;&#x ]00;&#x/BBo;&#xx [2;'.7;և ;բ.;㆘&#x 228;&#x.479; 57;.36;&#x ]00;I &#x/BBo;&#xx [2;0.3;ই ;բ.;禙&#x 238;&#x.079; 57;.87;™ ];&#x/BBo;&#xx [2;0.3;ই ;բ.;禙&#x 238;&#x.079; 57;.87;™ ];bO + &#x/BBo;&#xx [2;H.8;ވ ;գ.;( 2;V.5;֓ ;ղ.;螙&#x ]00;&#x/BBo;&#xx [2;H.8;ވ ;գ.;( 2;V.5;֓ ;ղ.;螙&#x ]00;bl &#x/BBo;&#xx [2;X.2;Α ;բ.;嘄&#x 258;&#x.958; 57;.60; ];&#x/BBo;&#xx [2;X.2;Α ;բ.;嘄&#x 258;&#x.958; 57;.60; ];1 &#x/BBo;&#xx [2;f.3;ই ;գ.;( 2;†.3;Ƒ ;ճ.;ህ&#x ]00;&#x/BBo;&#xx [2;f.3;ই ;գ.;( 2;†.3;Ƒ ;ճ.;ህ&#x ]00;[S- &#x/BBo;&#xx [2;‰.1;আ ;գ.;( 3;.7;֔ ;ղ.;螙&#x ]00;&#x/BBo;&#xx [2;‰.1;আ ;գ.;( 3;.7;֔ ;ղ.;螙&#x ]00;(P2 &#x/BBo;&#xx [3;.9;Ɠ ;դ.;爃&#x 314;&#x.158; 57;
.20;
];&#x/BBo;&#xx [3;.9;Ɠ ;դ.;爃&#x 314;&#x.158; 57;
.20;
];+ &#x/BBo;&#xx [3;.0;Α ;գ.;分&#x 337;&#x.199;&#x 573;&#x.36 ;&#x]000;&#x/BBo;&#xx [3;.0;Α ;գ.;分&#x 337;&#x.199;&#x 573;&#x.36 ;&#x]000;g2)], (65) = &#x/BBo;&#xx [7;.71;’ 5;G.2; ;“.8;Ά ;Ֆ.;ࠇ&#x ]00;&#x/BBo;&#xx [7;.71;’ 5;G.2; ;“.8;Ά ;Ֆ.;ࠇ&#x ]00;UF + &#x/BBo;&#xx [1;.6;Γ ;Ն.;閘&#x 112;&#x.079; 55;.84;
];&#x/BBo;&#xx [1;.6;Γ ;Ն.;閘&#x 112;&#x.079; 55;.84;
];E = &#x/BBo;&#xx [1;&.9;֑ ;Մ.;嘄&#x 154;&#x.798; 55;.64; ];&#x/BBo;&#xx [1;&.9;֑ ;Մ.;嘄&#x 154;&#x.798; 55;.64; ];&rpU2 Re { - &#x/BBo;&#xx [1;ƒ.3;օ ;Մ.;㆘&#x 206;&#x.158; 55;.39;˜ ];&#x/BBo;&#xx [1;ƒ.3;օ ;Մ.;㆘&#x 206;&#x.158; 55;.39;˜ ];ja(b, + &#x/BBo;&#xx [2;.9;֑ ;Մ.;嘄&#x 227;&#x.758; 55;.39;˜ ];&#x/BBo;&#xx [2;.9;֑ ;Մ.;嘄&#x 227;&#x.758; 55;.39;˜ ];b,) &#x/BBo;&#xx [2;0.1;֒ ;Մ.;嘄&#x 247;&#x.679;
55;.36;&#x ]00;&#x/BBo;&#xx [2;0.1;֒ ;Մ.;嘄&#x 247;&#x.679;
55;.36;&#x ]00;[(P: - &#x/BBo;&#xx [2;X.4;ކ ;Մ.;嘄&#x 273;&#x.119;&#x 555;&#x.120; ]0;&#x/BBo;&#xx [2;X.4;ކ ;Մ.;嘄&#x 273;&#x.119;&#x 555;&#x.120; ]0;P:) &#x/BBo;&#xx [2;u.7;Y 5;D.7;ঙ ;ʕ.;枔&#x 554;&#x.399; ]0;&#x/BBo;&#xx [2;u.7;Y 5;D.7;ঙ ;ʕ.;枔&#x 554;&#x.399; ]0;@(a) &#x/BBo;&#xx [2;˜.3;ƕ ;Մ.;㆘&#x 327;&#x.358; 55;.36;&#x ]00;&#x/BBo;&#xx [2;˜.3;ƕ ;Մ.;㆘&#x 327;&#x.358; 55;.36;&#x ]00;+P:]), (66) &#x/BBo;&#xx [1;.43;… 5;&.5;؄ ;&.8;ޒ ;Դ.;䠅&#x ]00;&#x/BBo;&#xx [1;.43;… 5;&.5;؄ ;&.8;ޒ ;Դ.;䠅&#x ]00;the final expressions being obtained upon &#x/BBo;&#xx [1;.43;… 5;&.5;؄ ;&.8;ޒ ;Դ.;䠅&#x ]00;&#x/BBo;&#xx [1;.43;… 5;&.5;؄ ;&.8;ޒ ;Դ.;䠅&#x ]00;(61), (62). This solution agrees &#x/BBo;&#xx [1;.43;… 5;&.5;؄ ;&.8;ޒ ;Դ.;䠅&#x ]00;&#x/BBo;&#xx [1;.43;… 5;&.5;؄ ;&.8;ޒ ;Դ.;䠅&#x ]00;with the previous result (1961, in &#x/BBo;&#xx [1;.43;… 5;&.5;؄ ;&.8;ޒ ;Դ.;䠅&#x ]00;&#x/BBo;&#xx [1;.43;… 5;&.5;؄ ;&.8;ޒ ;Դ.;䠅&#x ]00;bn were written as &#x/BBo;&#xx [3;.0;Δ ;ԓ.;萅&#x 335;&#x.758; 52;.67;˜ ];&#x/BBo;&#xx [3;.0;Δ ;ԓ.;萅&#x 335;&#x.758; 52;.67;˜ ];-A,). To this &#x/BBo;&#xx [1;.43;… 5;
.6; ;(.7;অ ;Ԑ.;&#x ]00;&#x/BBo;&#xx [1;.43;… 5;
.6; ;(.7;অ ;Ԑ.;&#x ]00;end we observe that the above has the property E &#x/BBo;&#xx [2;S.6;އ ;Ԃ.;喗&#x 259;&#x.438; 50; .28;&#x ]00;&#x/BBo;&#xx [2;S.6;އ ;Ԃ.;喗&#x 259;&#x.438; 50; .28;&#x ]00;c &#x/BBo;&#xx [2;c.9;ও ;ԃ.;⠄&#x 270;&#x.959;
51;�.24; ];&#x/BBo;&#xx [2;c.9;ও ;ԃ.;⠄&#x 270;&#x.959;
51;�.24; ];0. In fact, Theodorsen's function &#x/BBo;&#xx [5;.19;† 4;ˆ.6; 75;&#x.119;&#x 498;&#x.000; ]0;&#x/BBo;&#xx [5;.19;† 4;ˆ.6; 75;&#x.119;&#x 498;&#x.000; ]0;@(a) = &#x/BBo;&#xx [8; .51;ˆ 4;.0;ࠃ ;Ć.;禒&#x 498;&#x.480; ]0;&#x/BBo;&#xx [8; .51;ˆ 4;.0;ࠃ ;Ć.;禒&#x 498;&#x.480; ]0;F+ &#x/BBo;&#xx [1;.9;֑ ;Ґ.;禙&#x 119;&#x.999; 49;.23;™ ];&#x/BBo;&#xx [1;.9;֑ ;Ґ.;禙&#x 119;&#x.999; 49;.23;™ ];iY possesses the property &#x/BBo;&#xx [2;3.9;উ ;Ґ.;嘄&#x 244;&#x.319; 49;.23;™ ];&#x/BBo;&#xx [2;3.9;উ ;Ґ.;嘄&#x 244;&#x.319; 49;.23;™ ];F &#x/BBo;&#xx [2;F.7;Ɖ ;҉.;怂&#x 252;&#x.718; 49;.80; ];&#x/BBo;&#xx [2;F.7;Ɖ ;҉.;怂&#x 252;&#x.718; 49;.80; ];&#x/BBo;&#xx [2;F.7;Ɖ ;҉.;怂&#x 252;&#x.718; 49;.80; ]; (F2 &#x/BBo;&#xx [2;v.2;Α ;Ґ.;ࠃ&#x 282;&#x.718; 49;.56;
];&#x/BBo;&#xx [2;v.2;Α ;Ґ.;ࠃ&#x 282;&#x.718; 49;.56;
];+ &#x/BBo;&#xx [2;….3;։ ;҈.;d 2;™.2;ކ ;Ҙ.;⎙&#x ]00;&#x/BBo;&#xx [2;….3;։ ;҈.;d 2;™.2;ކ ;Ҙ.;⎙&#x ]00;Y2) for a &#x/BBo;&#xx [3;2.8;ޒ ;Ґ.;ࠃ&#x 338;&#x.878; 49;.04; ];&#x/BBo;&#xx [3;2.8;ޒ ;Ґ.;ࠃ&#x 338;&#x.878; 49;.04; ];3 0, the equality holding only when a = &#x/BBo;&#xx [1;i.6;ޔ ;Ѹ.; 1;v.6;Γ ;҅.;Θ&#x ]00;&#x/BBo;&#xx [1;i.6;ޔ ;Ѹ.; 1;v.6;Γ ;҅.;Θ&#x ]00;0. Therefore &#x/BBo;&#xx [2;4.4;y 4;x.3;Ȇ ;Ɂ.;醉&#x 486;&#x.960; ]0;&#x/BBo;&#xx [2;4.4;y 4;x.3;Ȇ ;Ɂ.;醉&#x 486;&#x.960; ]0;B &#x/BBo;&#xx [2;E.9;ও ;ѷ.;ᆘ&#x 251;&#x.759; 48;.08; ];&#x/BBo;&#xx [2;E.9;ও ;ѷ.;ᆘ&#x 251;&#x.759; 48;.08; ];&#x/BBo;&#xx [2;E.9;ও ;ѷ.;ᆘ&#x 251;&#x.759; 48;.08; ]; &#x/BBo;&#xx [2;V.3;Ƈ ;Ѹ.;㈆&#x 261;&#x.118; 48;.28; ];&#x/BBo;&#xx [2;V.3;Ƈ ;Ѹ.;㈆&#x 261;&#x.118; 48;.28; ];0 in general, &#x/BBo;&#xx [2;V.3;Ƈ ;Ѹ.;㈆&#x 261;&#x.118; 48;.28; ];&#x/BBo;&#xx [2;V.3;Ƈ ;Ѹ.;㈆&#x 261;&#x.118; 48;.28; ];B &#x/BBo;&#xx [3;U.9;Ɔ ;ѹ.;刃&#x 362;&#x.638; 48;.16; ];&#x/BBo;&#xx [3;U.9;Ɔ ;ѹ.;刃&#x 362;&#x.638; 48;.16; ];= &#x/BBo;&#xx [3;f.9;և ;Ѹ.;㈆&#x 371;&#x.758; 48;.28; ];&#x/BBo;&#xx [3;f.9;և ;Ѹ.;㈆&#x 371;&#x.758; 48;.28; ];0 holds either when a= &#x/BBo;&#xx [1;.3;֓ ;Ѥ.;ᘃ&#x 124;&#x.558; 47;.56; ];&#x/BBo;&#xx [1;.3;֓ ;Ѥ.;ᘃ&#x 124;&#x.558; 47;.56; ];0, a trivial &#x/BBo;&#xx [1;.3;֓ ;Ѥ.;ᘃ&#x 124;&#x.558; 47;.56; ];&#x/BBo;&#xx [1;.3;֓ ;Ѥ.;ᘃ&#x 124;&#x.558; 47;.56; ];b, &#x/BBo;&#xx [3;$.2;Ε ;ѥ.;6 3;0.7;ƒ ;ѱ.;莗&#x ]00;&#x/BBo;&#xx [3;$.2;Ε ;ѥ.;6 3;0.7;ƒ ;ѱ.;莗&#x ]00;+ &#x/BBo;&#xx [3;2.8;ޒ ;ѣ.;䐇&#x 340;&#x.558; 47;.52; ];&#x/BBo;&#xx [3;2.8;ޒ ;ѣ.;䐇&#x 340;&#x.558; 47;.52; ];b, = &#x/BBo;&#xx [3;U.1;™ 4;d.1;؃ ;͢.;掉&#x 472;&#x.799; ]0;&#x/BBo;&#xx [3;U.1;™ 4;d.1;؃ ;͢.;掉&#x 472;&#x.799; ]0;0, a special case acceleration; small a typical &#x/BBo;&#xx [3;U.1;™ 4;d.1;؃ ;͢.;掉&#x 472;&#x.799; ]0;&#x/BBo;&#xx [3;U.1;™ 4;d.1;؃ ;͢.;掉&#x 472;&#x.799; ]0;U(t) we consider the case in which a flexible plate moves forward with a constant acceleration from represented by form in x, &#x/BBo;&#xx [3;U.1;™ 4;d.1;؃ ;͢.;掉&#x 472;&#x.799; ]0;&#x/BBo;&#xx [3;U.1;™ 4;d.1;؃ ;͢.;掉&#x 472;&#x.799; ]0;h(x, t) = &#x/BBo;&#xx [9;.87;’ 3;&.1;֙ ;Ĵ.;Ι ;̶.;⎙&#x ]00;&#x/BBo;&#xx [9;.87;’ 3;&.1;֙ ;Ĵ.;Ι ;̶.;⎙&#x ]00;+p,(t)+ &#x/BBo;&#xx [1;R.6;΅ ;̦.;䀅&#x 173;&#x.758; 33;.23;™ ];&#x/BBo;&#xx [1;R.6;΅ ;̦.;䀅&#x 173;&#x.758; 33;.23;™ ];p,(t) &#x/BBo;&#xx [1;w.5;ক ;̨.;嘄&#x 191;&#x.039;
33;.84; ];&#x/BBo;&#xx [1;w.5;ক ;̨.;嘄&#x 191;&#x.039;
33;.84; ];cos &#x/BBo;&#xx [1;“.9;Ɔ ;̨.;禙&#x 204;&#x.958; 33;.48; ];&#x/BBo;&#xx [1;“.9;Ɔ ;̨.;禙&#x 204;&#x.958; 33;.48; ];n0 (X = &#x/BBo;&#xx [2;9.0;Δ ;̨.;䐇&#x 252;&#x.479;&#x 333;&#x.480;
]0;&#x/BBo;&#xx [2;9.0;Δ ;̨.;䐇&#x 252;&#x.479;&#x 333;&#x.480;
]0;cos 0, &#x/BBo;&#xx [2;s.5;঑ ;̨.;鈈&#x 278;&#x.399;&#x 335;&#x.880; ]0;&#x/BBo;&#xx [2;s.5;঑ ;̨.;鈈&#x 278;&#x.399;&#x 335;&#x.880; ]0;0 &#x/BBo;&#xx [2;‚.4;ޔ ;̨.;栂&#x 288;&#x.239; 33;.96; ];&#x/BBo;&#xx [2;‚.4;ޔ ;̨.;栂&#x 288;&#x.239; 33;.96; ];&#x/BBo;&#xx [2;’.5;֓ ;̨.;栂&#x 297;&#x.598; 33;.36; ];&#x/BBo;&#xx [2;’.5;֓ ;̨.;栂&#x 297;&#x.598; 33;.36; ];0 &#x/BBo;&#xx [3;
.6;ޑ ;̨.;䐇&#x 307;&#x.678; 33;.72; ];&#x/BBo;&#xx [3;
.6;ޑ ;̨.;䐇&#x 307;&#x.678; 33;.72; ];&#x/BBo;&#xx [3;.5;Ƅ ;̦.;刃&#x 323;&#x.038; 33;.12;
];&#x/BBo;&#xx [3;.5;Ƅ ;̦.;刃&#x 323;&#x.038; 33;.12;
];7~). &#x/BBo;&#xx [1;6.0;މ ;̠.;搄&#x 146;&#x.399; 32;.24; ];&#x/BBo;&#xx [1;6.0;މ ;̠.;搄&#x 146;&#x.399; 32;.24; ];n= &#x/BBo;&#xx [1;H.0;ޒ ;̠.;搄&#x 150;&#x.719; 32;.67;˜ ];&#x/BBo;&#xx [1;H.0;ޒ ;̠.;搄&#x 150;&#x.719; 32;.67;˜ ];1 &#x/BBo;&#xx [3;T.4;ޔ ;̦.;䀅&#x 371;&#x.278; 33;.00; ];&#x/BBo;&#xx [3;T.4;ޔ ;̦.;䀅&#x 371;&#x.278; 33;.00; ];(68) This profile provides enough in a time interval &#x/BBo;&#xx [3;T.4;ޔ ;̦.;䀅&#x 371;&#x.278; 33;.00; ];&#x/BBo;&#xx [3;T.4;ޔ ;̦.;䀅&#x 371;&#x.278; 33;.00; ];Pn can be for small from t2 initial plate &#x/BBo;&#xx [3;T.4;ޔ ;̦.;䀅&#x 371;&#x.278; 33;.00; ];&#x/BBo;&#xx [3;T.4;ޔ ;̦.;䀅&#x 371;&#x.278; 33;.00; ];V(x, &#x/BBo;&#xx [3;#.0;· ;Ȧ.;㈆&#x 330;&#x.719; 23;.92; ];&#x/BBo;&#xx [3;#.0;· ;Ȧ.;㈆&#x 330;&#x.719; 23;.92; ];0) = &#x/BBo;&#xx [3;E.5;঑ ;Ȩ.;䠅&#x 352;&#x.559;&#x 235;&#x.440; ]0;&#x/BBo;&#xx [3;E.5;঑ ;Ȩ.;䠅&#x 352;&#x.559;&#x 235;&#x.440; ]0;0. We shall, however, in t, from which information can small time behaviour The corresponding Fourier Consequently, the Laplace (with respect 7) of tm is BY (3% a, (s) = 6, - (6, + 61) H(s). Now, by using the asymptotic expansion of K,(s) for large Is1 in (32b), Hence 6,(s) = i (61-60)-(1/~~)(b"0+61) [I +0(lsl-l)]. The Laplace last term two orders smaller t than (b, - b,), according to (73). Consequently, a, (t) = i(bl - b,) + 0(t3). (74) shall assume inerbial resulting lift be small so (40), (41)) shows clearly L = &np(b2 - 6,) + 0(t2), M = +np(6, - 6,) + 0(t2). behave like 0 unless, up to 0(t2), b2 = bo, b3 = b1; or P2 = Po, P3 = Pl. (75) Under this deduce from (44)' also using (71), (74) : The expressions for T, P in terms of the coefficients P,, are 2Th-p = (Pt2 + P?2 - 4P02P12) t2 f 3[P03 (Po2 - 3P12) +Pi3 (P12 - $PO2)1 t3 + 0(t4), (77) ~PI~P = (ui2 + 4~:~) t + 3rh2pO3 + ~~PI~PI~I t2 + w3). (78) It is of interest to note that the thrust is produced by the time of O(t2), power is O(t), being positive definite. Another point interest is rectilinear acceleration first two order A qualitative evaluation Ohe initial stage is maximum for P if 5 Pl2/PO2 = - (1 +1/193)/12 = - 1.24, (79) which can readily undetermined multipliers. Deter- not as motion can to vanish, then be positive, under condition (79), if /3,,/PO3 = 1.07 and /303/P02 is sufficiently large and positive. Under condition (75), the body has the S-shape form, the maximum and minimum of h are given by ahlax = 12,8,x2+4P,x-2,8, = 0, propulsion. Part ��1 355 &#x/BBo;&#xx [1;.07;… 6;&.6;Γ ;&.8;ޒ ;س.;ሁ&#x ]00;&#x/BBo;&#xx [1;.07;… 6;&.6;Γ ;&.8;ޒ ;س.;ሁ&#x ]00;or, &#x/BBo;&#xx [2; .99;“ 6;&.8;ޙ ;A.2;މ ;ش.;㆘&#x ]00;&#x/BBo;&#xx [2; .99;“ 6;&.8;ޙ ;A.2;މ ;ش.;㆘&#x ]00;up to the &#x/BBo;&#xx [2; .99;“ 6;&.8;ޙ ;A.2;މ ;ش.;㆘&#x ]00;&#x/BBo;&#xx [2; .99;“ 6;&.8;ޙ ;A.2;މ ;ش.;㆘&#x ]00;first-order term, &#x/BBo;&#xx [9;.99;… 6; 1;.1; 6; .4;
;&#x]000;&#x/BBo;&#xx [9;.99;… 6; 1;.1; 6; .4;
;&#x]000;xl,Z &#x/BBo;&#xx [1;.9;և ;ؖ.;喓&#x 121;&#x.438; 61; .19;“ ];&#x/BBo;&#xx [1;.9;և ;ؖ.;喓&#x 121;&#x.438; 61; .19;“ ];= [ - &#x/BBo;&#xx [1;9.9;Ɔ ;ؕ.;怂&#x 143;&#x.279; 62;.31;• ];&#x/BBo;&#xx [1;9.9;Ɔ ;ؕ.;怂&#x 143;&#x.279; 62;.31;• ];1 &#x/BBo;&#xx [1;F.6;Ή ;ؓ.;醓&#x 152;&#x.158; 62;
.36; ];&#x/BBo;&#xx [1;F.6;Ή ;ؓ.;醓&#x 152;&#x.158; 62;
.36; ];+ (1 &#x/BBo;&#xx [1;e.1; 6;.3;ঔ ;ű.;莔&#x 621;&#x.119; ]0;&#x/BBo;&#xx [1;e.1; 6;.3;ঔ ;ű.;莔&#x 621;&#x.119; ]0;4- &#x/BBo;&#xx [1;t.2;Έ ;ؒ.;䠁&#x 211;&#x.679;
62;.00; ];&#x/BBo;&#xx [1;t.2;Έ ;ؒ.;䠁&#x 211;&#x.679;
62;.00; ];6[2)t]/6c = &#x/BBo;&#xx [2;'.0;Α ;ؗ.;Δ&#x 233;&#x.759; 61;.99;– ];&#x/BBo;&#xx [2;'.0;Α ;ؗ.;Δ&#x 233;&#x.759; 61;.99;– ];- 0.564, &#x/BBo;&#xx [2;e.6;ޑ ;ؕ.;ሁ&#x 291;&#x.358; 62;.08;&#x ]00;&#x/BBo;&#xx [2;e.6;ޑ ;ؕ.;ሁ&#x 291;&#x.358; 62;.08;&#x ]00;0.295. &#x/BBo;&#xx [3;W.3;։ ;ؒ.;熖&#x 373;&#x.918; 62;.31;• ];&#x/BBo;&#xx [3;W.3;։ ;ؒ.;熖&#x 373;&#x.918; 62;.31;• ];(80) &#x/BBo;&#xx [1;.83; 59;.96; 4;
.27;‰ 6;.3;؄ ;&#x]000;&#x/BBo;&#xx [1;.83; 59;.96; 4;
.27;‰ 6;.3;؄ ;&#x]000;When the higher terms in t are included, these points are seen to move back &#x/BBo;&#xx [1;.59;• 5;‚.7;ȃ ;P.1;֒ ;֒.;ޖ&#x ]00;&#x/BBo;&#xx [1;.59;• 5;‚.7;ȃ ;P.1;֒ ;֒.;ޖ&#x ]00;towards the trailing edge with increasing time. &#x/BBo;&#xx [2;.67;” 5;p.9;֕ ;c.1;Ƈ ;չ.;怂&#x ]00;&#x/BBo;&#xx [2;.67;” 5;p.9;֕ ;c.1;Ƈ ;չ.;怂&#x ]00;Withrout giving the detail, the vortex sheet strength at the trailing edge is found to &#x/BBo;&#xx [5;.67;‡ 5;`.1;֙ ;e.9;ও ;է.;妙&#x ]00;&#x/BBo;&#xx [5;.67;‡ 5;`.1;֙ ;e.9;ও ;է.;妙&#x ]00;be &#x/BBo;&#xx [1;3.9;Ɖ ;Ք.;掓&#x 149;&#x.039; 56;
.60; ];&#x/BBo;&#xx [1;3.9;Ɖ ;Ք.;掓&#x 149;&#x.039; 56;
.60; ];lslr 6 &#x/BBo;&#xx [1;.9;Ɔ ;Յ.;首&#x 109;&#x.678; 55;.36;&#x ]00;&#x/BBo;&#xx [1;.9;Ɔ ;Յ.;首&#x 109;&#x.678; 55;.36;&#x ]00;t) = &#x/BBo;&#xx [1;$.5;ֆ ;Թ.;写&#x 296;&#x.158; 56;.08; ];&#x/BBo;&#xx [1;$.5;ֆ ;Թ.;写&#x 296;&#x.158; 56;.08; ];---[(~oz+8,z)+s(Bo3+~3t+~(t2) &#x/BBo;&#xx [1;9.9;Ɔ ;Ը.;喗&#x 151;&#x.918; 54; .59;™ ];&#x/BBo;&#xx [1;9.9;Ɔ ;Ը.;喗&#x 151;&#x.918; 54; .59;™ ];4. &#x/BBo;&#xx [2;—.3;֓ ;Թ.;➓&#x 300;&#x.718; 56;
.83;— ];&#x/BBo;&#xx [2;—.3;֓ ;Թ.;➓&#x 300;&#x.718; 56;
.83;— ];1 , &#x/BBo;&#xx [3;W.1;Ɣ ;Յ.;冕&#x 373;&#x.679;
55;.11;” ];&#x/BBo;&#xx [3;W.1;Ɣ ;Յ.;冕&#x 373;&#x.679;
55;.11;” ];(81) &#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;&#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;which shows that, immediately after the motion is vortex shed from under condition &#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;&#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;(79), is such that the fluid near the trailing edge is propelled downstream. very much Lighthill for interesting &#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;&#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;stimulating discussions, and particularly for his kindness &#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;&#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;privilege of knowing his publication. This work &#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;&#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;was partially sponsored by Foundation, under &#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;&#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;GK 10216, and by the Naval Research, under &#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;&#x/BBo;&#xx [1;.83; 52;�.32; 4;
.27;‰ 5;).1;ও ;&#x]000;N00014- 67-AOO94-0012. REFERENCES &#x/BBo;&#xx [1;.35;‰ 3;„.9;֘ ;9.1; 3;‘.9;Ɨ ;&#x]000;&#x/BBo;&#xx [1;.35;‰ 3;„.9;֘ ;9.1; 3;‘.9;Ɨ ;&#x]000;FUNG, Y. C. 1967 Am. J. Physiology, 213, 1532. &#x/BBo;&#xx [1;.35;‰ 3;s.4;Ζ ;8.8;ބ ;΀.;d ];&#x/BBo;&#xx [1;.35;‰ 3;s.4;Ζ ;8.8;ބ ;΀.;d ];GRAY, J. 1948 Nature, Lond. 161, 199. GRAY, J. 1949 Nature, Lond. 164, 1073. &#x/BBo;&#xx [1;.35;‰ 3;s.4;Ζ ;8.8;ބ ;΀.;d ];&#x/BBo;&#xx [1;.35;‰ 3;s.4;Ζ ;8.8;ބ ;΀.;d ];& Nicholson. GRAY, J. &#x/BBo;&#xx [5;.51;ˆ 3;@.0;ޖ ;Y.5;Ƅ ;͆.;ࠃ&#x ]00;&#x/BBo;&#xx [5;.51;ˆ 3;@.0;ޖ ;Y.5;Ƅ ;͆.;ࠃ&#x ]00;& &#x/BBo;&#xx [6;.87;’ 3;9.1;Ɣ ;ă.;䎅&#x 346;&#x.559; ]0;&#x/BBo;&#xx [6;.87;’ 3;9.1;Ɣ ;ă.;䎅&#x 346;&#x.559; ]0;HANCOCK, G. J. 1955 J. Exp. Biol. 32, 802. &#x/BBo;&#xx [1;.35;‰ 3;'.3;֖ ;T.4;y 3;4.5;؁ ;&#x]000;&#x/BBo;&#xx [1;.35;‰ 3;'.3;֖ ;T.4;y 3;4.5;؁ ;&#x]000;HANCOCK, G. J. 1953 Proc. Roy. &#x/BBo;&#xx [1;Q.4;Έ ;̨.;喓&#x 166;&#x.798; 33;.04; ];&#x/BBo;&#xx [1;Q.4;Έ ;̨.;喓&#x 166;&#x.798; 33;.04; ];Soc. A &#x/BBo;&#xx [1;y.9;উ ;̥.;ᦗ&#x 210;&#x.479; 33;.04; ];&#x/BBo;&#xx [1;y.9;উ ;̥.;ᦗ&#x 210;&#x.479; 33;.04; ];217,,96. JOHANNESSEN, C. L. &#x/BBo;&#xx [9; .35;“ 3;.5;ȃ ;ą.;㖉&#x 323;&#x.519; ]0;&#x/BBo;&#xx [9; .35;“ 3;.5;ȃ ;ą.;㖉&#x 323;&#x.519; ]0;& HARDER, J. A. 1960 Science, 132, 1550. &#x/BBo;&#xx [1;.59;• 3;.3;Ȃ ;Q.1;Ɣ ;̑.;首&#x ]00;&#x/BBo;&#xx [1;.59;• 3;.3;Ȃ ;Q.1;Ɣ ;̑.;首&#x ]00;-MAN, T. VON &#x/BBo;&#xx [8;.67;‘ 3; 9;
.67;‡ 3;.9;খ ;&#x]000;&#x/BBo;&#xx [8;.67;‘ 3; 9;
.67;‡ 3;.9;খ ;&#x]000;& BURGERS, J. &#x/BBo;&#xx [1;H.3;Ƈ ;̅.;疔&#x 158;&#x.159; 31;
.99;– ];&#x/BBo;&#xx [1;H.3;Ƈ ;̅.;疔&#x 158;&#x.159; 31;
.99;– ];M. 1943 General aerodynamic theory: perfect fluids. Aerodynamic Theory E2 (ed. W. F. Durand). LANG, T. G. 1966 Hydrodynamic analysis of cetacean performance. Whales, Dolphins &#x/BBo;&#xx [1;H.3;Ƈ ;̅.;疔&#x 158;&#x.159; 31;
.99;– ];&#x/BBo;&#xx [1;H.3;Ƈ ;̅.;疔&#x 158;&#x.159; 31;
.99;– ];& &#x/BBo;&#xx [7;.59;‘ 2;a.5;ঙ ;ĕ.;ᦓ&#x 269;&#x.039; ]0;&#x/BBo;&#xx [7;.59;‘ 2;a.5;ঙ ;ĕ.;ᦓ&#x 269;&#x.039; ]0;DAYBELL, D. A. 1963 NAVWEPS Rep. 8060; NOTS Tech. Publ. &#x/BBo;&#xx [7;.59;‘ 2;a.5;ঙ ;ĕ.;ᦓ&#x 269;&#x.039; ]0;&#x/BBo;&#xx [7;.59;‘ 2;a.5;ঙ ;ĕ.;ᦓ&#x 269;&#x.039; ]0;LANG, T. G. &#x/BBo;&#xx [6;.47;” 2;Q.2;ޓ ;r.4;y 2;W.2; ]0;&#x/BBo;&#xx [6;.47;” 2;Q.2;ޓ ;r.4;y 2;W.2; ]0;& NORRIS, K. S. 1966 Science, 151, 588. &#x/BBo;&#xx [6;.47;” 2;Q.2;ޓ ;r.4;y 2;W.2; ]0;&#x/BBo;&#xx [6;.47;” 2;Q.2;ޓ ;r.4;y 2;W.2; ]0;LANG, T. G. &#x/BBo;&#xx [6;.71;‰ 2;9.6;Γ ;r.7;ƅ ;Ʌ.;d ];&#x/BBo;&#xx [6;.71;‰ 2;9.6;Γ ;r.7;ƅ ;Ʌ.;d ];& &#x/BBo;&#xx [7;.07;’ 2;8.6;ࠂ ;ą.;㖉&#x 246;&#x.120;
]0;&#x/BBo;&#xx [7;.07;’ 2;8.6;ࠂ ;ą.;㖉&#x 246;&#x.120;
]0;PRYOR, K. 1966 Science, 152, 531. Comm. Pure &#x/BBo;&#xx [7;.07;’ 2;8.6;ࠂ ;ą.;㖉&#x 246;&#x.120;
]0;&#x/BBo;&#xx [7;.07;’ 2;8.6;ࠂ ;ą.;㖉&#x 246;&#x.120;
]0;Appl. Math. 5, 109. LIGHTHILL, M. J. 1960 J. Fluid Mech. 9, 305. LIGHTHILL, M. J. 1969 Ann. Rev. Fluid &#x/BBo;&#xx [7;.07;’ 2;8.6;ࠂ ;ą.;㖉&#x 246;&#x.120;
]0;&#x/BBo;&#xx [7;.07;’ 2;8.6;ࠂ ;ą.;㖉&#x 246;&#x.120;
]0;LIGHTHILL, M. J. 1970 J. Fluid Mech. 44, 265. &#x/BBo;&#xx [1;.59;• 1;.1;গ ;ƒ.5;ƒ ;Ɖ.;㘄&#x ]00;&#x/BBo;&#xx [1;.59;• 1;.1;গ ;ƒ.5;ƒ ;Ɖ.;㘄&#x ]00;MUSKHELISHVILI, N. I. 1953 Singular Integral Equations. Groningen, Holland: Noordhoff. &#x/BBo;&#xx [1;.07;… 1;i.6;ޔ ;S.2;ޓ ;Ÿ.;ޖ&#x ]00;&#x/BBo;&#xx [1;.07;… 1;i.6;ޔ ;S.2;ޓ ;Ÿ.;ޖ&#x ]00;OSBORNE, M. F. M. &#x/BBo;&#xx [9;.63;… 1;p.8;ࠂ ;ĕ.;醉&#x 177;&#x.119; ]0;&#x/BBo;&#xx [9;.63;… 1;p.8;ࠂ ;ĕ.;醉&#x 177;&#x.119; ]0;1960 J. Exp. Biol. 38, 365. REYNOLDS, A. J. 1965 J. &#x/BBo;&#xx [1;".8;ވ ;ř.;妕&#x 144;&#x.718; 16;.83;— ];&#x/BBo;&#xx [1;".8;ވ ;ř.;妕&#x 144;&#x.718; 16;.83;— ];Fluid Mech. 23, 241. &#x/BBo;&#xx [1;.07;… 1;F.8;ޕ ;R.7;঒ ;ŕ.;➗&#x ]00;&#x/BBo;&#xx [1;.07;… 1;F.8;ޕ ;R.7;঒ ;ŕ.;➗&#x ]00;SUBMAN, P. G. 1967 J. Fluid Mech. 28, 385. &#x/BBo;&#xx [1;.07;… 1;5.1;Ƙ ;W.8;Ά ;Ń.;疔&#x ]00;&#x/BBo;&#xx [1;.07;… 1;5.1;Ƙ ;W.8;Ά ;Ń.;疔&#x ]00;SIEKMANN, J. 1963 J. Fluid Mech. 15, 399. &#x/BBo;&#xx [1;.83; 12;.95;˜ 4;.83;” 1;2.3; ]0;&#x/BBo;&#xx [1;.83; 12;.95;˜ 4;.83;” 1;2.3; ]0;S~O~S, G. G. &#x/BBo;&#xx [1;.83; 12;.95;˜ 4;.83;” 1;2.3; ]0;&#x/BBo;&#xx [1;.83; 12;.95;˜ 4;.83;” 1;2.3; ]0;Soc. 9, 8. TAYLOR, G. I. 1951 Proc. Roy. &#x/BBo;&#xx [1;C.5;ƈ ;ē.;冕&#x 159;&#x.119; 12;�.00; ];&#x/BBo;&#xx [1;C.5;ƈ ;ē.;冕&#x 159;&#x.119; 12;�.00; ];Soc. A 209, 447. &#x/BBo;&#xx [1;C.5;ƈ ;ē.;冕&#x 159;&#x.119; 12;�.00; ];&#x/BBo;&#xx [1;C.5;ƈ ;ē.;冕&#x 159;&#x.119; 12;�.00; ];1952a Proc. Roy. &#x/BBo;&#xx [1;I.2;މ ;Ă.;&#x 164;&#x.638; 10;.48;
];&#x/BBo;&#xx [1;I.2;މ ;Ă.;&#x 164;&#x.638; 10;.48;
];Soc. A 211, 225.