Mathematical theory of irrotational translation waves

Abstract

This paper is the first of a series dealing with the motion of flood waves and other waves of translation in open channels. The case treated is that of waves for which the forces of fluid friction are negligible with r espect to the inertia and gravitational forces. The irrotational motion of a perfect liquid in a horizontal rectangular canal when the original surface is disturbed is investigated on the assumption that the horizontal velocity in a cross section is approximately uniform. The results are also applicable to motion in a canal of uniform slope containing water originally moving with a uniform velocity. Special emphasis is laid on disturbances which are propagated without change of form, and in these cases formulas are derived for the wave profile and velocity of propagation. Formulas are also derived which give the deformation, energy, motion of the center of gravity, and moment of instability of an arbitrary intumescence. Consideration is given to the maximum height of a wave of permanent form. Formulas have been compared with the available experimental data. Of special interest is the comparison of the shape of the undulations composing the head of an initial sur6e with the characteristics of the cnoidal wave. CONTENTS Page List of symbols____ __ __ _ _ __ _ _ _ _ __ _ __ _ _ _ ___ _ __ _ _ _ _ __ _ _ _ _ _ _ _ _ _ __ _ ___ _ __ 48 I. Introduction____ __ _ _ __ __ _ _ __ _ __ _ __ __ __ _ _ ___ _ _ _ _ _ __ __ __ _ __ _ _ _ __ _ _ 49 II. Irrotational motion_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 53 1. Dynamical equations of motion_________ ____________________ 53 2. Displacement and deformation of a liquid element_ _ __ __ _ _ __ __ 54 3. Kinematical relations______ __ ____ _ _ _ _ _ _ _ __ __ _ _ _ _ __ _ _ _ _ _ _ _ __ 55 4. Irrotational motion __ ____ _ _ ____ _ ____ ____ ___ _ __ _ _ _ _ _ ______ __ 56 5. Invariance of the circulation in a moving circuit_ _ _ _ _ _ _ _ _ _ _ _ _ _ 57 6. Impulsive generation of motion________________ _____________ 59 7. Pressure equation_ ____ _ __ _ __ ____ __ _ _ _ _ _ _ _ __ ___ _ _ __ _ _ _ _ __ _ _ 60 8. Solution of problems in irrotational motion____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 61 III. Formulation of the wave problem-Waves in water of small depth ____ 61 IV. Long waves of negligible height and curvature_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 63 1. Velocity of propagation__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 63 2. Energy of an intumescence__ ___________ ____________________ 65 3. Displacement of the particles_______________________________ 66 4. Effect of an arbitrary initial disturbance_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 66 5. Reflection and transmission of waves___________ _____________ 67 47 J98881-40-;1 1 48 Journal oj Research oj the National Bureau oj Standards {Vol..t4 Page V. Waves of appreciable height and curvature_______ __ __ ___ ___________ 69 1. Two definitions of velocity of propagation_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 69 2. Mean velocity of particles _________________________________ 70 3. Effect of curvature and height on velocity of propagation of a volume elemen t_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 72 4. Deformation of the wave profile______________________ ______ 74 5. Motion of the center of gravity of a wave__________________ __ 75 6. Variation in energy of an intumescence______________________ 76 7. The moment of instability of an intumescence_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 77 , 8.' Solitary waves______ _____ _ _ _ _ _ _ _ _ __ __ __ _ _ __ __ _ _ _ __ __ _ _ __ __ 78 R' 9. Cnoidal waves____________________________________________ 81 10. Moment of instability of a solitary wave_____________________ 85 11. Moment of instability and wave formation_______ ____________ 87 12. Negative waves_____ _ ____ _ _ ___ ____ _ __ _ _ __ __ __ _ _ __ _ ___ _ __ __ 88 13. Intumescences of finite height but negligible curvature ________ 89 14. Horizontal channel in communication with the ocean__________ 90 15. Waves due to sudden increase of discharge___________________ 91 16. Effect of channel velocity distribution_______________________ 91 17. Initial waves_ _ __ ___ ____ _____ _ _______ _ _ __ __ __ __ ___ _ ___ __ __ 95 18. Favre's experiments on the undulations in a positive surge_ _ _ _ _ 97 19. Theory of breakers____________________________________ ____ 99 VI. Fteferences ______________________________________________________ 100 f (g (LIST OF SYMBOLS c velocity of propagation of long wave of negligible height and curvature. CI velocity of wave approaching a discontinuity in a channel cross section. Also a constant of integration. C2 velocity of transmitted wave after passing a discontinuity in cn (x, k) pgE El (k)), F (), F (x, k) Fl (k) g), G () h a channel cross section. Also a constant of integration. Jacobian elliptic function, cosine amplitude. total energy of a wave per unit width of channel [MLT-2]. complete elliptic integral of the second kind. with or without subscripts-functional symbols. incomplete elliptic integral of the first kind. complete elliptic integral of the first kind. acceleration of gravity. functional symbol. height of main body of discharge wave (section V-17); vertical displacement of water surface. hi vertical displacement of surface as wave approaches discon-tinuity in channel cross section (section IV-5). h2 vertical displacement of surface after wave passes discon-tinuity in channel cross section (section IV-5). hi maximum height of solitary wave, maximum depth of negative wave, maximum height of initial discharge wave. hb h2 maximum height and depth, respectively, of cnoidal wave. ha=H1Ja. hG maximum height of initial discharge wave. h' average height of discharge wave. H undisturbed depth of liquid in channel. H () functional symbol. k modulus of elliptic integral. l length of section of channel (section IV-4). I, m, n direction cosines. M. moment of instability [L]. M'b M 2, M'2, M", momenta per unit width [MT-I). P pressure [ML-IT-2). PG atmospheric pressure. q speed of particle (section II-5, 7, 8). q discharge per unit width in a cross section [DT-I) (section V-2). Q total volume per unit width of a solitary wave [D). s length of arc. Keuhgan] Patter80n I rrotational Translation Waves sn (x, k) Jacobian elliptic function, sine amplitude. t time. 49 u, v, w x component, y component, z component, respectively, of velocity. Uo x component of velocity at bottom of channel, also first approximation to velocity in cross section. Uo, vo, Wo velocity components prior to impulse (section II-6). UIJ Uz velocities of particles in a wave before and after passing a discontinuity in a channel cross section (section IV-5). UIJ Uz , Vt, V 2 local and mean velocities in two cross sections (section V-16). Ul velocity of particle at apex of wave (section V-19). V mean velocity of particles in cross section. Vo mean velocity in a section where h=O. V volume [L3] (section II-3); volume per unit width [V] (sec-tions IV-3, V-5). x Cartesian coordinate, specifically, parallel to channel in bottom plane. y Cartesian coordinate, specifically, lateral to channel in the horizontal plane. z Cartesian coordinate, specifically, drawn upward with origin at bottom of channel. a=3/H3 (section V-I0); a ratio (section V-14). ' Yt, 'Yz, ' Y3 rates of angular dilatation rr-lj. r circulation [L2T-l]. fl, f2, <3 rates of linear dilatation [T-l]. velocity potential [VT-l]. < velocity of propagation of 8. cnoidal wave. w. velocity of propagation of discharge. '" impulsive pressure [ML-IT-l]. n gravitational potential [L2T-2].