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Transient fluid motion due to the forced horizontal oscillations of a vertical cylinder
Applied OceanResearch16 (1994)
347 351 Elsevier Science Limited Printed in Great Britain. 0141 - 1187(94)$07.00
~rN!1 ELSEVIER
0141-1187(94)00025-5
Transient fluid motion due to the forced horizontal oscillations of a vertical cylinder P. Mclver Department of Mathematieal Sciences, Loughborough University, Loughborough, Leics, UK, LEI1 3TU
(Received 20 June 1994) The problem of the forced horizontal oscillations of a vertical cylinder extending throughout the fluid depth is considered on the basis of the linearised theory of water waves. A new integral form is given for the frequency-domainsolution and the procedure is then used to obtain an explicit time-domain solution. condition, the reverse situation to the conventional approach. Eigenfunctions that satisfy the body condition are combined to satisfy the free-surface condition. This approach to the frequency-domain analysis was suggested by the standard approach to time-domain problems which employs what is essentially the same construction. The result of this frequency-domain analysis is a form of the solution in which the frequency appears explicitly allowing solutions in the time domain to be obtained analytically by the Fourier transform. For completeness, the same procedure is then used in the time domain to obtain solutions directly without reference to the Fourier transform of the frequencydomain solution. Finally, for the purposes of illustration, a solution is given for the particular case of a cylinder moving under a sinusoidal force after the fluid is initially at rest. The vertical circular cylinder is of radius a and extends throughout water of constant depth h. Cylindrical polar coordinates (r, 0, z) are chosen with the origin of z in the mean free surface, and z directed vertically upwards, and the origin of (r, 0) being the cylinder axis. The cylinder is forced to perform horizontal oscillations, with a prescribed time-dependent velocity, in the direction 0 -- 0. Under the usual assumptions of the linearised theory of water waves the motion may be described by a velocity potential satisfying Laplace's equation throughout the fluid domain.
1 INTRODUCTION Within the context of the linearised theory of water waves, there is a small number of geometries for which explicit analytical solutions are possible. One such geometry is a vertical cylinder extending throughout the depth of the fluid and this paper is concerned with the wave field resulting from the forced horizontal motion of this type of cylinder. The solution in the frequency domain is usually expressed in terms of an infinite series (Dean & Dalrymple, 1 Section 6.4) and solutions in the time domain may then be obtained with the aid of the Fourier transform. However, this series form of the frequency-domain solution contains the frequency only implicitly through the solutions of the dispersion relation and so the Fourier transform must be carried out numerically. The main result of the present work is an integral form for the time-domain solution corresponding to an impulsive motion of the cylinder. This may be used to construct time-domain solutions for more general motions of the cylinder without the need for numerical transforms between the frequency and time domains. To illustrate the solution procedure, in what is perhaps a more familiar context, a new integral form for the frequency-domain solution is first obtained. In the conventional frequency-domain solution procedure, eigenfunctions are constructed that satisfy the homogeneous free-surface condition and then these are combined to satisfy the non-homogeneous body boundary condition. Here, the infinite-frequency limit of the solution is first subtracted out which leads to a new boundary-value problem with a non-homogeneous free-surface condition and a homogeneous body
2 FREQUENCY DOMAIN For the case of steady time-harmonic oscillations of angular frequency ~vand velocity amplitude U persisting for all time, the velocity potential may be written (r, 0, z, t) = Re{ UfA(r, 0, r)e -'~t }
© Crown copyright (1995). 347
(1)
P. McIver
348
The time-dependent potential ¢ must satisfy the freesurface boundary condition
satisfying the boundary conditions
0~
0f~
Oz
0O _- - Kq5 o n , 7 = 0
(2)
- -
Oz
where the frequency parameter K = w2/g and g is the acceleration due to gravity, the bed condition
00
O~" = 0 on z = - h
(3)
Or
cos 0 on r = a
(4)
and the radiation condition
rl/2(O~rr-ikO )
0O
Or
0 on z = - h
(15)
0 on r
(16)
a
and
(5)
kh
In what follows, the limiting potential as the frequency tends to infinity is required. This 'infinitefrequency' potential will be denoted by 9t(r, 0, z) and will satisfy the above conditions in the limit K ~ oc so that the free-surface condition (2) is replaced by (7)
and the radiation condition (5) by Q ---+0 as r ~ oc
(8)
Separation of variables and satisfaction of all boundary conditions except that on the body gives oc
f~ = cosO Z A,Kl(p,r ) cospn(z +
h)
Oz K~
where (10)
and Kl denotes the modified Bessel function of the second kind and order one. The body boundary condition now requires oc
h) = 1
on z
0
R(r) = -2 ~7" K,(p,r) n~=l pnhK'l ( Pna)
(18)
(19)
In contrast to 4), which must satisfy a homogeneous free-surface condition and a non-homogeneous body boundary condition, the new potential ~ is required to satisfy a non-homogeneous free-surface condition and a homogeneous body boundary condition. The solution proceeds by separation of variables, first noting that the body boundary condition is satisfied identically by
C,(qr) = J,(qr) Y~(qa) - Y,(qr)J~(qa)
(20)
where Jn and 1I, denote Bessel functions and a prime denotes differentiation with respect to the function argument. Thus
dc,(qr)
Z A'pnK'I (Pna) c°sp" (z +
R(r)cosO
(9)
n=l
p, = ( 2 n - 1)Tr/2h
(17)
where (6)
f~ = 0 on z = 0
---* 0 as r ---~oc
The solution for It given by (9) and (12) may be substituted into the free-surface boundary condition (14) to obtain
0~, ---+0 as r ---* oc
Here k is the root of the dispersion relation K = k tanh
Oz
(14)
rl/2(O-~F--ikO)
the body condition
oo
o~
K~ = -0-f on z = 0
(11)
= 0 on r = a
(21)
A solution for ~ satisfying all except the free-surface and radiation conditions is = cos 0
r A(q)C 1(qr) cosh q(z + h)dq
j0
(22)
The unknown function A(q) is found by applying a modified version of Weber's integral theorem (Hunt & Baddour, 2 eqn (54)) which states that
n=l
and the expansion coefficients follow from the orthogonality of the set {cosp,(z + h); n = 1,2,...} over the depth so that 2 ( - 1 ) "+~
An p2nhK,l(p,a),
n = 1,2,...
(12)
To obtain the solution for arbitrary frequency consider the harmonic function ~/,= ~ - Q
(13)
f(r)=Io
Cn(q(qa)12 r)lHl qoq J ] f Cn(qr')r'fl(r')dr'.
(23)
where H1 denotes a Hankel function of the first kind, provided r l/2f(r) is integrable over [a, oc). Substitution of (22) into the free-surface condition (18) and application of (23) gives A (q) (K cosh qh =
q sinh qh)
q , lim [o~ IH[(qa)[ ~ ~ o J , . ,
Cl(qr)rR(r)dr
(24)
Forced horizontal oscillations of vertical cylinder where e has been introduced to allow the interchange of the integration and summation when (19) is substituted into (24). Now from Abramowitz & Stegun, 3 eqn (11.3.29), lim [~ Cl(qr)K~(pr)rdr -
e~O Ja+e
P
!
7rq(p2 + q2) K, (pa)
generated by a cylinder started from rest; the formulation follows the description of the standard time-domain theory as given, for example, in Section 7.11 of Mei. 5 The fluid response to an impulsive velocity V(r)6(t - r) imposed at time t = r is described by the potential • (r, 0, z, t) = V(r)[Q(r, 0, z)6(t - "r)
(25) so that A (q) (K cosh qh - q sinh qh) 4
oo
_ 7ch[Hf(qa)l2 ~_lp2 1 _ + q2
2tanhqh
Irq[H~(qa)12
(26)
where the series has been summed using eqn (1.421(2)) of Gradshteyn & Ryzhik. 4 The solution for ¢ is therefore 2 cos 0
+ p(r,
~0~
Ot
2 ( - 1)'Kl (p,r) cosp,(z +
K{(pna)p 2h
(28)
where {kn; n = 1,2,...} are the real, positive roots of
K = -kn tan k, h (
sin 2k, h)
0f~
OF
Ot 2 ~ - g ~ z = 0 ° n z = 0 f ° r 0 < t < ° °
OF Oz
0onz
-h
(32) (33) (34)
(35)
and OF
Or
0 on r = a
F = cos 0
h)]
sinh kh H1 (kr) cosh k(z + h) + ~(~a)~ /
(31)
(36)
The solution proceeds in a similar way to the frequencydomain solution given in the previous section. The body and bed conditions (35)-(36) are satisfied by taking
~ = c°sO{~=l [sinknh K'(knr) (kna)kc°skn(z n2hN n2 +
+
r)]
g-~z on z = 0 at t = 0
02F (27)
where the path of integration has been chosen to run beneath the pole at the real root of K = q tanh qh in order to satisfy the radiation condition. This can be confirmed by a standard residue calculus technique as described, for example, by Mei, 5 p. 379. Indeed this solution may be written in series form as
-
F = 0 on z = 0 at t = 0 OF
tanh qh C l (qr) cosh q(z + h) ]H~(qa)lZq(Kcoshqh- qsinhqh) dq
O, z , t - r ) H ( t
The first term on the right-hand side of (31) describes the initial pressure impulse transmitted throughout the fluid and the second term describes the subsequent fluid motion. Here 6(t) is the delta function, H(t) is the Heaviside step function and F(r, 0,z, t) is a harmonic function satisfying the initial and boundary conditions
71"
x
349
(29) (30)
f0o A(q, t)Cl (qr) cosh q(z + h) dq
(37)
where C1 is defined in eqn (20). The free-surface condition (34) is satisfied provided d2A dt 2 + W2A = 0
(38)
which has a general solution
A = a(q) cos Wt + ~(q) sin Wt
(39)
where and k0 = - i k . When combined with the infinitefrequency solution 9t this gives the 'well-known' form of the frequency-domain solution for radiation by a cylinder as presented in Section 6.4 of Dean & Dalrymple. 1 For the purpose of obtaining solutions in the time domain by Fourier transform, the advantage of (27) is that the frequency appears explicitly through K = w2/g. In the form (28), the frequency appears only implicitly through the solutions of (29).
3 IMPULSIVE MOTION Attention
is now turned to the transient motion
W 2 = gq tanh qh
(40)
The initial conditions (32)-(33) require that
Jo
~a(q) C 1(qr) cosh qh dq = 0
(41)
and
lo
Zfl(q) W(q)C1 (qr) cosh qh dq = -gR(r)
(42)
where R(r) is defined in eqn (19). It follows from the modified version of Weber's integral theorem, eqn (23), and following similar manipulations as carried out in eqns (24)-(26) that
a(q) = 0
(43)
P. Mclver
350 and 2g tanh q h
7rqWc°shqhIH~(qa)[ 2
/3(q) =
(44)
This now has given
q5 = ~V(t) +
2gc_os0 [X sin Wttanhqh Cl(qr) coshq(z + h) dq 7r Jo qWcoshqh[H~(qa)[ 2
F -
(45) This is related to the Fourier transform of the frequency-domain solution through lI~ F = ~
~e-i~'tdco
where ~ is given by (27) and may be written in the form _ 2gcos0 ~ ¢ tanhqh G(qr) coshq(z + h)
Jo q T J2 ~ ~
)
~
i
2dq (47)
To recover (45) from (46), co is replaced by co + ie in (47), so that the pole on the real q axis is moved into the upper half of the complex q plane, and then the result
fOO
lim e i~t 27r (~0 _~ (co+ie) 2 - W 2dco= - ~ s i n
Wt
vV
F(t - 7-) sin aJ~- d~-
(51)
0
where f~ is given by (9) and (12) and F is given by (45). The result is q5 = I)[(~ + Re~) sin~t + ~], t> 0 (52) where R e ~ is the principal value equivalent of (27) and qj _ 2cocos sin Wt 7r
(46)
c3~
;
which is turned on at t = 0; here H(t) is the Heaviside step function. From (49) and (50) the potential for the resulting disturbance is
0
W
tanh qh C 1(qr) cosh q(z + h) x iH[(qa)lZq(Kcoshqh _ qsinhqh)dq
(53)
It is expected that purely time-harmonic motion will be achieved after a long time, that is q~ ~ Re{ii20e i~'t} = /2[(f~ + Re~) sin a;t - Im~ cos cot] as t ~ oe (54) where q~is defined in eqn (1) and I m + is the contribution to (27) from the integration around the pole and is given by
(48)
sinh kh Ct (kr) cosh k(z + h) cos 0
(55)
]H~(ka)[2kZhN2 °
Im~ = is used.
This limiting behaviour (54) can be confirmed by writing ~, which appears in (52), as 4 R E S P O N S E TO A SINUSOIDAL FORCING
According to Mei, 5 p. 374, the response to a continuous velocity V(t) imposed on the cylinder is given by
= av(t) +
r ( t - ,-)
(49)
Consider, for example, the sinusoidal forcing
V(t) = H(t)Vsincot
(50)
i0
5 (I)
D O
coc°S0Im.~ ~ ~ - -~c eiwtW
~-
tanh qh C1 (qr) cosh q(z + h) x ]H[(qa)[Zq(Kcoshqh _ qsinhqh) dq
(56)
and shifting the path of integration into the upper half of the complex q plane by allowing q to have a small positive imaginary part. The limiting behaviour arises from the poles on the real axis while the remaining part of the integral decays to zero as t ~ oc. The details are straightforward but messy and so are omitted here. The solution found above may be used, for example, to calculate the free-surface elevation at an arbitrary point or the hydrodynamic force on the cylinder. The latter is used for purposes of illustration. The timedependent force in the direction 0 = 0 is
0
F(t) = - p
JI ~°°- c o s 0 d S
(57)
s -5
-I0
where S denotes the wetted surface of the cylinder. Carrying out the integrations gives .............................. 0 2 4 6 8
i0
12
F(t) pTra3co(/
2a Kl (p,a) cos cot + -h = (pna)3Kl(p.a) 7ra2
T
Fig. 1. Non-dimensional force F(t)/(praawlT"), due to the sinusoidal forcing (50), versus non-dimensional time T = wt for a/h = 0"1, Ka = 0'5: (o • • •) time-domain solution; ( - - - ) frequency-domain solution.
fro (cos Wt - cos cot) tanh qh C 1(qa)
x
]n~(qa)12q2(K~--_- ~
dq(58)
This expression is readily evaluated by standard
Forced horizontal oscillations of vertical cylinder numerical techniques and a sample calculation is given in Fig. 1 showing the rapid approach to the frequencydomain solution.
REFERENCES 1. Dean, R. G. & Dalrymple, R. A., Water Wave Mechanics for Engineers and Scientists. World Scientific, 1991.
351
2. Hunt, J. N. & Baddour, R. E., Nonlinear standing waves bounded by cylinders. Quarterly Journal of Mechanics and Applied Mathematics, 33, 357-371. 3. Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC, 1964. 4. Gradshteyn, I. S. & Ryzhik, I. M., Tables of Integrals, Series and Products. Academic Press, New York, 1980. 5. Mei, C. C., The Applied Dynamics of Ocean Surface Waves. John Wiley, New York, 1983.
347 351 Elsevier Science Limited Printed in Great Britain. 0141 - 1187(94)$07.00
~rN!1 ELSEVIER
0141-1187(94)00025-5
Transient fluid motion due to the forced horizontal oscillations of a vertical cylinder P. Mclver Department of Mathematieal Sciences, Loughborough University, Loughborough, Leics, UK, LEI1 3TU
(Received 20 June 1994) The problem of the forced horizontal oscillations of a vertical cylinder extending throughout the fluid depth is considered on the basis of the linearised theory of water waves. A new integral form is given for the frequency-domainsolution and the procedure is then used to obtain an explicit time-domain solution. condition, the reverse situation to the conventional approach. Eigenfunctions that satisfy the body condition are combined to satisfy the free-surface condition. This approach to the frequency-domain analysis was suggested by the standard approach to time-domain problems which employs what is essentially the same construction. The result of this frequency-domain analysis is a form of the solution in which the frequency appears explicitly allowing solutions in the time domain to be obtained analytically by the Fourier transform. For completeness, the same procedure is then used in the time domain to obtain solutions directly without reference to the Fourier transform of the frequencydomain solution. Finally, for the purposes of illustration, a solution is given for the particular case of a cylinder moving under a sinusoidal force after the fluid is initially at rest. The vertical circular cylinder is of radius a and extends throughout water of constant depth h. Cylindrical polar coordinates (r, 0, z) are chosen with the origin of z in the mean free surface, and z directed vertically upwards, and the origin of (r, 0) being the cylinder axis. The cylinder is forced to perform horizontal oscillations, with a prescribed time-dependent velocity, in the direction 0 -- 0. Under the usual assumptions of the linearised theory of water waves the motion may be described by a velocity potential satisfying Laplace's equation throughout the fluid domain.
1 INTRODUCTION Within the context of the linearised theory of water waves, there is a small number of geometries for which explicit analytical solutions are possible. One such geometry is a vertical cylinder extending throughout the depth of the fluid and this paper is concerned with the wave field resulting from the forced horizontal motion of this type of cylinder. The solution in the frequency domain is usually expressed in terms of an infinite series (Dean & Dalrymple, 1 Section 6.4) and solutions in the time domain may then be obtained with the aid of the Fourier transform. However, this series form of the frequency-domain solution contains the frequency only implicitly through the solutions of the dispersion relation and so the Fourier transform must be carried out numerically. The main result of the present work is an integral form for the time-domain solution corresponding to an impulsive motion of the cylinder. This may be used to construct time-domain solutions for more general motions of the cylinder without the need for numerical transforms between the frequency and time domains. To illustrate the solution procedure, in what is perhaps a more familiar context, a new integral form for the frequency-domain solution is first obtained. In the conventional frequency-domain solution procedure, eigenfunctions are constructed that satisfy the homogeneous free-surface condition and then these are combined to satisfy the non-homogeneous body boundary condition. Here, the infinite-frequency limit of the solution is first subtracted out which leads to a new boundary-value problem with a non-homogeneous free-surface condition and a homogeneous body
2 FREQUENCY DOMAIN For the case of steady time-harmonic oscillations of angular frequency ~vand velocity amplitude U persisting for all time, the velocity potential may be written (r, 0, z, t) = Re{ UfA(r, 0, r)e -'~t }
© Crown copyright (1995). 347
(1)
P. McIver
348
The time-dependent potential ¢ must satisfy the freesurface boundary condition
satisfying the boundary conditions
0~
0f~
Oz
0O _- - Kq5 o n , 7 = 0
(2)
- -
Oz
where the frequency parameter K = w2/g and g is the acceleration due to gravity, the bed condition
00
O~" = 0 on z = - h
(3)
Or
cos 0 on r = a
(4)
and the radiation condition
rl/2(O~rr-ikO )
0O
Or
0 on z = - h
(15)
0 on r
(16)
a
and
(5)
kh
In what follows, the limiting potential as the frequency tends to infinity is required. This 'infinitefrequency' potential will be denoted by 9t(r, 0, z) and will satisfy the above conditions in the limit K ~ oc so that the free-surface condition (2) is replaced by (7)
and the radiation condition (5) by Q ---+0 as r ~ oc
(8)
Separation of variables and satisfaction of all boundary conditions except that on the body gives oc
f~ = cosO Z A,Kl(p,r ) cospn(z +
h)
Oz K~
where (10)
and Kl denotes the modified Bessel function of the second kind and order one. The body boundary condition now requires oc
h) = 1
on z
0
R(r) = -2 ~7" K,(p,r) n~=l pnhK'l ( Pna)
(18)
(19)
In contrast to 4), which must satisfy a homogeneous free-surface condition and a non-homogeneous body boundary condition, the new potential ~ is required to satisfy a non-homogeneous free-surface condition and a homogeneous body boundary condition. The solution proceeds by separation of variables, first noting that the body boundary condition is satisfied identically by
C,(qr) = J,(qr) Y~(qa) - Y,(qr)J~(qa)
(20)
where Jn and 1I, denote Bessel functions and a prime denotes differentiation with respect to the function argument. Thus
dc,(qr)
Z A'pnK'I (Pna) c°sp" (z +
R(r)cosO
(9)
n=l
p, = ( 2 n - 1)Tr/2h
(17)
where (6)
f~ = 0 on z = 0
---* 0 as r ---~oc
The solution for It given by (9) and (12) may be substituted into the free-surface boundary condition (14) to obtain
0~, ---+0 as r ---* oc
Here k is the root of the dispersion relation K = k tanh
Oz
(14)
rl/2(O-~F--ikO)
the body condition
oo
o~
K~ = -0-f on z = 0
(11)
= 0 on r = a
(21)
A solution for ~ satisfying all except the free-surface and radiation conditions is = cos 0
r A(q)C 1(qr) cosh q(z + h)dq
j0
(22)
The unknown function A(q) is found by applying a modified version of Weber's integral theorem (Hunt & Baddour, 2 eqn (54)) which states that
n=l
and the expansion coefficients follow from the orthogonality of the set {cosp,(z + h); n = 1,2,...} over the depth so that 2 ( - 1 ) "+~
An p2nhK,l(p,a),
n = 1,2,...
(12)
To obtain the solution for arbitrary frequency consider the harmonic function ~/,= ~ - Q
(13)
f(r)=Io
Cn(q(qa)12 r)lHl qoq J ] f Cn(qr')r'fl(r')dr'.
(23)
where H1 denotes a Hankel function of the first kind, provided r l/2f(r) is integrable over [a, oc). Substitution of (22) into the free-surface condition (18) and application of (23) gives A (q) (K cosh qh =
q sinh qh)
q , lim [o~ IH[(qa)[ ~ ~ o J , . ,
Cl(qr)rR(r)dr
(24)
Forced horizontal oscillations of vertical cylinder where e has been introduced to allow the interchange of the integration and summation when (19) is substituted into (24). Now from Abramowitz & Stegun, 3 eqn (11.3.29), lim [~ Cl(qr)K~(pr)rdr -
e~O Ja+e
P
!
7rq(p2 + q2) K, (pa)
generated by a cylinder started from rest; the formulation follows the description of the standard time-domain theory as given, for example, in Section 7.11 of Mei. 5 The fluid response to an impulsive velocity V(r)6(t - r) imposed at time t = r is described by the potential • (r, 0, z, t) = V(r)[Q(r, 0, z)6(t - "r)
(25) so that A (q) (K cosh qh - q sinh qh) 4
oo
_ 7ch[Hf(qa)l2 ~_lp2 1 _ + q2
2tanhqh
Irq[H~(qa)12
(26)
where the series has been summed using eqn (1.421(2)) of Gradshteyn & Ryzhik. 4 The solution for ¢ is therefore 2 cos 0
+ p(r,
~0~
Ot
2 ( - 1)'Kl (p,r) cosp,(z +
K{(pna)p 2h
(28)
where {kn; n = 1,2,...} are the real, positive roots of
K = -kn tan k, h (
sin 2k, h)
0f~
OF
Ot 2 ~ - g ~ z = 0 ° n z = 0 f ° r 0 < t < ° °
OF Oz
0onz
-h
(32) (33) (34)
(35)
and OF
Or
0 on r = a
F = cos 0
h)]
sinh kh H1 (kr) cosh k(z + h) + ~(~a)~ /
(31)
(36)
The solution proceeds in a similar way to the frequencydomain solution given in the previous section. The body and bed conditions (35)-(36) are satisfied by taking
~ = c°sO{~=l [sinknh K'(knr) (kna)kc°skn(z n2hN n2 +
+
r)]
g-~z on z = 0 at t = 0
02F (27)
where the path of integration has been chosen to run beneath the pole at the real root of K = q tanh qh in order to satisfy the radiation condition. This can be confirmed by a standard residue calculus technique as described, for example, by Mei, 5 p. 379. Indeed this solution may be written in series form as
-
F = 0 on z = 0 at t = 0 OF
tanh qh C l (qr) cosh q(z + h) ]H~(qa)lZq(Kcoshqh- qsinhqh) dq
O, z , t - r ) H ( t
The first term on the right-hand side of (31) describes the initial pressure impulse transmitted throughout the fluid and the second term describes the subsequent fluid motion. Here 6(t) is the delta function, H(t) is the Heaviside step function and F(r, 0,z, t) is a harmonic function satisfying the initial and boundary conditions
71"
x
349
(29) (30)
f0o A(q, t)Cl (qr) cosh q(z + h) dq
(37)
where C1 is defined in eqn (20). The free-surface condition (34) is satisfied provided d2A dt 2 + W2A = 0
(38)
which has a general solution
A = a(q) cos Wt + ~(q) sin Wt
(39)
where and k0 = - i k . When combined with the infinitefrequency solution 9t this gives the 'well-known' form of the frequency-domain solution for radiation by a cylinder as presented in Section 6.4 of Dean & Dalrymple. 1 For the purpose of obtaining solutions in the time domain by Fourier transform, the advantage of (27) is that the frequency appears explicitly through K = w2/g. In the form (28), the frequency appears only implicitly through the solutions of (29).
3 IMPULSIVE MOTION Attention
is now turned to the transient motion
W 2 = gq tanh qh
(40)
The initial conditions (32)-(33) require that
Jo
~a(q) C 1(qr) cosh qh dq = 0
(41)
and
lo
Zfl(q) W(q)C1 (qr) cosh qh dq = -gR(r)
(42)
where R(r) is defined in eqn (19). It follows from the modified version of Weber's integral theorem, eqn (23), and following similar manipulations as carried out in eqns (24)-(26) that
a(q) = 0
(43)
P. Mclver
350 and 2g tanh q h
7rqWc°shqhIH~(qa)[ 2
/3(q) =
(44)
This now has given
q5 = ~V(t) +
2gc_os0 [X sin Wttanhqh Cl(qr) coshq(z + h) dq 7r Jo qWcoshqh[H~(qa)[ 2
F -
(45) This is related to the Fourier transform of the frequency-domain solution through lI~ F = ~
~e-i~'tdco
where ~ is given by (27) and may be written in the form _ 2gcos0 ~ ¢ tanhqh G(qr) coshq(z + h)
Jo q T J2 ~ ~
)
~
i
2dq (47)
To recover (45) from (46), co is replaced by co + ie in (47), so that the pole on the real q axis is moved into the upper half of the complex q plane, and then the result
fOO
lim e i~t 27r (~0 _~ (co+ie) 2 - W 2dco= - ~ s i n
Wt
vV
F(t - 7-) sin aJ~- d~-
(51)
0
where f~ is given by (9) and (12) and F is given by (45). The result is q5 = I)[(~ + Re~) sin~t + ~], t> 0 (52) where R e ~ is the principal value equivalent of (27) and qj _ 2cocos sin Wt 7r
(46)
c3~
;
which is turned on at t = 0; here H(t) is the Heaviside step function. From (49) and (50) the potential for the resulting disturbance is
0
W
tanh qh C 1(qr) cosh q(z + h) x iH[(qa)lZq(Kcoshqh _ qsinhqh)dq
(53)
It is expected that purely time-harmonic motion will be achieved after a long time, that is q~ ~ Re{ii20e i~'t} = /2[(f~ + Re~) sin a;t - Im~ cos cot] as t ~ oe (54) where q~is defined in eqn (1) and I m + is the contribution to (27) from the integration around the pole and is given by
(48)
sinh kh Ct (kr) cosh k(z + h) cos 0
(55)
]H~(ka)[2kZhN2 °
Im~ = is used.
This limiting behaviour (54) can be confirmed by writing ~, which appears in (52), as 4 R E S P O N S E TO A SINUSOIDAL FORCING
According to Mei, 5 p. 374, the response to a continuous velocity V(t) imposed on the cylinder is given by
= av(t) +
r ( t - ,-)
(49)
Consider, for example, the sinusoidal forcing
V(t) = H(t)Vsincot
(50)
i0
5 (I)
D O
coc°S0Im.~ ~ ~ - -~c eiwtW
~-
tanh qh C1 (qr) cosh q(z + h) x ]H[(qa)[Zq(Kcoshqh _ qsinhqh) dq
(56)
and shifting the path of integration into the upper half of the complex q plane by allowing q to have a small positive imaginary part. The limiting behaviour arises from the poles on the real axis while the remaining part of the integral decays to zero as t ~ oc. The details are straightforward but messy and so are omitted here. The solution found above may be used, for example, to calculate the free-surface elevation at an arbitrary point or the hydrodynamic force on the cylinder. The latter is used for purposes of illustration. The timedependent force in the direction 0 = 0 is
0
F(t) = - p
JI ~°°- c o s 0 d S
(57)
s -5
-I0
where S denotes the wetted surface of the cylinder. Carrying out the integrations gives .............................. 0 2 4 6 8
i0
12
F(t) pTra3co(/
2a Kl (p,a) cos cot + -h = (pna)3Kl(p.a) 7ra2
T
Fig. 1. Non-dimensional force F(t)/(praawlT"), due to the sinusoidal forcing (50), versus non-dimensional time T = wt for a/h = 0"1, Ka = 0'5: (o • • •) time-domain solution; ( - - - ) frequency-domain solution.
fro (cos Wt - cos cot) tanh qh C 1(qa)
x
]n~(qa)12q2(K~--_- ~
dq(58)
This expression is readily evaluated by standard
Forced horizontal oscillations of vertical cylinder numerical techniques and a sample calculation is given in Fig. 1 showing the rapid approach to the frequencydomain solution.
REFERENCES 1. Dean, R. G. & Dalrymple, R. A., Water Wave Mechanics for Engineers and Scientists. World Scientific, 1991.
351
2. Hunt, J. N. & Baddour, R. E., Nonlinear standing waves bounded by cylinders. Quarterly Journal of Mechanics and Applied Mathematics, 33, 357-371. 3. Abramowitz, M. & Stegun, I. A., Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC, 1964. 4. Gradshteyn, I. S. & Ryzhik, I. M., Tables of Integrals, Series and Products. Academic Press, New York, 1980. 5. Mei, C. C., The Applied Dynamics of Ocean Surface Waves. John Wiley, New York, 1983.