ON OBLIQUE WAVES FORCING BY A POROUS CYLINDRICAL WALL

The problem of oblique cylindrical linearized wave motion is considered for a fluid of infinite depth or finite constant depth in the presence of an impermeable cylindrical wall and coaxial porous wall immersed vertically in the fluid The motion is generated once by the oscillations, which are periodic in time and in 0-direction, of the impermeable wall and next by the porous wall. The velocity .potentials have been found in closed forms in the different regions of the fluid and then calculating the hydrodynamic pressure distribution on the porous wall and the profile of the free surface. The scattering problem of oblique waves is then considered A wave trapping phenomenon is investigated. Numerical results are given to the case of radial incident waves and the case when the angle of incident waves is 30 to the radial direction.


INTRODUCTION.
The scattering of surface waves obliquely incident on partially immersed or completely submerged vertical barriers and plates in infinite fluid were investigated by Faulkner [1,2], Jarvis and Taylor [3], Evans and Morris [4], Rhodes-Robinson [5] and Mandal and Goswami [6].Levine  [7] considered the scattering of surface waves obliquely incident on a submerged circular cylinder.The problem of scattering of oblique waves by a shallow draft cylinder at the free surface was solved by Garrison [8].Subsequently, Bai [9] studied the more general problem of scattering of oblique waves by a partially immersed cylinder.In all of such works the immersed bodies are assumed to be impermeable.Chwang  10] considered a porous wavemaker oscillating normally to its plane with a constant amplitude.In his linearized analysis, the wavemaker is located in the middle of an infinitely long channel with constant depth.Chwang and Li [11] applied the linearized porous wavemaker method developed in [10] to investigate the small amplitude surface waves produced by a piston-type porous wavemaker near the end of a semi-infinitely long channel of constant depth.Chwang and Dong [12] studied the problem of reflection and transmission of small amplitude surface waves by a vertical porous plate fixed near the end of a semi-infinitely long open channel of constant depth.Gorgui and Faltas [13] extended Chwang's work to include the study of wave motion for a fluid of infinite horizontal extend and of infinite or finite constant depth in the presence of an impermeable plate and a porous wall immersed in the fluid parallel to each other.The waves are generated by arbitrary prescribed horizontal oscillations performed by the impermeable plate or the porous wall.

M S FALTAS
In the present paper we investigate the case of oblique cylindrical wave motion in fluids of infinite depth or finite constant depth The linearized theory for waves of small amplitude is used to analyze the forced motion in fluids bounded internally by an impermeable vertical circular cylinder surrounded by a coaxial cylindrical porous wall The waves are generated by arbitrary prescribed oscillations, which are periodic in time and in 0-direction, first performed by the impermeable wall and later by the porous wall It is assumed that the pores of the wall are of such nature as to allow the application of Darcy's law that the fluid velocity normal to the wall is linearly proportional to the difference in pressure between its two sides The method of separation of variables is applied to find analytic solution in closed forms for the linearized boundary value problem in the different regions of the fluid The results of Sections 3 and 4 are used to find the reflection coefficient of the reflected waves due to the scattering of time and 0 periodic waves incident with angle/ to the radial direction In the last section numerical results are presented for the two cases of radial oscillations and the case of 30 2.

BOUNDARY VALUE PROBLEM
We consider here the excitation of gravity waves on the surface of a fluid by an impermeable vertical cylindrical wall of circular cross-section of radius a that performs oscillations which are periodic in time and in 0-direction.A coaxial cylindrical porous wall of circular cross-section of radius b( > a) is fix.ed in the fluid (see Fig. 1).Let (r, 0, y) be cylindrical coordinates with the origin 0 in the undisturbed free surface such that 0y pointing down into the fluid coinciding with the axis of the impermeable and porous walls.Let the velocity of the impermeable wall at time is U()exp(-icat + ivO), where v sin, / is the angle that the produced train of waves makes with the radial direction and U (/) is a complex valued and suitably limited.The resulting motion is therefore time and 0 harmonics with the same w and v as of the impermeable wall.
We assume that the fluid is incompressible and inviscid and that the motion originates from rest, by virtue of which there exist velocity potentials Cj(r, O, y; t) such that ej(r, 0, y; t) Re[;(r, y)exp( iw-t + zv0)] where the subscripts 3 1, 2 refer to the regions a < r < b and r > b respectively Also the motion is assumed small so that the linearization is permissible.We consider here first the case when the fluid is of infinite depth The functions es(r, y) satisfy The linearized free surface condition is 0 where K and g is the gravitational constant.( We shall also assume that the porous wall is made of material with very fine pores.Thus according to Taylor's  where g pwd/#, # is the dynac viscosity, p is the constt density of the fluid d d is a coefficient wch has the dimension of lenh.It should be noted here if the porous flow tough the will is sigficant, condition (2.5) may not be accurate enou.Hence we should cone our investigation to porous walls th fine pores.Finflly we have the condition for no motion at ite depth, 0 as y (2.6) d the radiation condition for the outgoing waves CH 1) (gr)e -gy as r (2.7) where C is multiple constt d H$1)(z) is Hel's Bessel nction of tMrd nd of order v.The peter G is a measure of the porous effect.G 0 mes the wMl is impeeable, wMle G approaches i the wall becomes completely peeable to the fluid. 3.
When the porous wall (r b) is completely permeable i.e.G oo, the velocity potential in the region r > a is ka(k)f(k,y) K(kr) dk-2rKA H(I),(Ka)e (3.12) Also when the porous wall at (r b) becomes impermeable, G 0, the results (3.7), (3.8) reduce to ka(k)f(k.V) K.(kb)I.(kr)I'.(kb)K(kr) 2 k2 + K2 K.(ka)F(kb K.(kb)F.(kadk H (1)' (gb)J,(gr) J(gb)H( 1) (gr) + 2KA H(),(ga)j(gb)_ H(l),(gb)j(ga) e 2 0 This solution is valid only when the quantity Y'(Ka)J(Kb) Y'(Kb)J:(Ka) (3.13) is different from zero.However, it indicates that when this quantity vanishes, resonance occurs and linearized theory for small motion cannot be applied.In the particular case when U (y) Ve-Kv, where V is a real constant, we have THE FINITE DEPTH CASE Now we consider the case of finite depth h.Using the same notation and coordinates, the complex potentials es, J 1, 2, for the motion in the fluid regions a < r < b, r > b are the solutions of the boundary value problem stated in Section 2 with conditions (2.6), (2.7) replaced by 0 O--es 0 on y h, (4.1) 2 CH(v 1) (kor) cosh ko(h y) as roo (4.2)   when C is a constant multiple, k0 is the real positive root of ksinhkh-Kcoshkh =0.

OBLIQUE WAVES GENERATED BY THE POROUS WALL
If we now let the porous wall oscillate obliquely with velocity U(V)exp(-iwt + ivO) while the impermeable wall at r a be kept fixed, then the new boundary value problem is the same as stated in Section 2 except that the boundary conditions (2 3), (2.Thus when the porous wall is the wave generator we have 4be_,v/2fo ka(k)f(k,y) [iv(kr)K,(ka)_Kv(kr)i,v(ka)]K,(kb)dk 7r (k + g2)A(ik) 7r2bgA [Jv(kr)H(v),(ga) ()(gr)J'(ga)]H( )' t, (gb)e -gv (5.3) /(g) (5.4)When M 0, the waves are trapped in the bounded region between the two cylinders a < r < b and no waves radiate away from the wall, liquid simply piles up around the wall.

WAVE TRAPPING
In this section we investigate an interesting application of the above results to the case of a time cylindrical wave CH(2)(kR)exp(ivO-Ky) incident obliquely, proceeding from infinity, the porous cylindrical wall at r b and the impermeable cylindrical wall at r a both fixed.The velocity potentials Ca(r, y) are functions that satisfy (2.1), (2.2) and (2.6).On the porous wall 2 = iG(1 ), and on the impermeable wall Here A (to be determined) is a complex constant relating to the amplitude and phase of the reflected wave Consider the functions (r, y)-2C J,(Kr)e -K These new functions satisfy equations (2 1) and the free surface boundary conditions On the porous wall (6.8) Since the present problem is linear, II/1, I,I/2 call be obtained by a suitable superposition of the results (3 7), (5.3) and (3.8), (5 4) respectively.Hence 2CG [j(Kr)H(I),(Ka j(Ka)H(l)(Kr)]e_ly (6.9) -A(K) /k,(K) ()(Kr)+CH(f)(Kr)]e -Ky (6.10)
The coefficient of reflection R is defined as the square of the ratio of the amplitude of the reflected wave to the amplitude to the incident wave i.e.
A*(K) /(K) 0 2M 2 G +/32M 2 a2 + 2M2G +/02M 2 (6.11)where a2 rb[j2(Ka)+ y2(Ka) f12 'rb[j2(Kb)+ Y2(Kb)] when the wall at r b is impermeable i.e., when G 0, the incident wave is totally reflected by it.We get the same situation when the wall (r b) is completely permeable but now the wave is totally reflected by the impermeable wall at r a.We note also that when M 0 i.e. when a and b has values satisfying the equation J(gb)Y'(ga) J'(ga)Y(gb) 0, (5.12) the incident wave is totally reflected (R 1) at r b irrespective of the value of G.By simple differentiation of (6.11) with respect to G for any fixed values of a and b, R reduces to a minimum, cq3-M (6.13) Rm,n a,t3 + M when G M__y. this minimum value vanishes when a/3 M i.e., when a and b satisfy the equation TI (Kb)J,(Ka) + (Kb)r((Ka) O, (6.14) That is R 0 when G =/3 and a, b has values satisfying equation (6.14).Under these circumstances the porous wall acts as an efficient wave absorber or eliminator for the incident waves, i.e., for G =/3

Fig. 1 :
Fig. 1: Schematic diagram of a horizontal cross-section of the physical problem.

Table 1
Values of Kb( > Ka) and G/K for wave trapping

Table 2
Values of Kb( > Ka) for complete reflection