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On some integrals of Glauert’s type
Applied Mathematics Letters 18 (2005) 631–633 www.elsevier.com/locate/aml
On some integrals of Glauert’s type Yves-Marie Scolan∗, Jacques Louis Ecole Supérieure d’Ingènieurs de Marseille, Technopôle Château Gombert, 13451 Marseille Cedex 20, France Received 1 January 2004; accepted 1 January 2004
Abstract Some principal value integrals of Glauert’s type are calculated analytically. These integrals appear when dealing with the hydrodynamic impact problem of a two-dimensional wedge onto a liquid free surface in the potential theory. © 2004 Elsevier Ltd. All rights reserved. Keywords: Analytic integration; Hydrodynamics; Glauert’s integrals
There are many situations in aerodynamics and in hydrodynamics as well, where the so-called Glauert’s integral must be calculated. Its usual expression is π sin |k|α cos kθ dθ = π , (1) PV sin α 0 cos θ − cos α where PV means the Principal Value of the integral. As recalled in [1], the usual way to calculate these integrals is to perform the integration of the following function in the complex plane z f (z) =
zn , z 2 − 2z cos α + 1
(2)
where z describes the unit circle z = eiθ . When studying the impact of an elastic two-dimensional beam or wedge onto a liquid free surface within the Wagner [2] approach (potential theory), we face the usual “lifting” problem which must be turned into a “thickness” problem (see [3, pp. 180–184]). Applications ∗ Corresponding author. Tel.: 33 4 91 05 44 29; fax: +33 4 91 05 46 15.
E-mail addresses: [email protected] (Y.-M. Scolan), [email protected] (J. Louis). 0893-9659/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2004.01.008
632
Y.-M. Scolan, J. Louis / Applied Mathematics Letters 18 (2005) 631–633
of such problems are described in [4]. As a result in the frame of a modal-based method, we end up with the computation of the following integral π/2 π/2 sg(θ) sin(2k + 1)θ sin(2k + 1)θ Ik (α) = PV dθ = −4 sin αPV dθ, (3) sin θ − sin α cos 2θ − cos 2α −π/2 0 where α ∈ [0, π/2] and k ≥ 0. Integral (3) looks like a Glauert’s integral but its calculation is more delicate due to the fact that the integrand has no parity and also the interval of integration is not the whole unit circle. It is difficult to calculate directly the integral Ik (α) by using a contour method in the complex plane, unless the same artifice – as presented here – is utilized. An alternate calculation of integral (3) where only a finite summation is necessary is proposed here. As a starting point the following identity is introduced sin(2k + 1)θ = sin(2k − 1)θ + 2 sin θ cos 2kθ, to express Ik (α) as a recursive scheme Ik (α) = Ik−1 (α) − 8Mk (α) sin α,
(4)
π/2
with Mk (α) = PV 0
sin θ cos 2kθ dθ, cos 2θ − cos 2α
(5)
and the initial value I0 (α) = −4M0 (α) sin α.
(6)
In order to calculate Mk (α), the key identity is used 1 ∞ 4 2 B cos 2α, B = sg(α) sin α = 1 π =0 1 − 42
=0
(7)
> 0.
By using Glauert integrals, Mk (α) is written as a series Mk (α) = M0 (α) cos 2kα + with Sc (α) , M0 (α) = − cos α
and
k π B sin 2(k − )α, 2 sin 2α =0
(8)
sin α 1 Sc (α) = log . 2 1 − cos α
(9)
Knowing that the double summation reduces according to k i i=1 m=1
Bi−m
k−1 sin 2mα sin(k − m + 1)α sin(k − m)α =2 , Bm cos α sin 2α m=0
finally the original integral has the form
k sin α sin(2k + 1)α sin2 α sin 2α log − 2π + , Bk− Ik (α) = 2 cos α 1 − cos α sin α 2 cos α =1
(10)
k ≥ 1.
(11)
Y.-M. Scolan, J. Louis / Applied Mathematics Letters 18 (2005) 631–633
633
Fig. 1. Variation of Ik (α) in the interval α ∈ [0, π/2] and k ∈ {0, 1, 2, 3, 4, 5} (up), k ∈ {6, 7, 8, 9, 10} (below).
The integral I0 (α) is given by Eq. (6). The limiting values of Ik (α) at the origin α = 0 and at α → easily extracted. Fig. 1 shows the variation of Ik (α) for the first eleven values of k.
π 2
are
References [1] [2] [3] [4]
F. Quori, A new computation technique of Glauert’s integrals, Applied Mathematics Letters 8 (5) (1995) 7–9. H. Wagner, Über Stoss- und Gleitvorgänge an der Oberfläche von Flüssigkeiten, ZAMM 12 (1932) 193–215. J.N. Newman, Marine Hydrodynamics, MIT Press, 1977. J. Kvålsvold, Hydroelastic modelling of wetdeck slamming on multihull vessels, Ph.D. Thesis, University of Trondheim, 1994.
On some integrals of Glauert’s type Yves-Marie Scolan∗, Jacques Louis Ecole Supérieure d’Ingènieurs de Marseille, Technopôle Château Gombert, 13451 Marseille Cedex 20, France Received 1 January 2004; accepted 1 January 2004
Abstract Some principal value integrals of Glauert’s type are calculated analytically. These integrals appear when dealing with the hydrodynamic impact problem of a two-dimensional wedge onto a liquid free surface in the potential theory. © 2004 Elsevier Ltd. All rights reserved. Keywords: Analytic integration; Hydrodynamics; Glauert’s integrals
There are many situations in aerodynamics and in hydrodynamics as well, where the so-called Glauert’s integral must be calculated. Its usual expression is π sin |k|α cos kθ dθ = π , (1) PV sin α 0 cos θ − cos α where PV means the Principal Value of the integral. As recalled in [1], the usual way to calculate these integrals is to perform the integration of the following function in the complex plane z f (z) =
zn , z 2 − 2z cos α + 1
(2)
where z describes the unit circle z = eiθ . When studying the impact of an elastic two-dimensional beam or wedge onto a liquid free surface within the Wagner [2] approach (potential theory), we face the usual “lifting” problem which must be turned into a “thickness” problem (see [3, pp. 180–184]). Applications ∗ Corresponding author. Tel.: 33 4 91 05 44 29; fax: +33 4 91 05 46 15.
E-mail addresses: [email protected] (Y.-M. Scolan), [email protected] (J. Louis). 0893-9659/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2004.01.008
632
Y.-M. Scolan, J. Louis / Applied Mathematics Letters 18 (2005) 631–633
of such problems are described in [4]. As a result in the frame of a modal-based method, we end up with the computation of the following integral π/2 π/2 sg(θ) sin(2k + 1)θ sin(2k + 1)θ Ik (α) = PV dθ = −4 sin αPV dθ, (3) sin θ − sin α cos 2θ − cos 2α −π/2 0 where α ∈ [0, π/2] and k ≥ 0. Integral (3) looks like a Glauert’s integral but its calculation is more delicate due to the fact that the integrand has no parity and also the interval of integration is not the whole unit circle. It is difficult to calculate directly the integral Ik (α) by using a contour method in the complex plane, unless the same artifice – as presented here – is utilized. An alternate calculation of integral (3) where only a finite summation is necessary is proposed here. As a starting point the following identity is introduced sin(2k + 1)θ = sin(2k − 1)θ + 2 sin θ cos 2kθ, to express Ik (α) as a recursive scheme Ik (α) = Ik−1 (α) − 8Mk (α) sin α,
(4)
π/2
with Mk (α) = PV 0
sin θ cos 2kθ dθ, cos 2θ − cos 2α
(5)
and the initial value I0 (α) = −4M0 (α) sin α.
(6)
In order to calculate Mk (α), the key identity is used 1 ∞ 4 2 B cos 2α, B = sg(α) sin α = 1 π =0 1 − 42
=0
(7)
> 0.
By using Glauert integrals, Mk (α) is written as a series Mk (α) = M0 (α) cos 2kα + with Sc (α) , M0 (α) = − cos α
and
k π B sin 2(k − )α, 2 sin 2α =0
(8)
sin α 1 Sc (α) = log . 2 1 − cos α
(9)
Knowing that the double summation reduces according to k i i=1 m=1
Bi−m
k−1 sin 2mα sin(k − m + 1)α sin(k − m)α =2 , Bm cos α sin 2α m=0
finally the original integral has the form
k sin α sin(2k + 1)α sin2 α sin 2α log − 2π + , Bk− Ik (α) = 2 cos α 1 − cos α sin α 2 cos α =1
(10)
k ≥ 1.
(11)
Y.-M. Scolan, J. Louis / Applied Mathematics Letters 18 (2005) 631–633
633
Fig. 1. Variation of Ik (α) in the interval α ∈ [0, π/2] and k ∈ {0, 1, 2, 3, 4, 5} (up), k ∈ {6, 7, 8, 9, 10} (below).
The integral I0 (α) is given by Eq. (6). The limiting values of Ik (α) at the origin α = 0 and at α → easily extracted. Fig. 1 shows the variation of Ik (α) for the first eleven values of k.
π 2
are
References [1] [2] [3] [4]
F. Quori, A new computation technique of Glauert’s integrals, Applied Mathematics Letters 8 (5) (1995) 7–9. H. Wagner, Über Stoss- und Gleitvorgänge an der Oberfläche von Flüssigkeiten, ZAMM 12 (1932) 193–215. J.N. Newman, Marine Hydrodynamics, MIT Press, 1977. J. Kvålsvold, Hydroelastic modelling of wetdeck slamming on multihull vessels, Ph.D. Thesis, University of Trondheim, 1994.