Algorithm to Calculate a Large Number of Roots of the Cross-Product of Bessel Functions

Sorolla, Edén; Mosig, Juan R.; Mattes, Michael

doi:10.1109/tap.2012.2231929

Cited by 9 publications

(6 citation statements)

References 26 publications

(27 reference statements)

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“…Here k 𝑝,𝑠 is the 𝑠th positive zero of 𝐽 𝑝 -the derivative of the Bessel function 𝐽 𝑝 with integer or half-integer 𝑝 ≥ 0 (again our notation differs from that in [14] for 𝑚 = 0). The twenty initial values in the increasing order are as follows: k1/2,1 = 1.1655..., k1,1 = 1.8411..., k3/2,1 = 2.4605..., k2,1 = 3.0542..., k5/2,1 = 3.6327..., k0,1 = 3.8317..., k3,1 = 4.2011..., k1/2,2 = 4.6042..., k7/2,1 = 4.7621..., k4,1 = 5.3175..., k1,2 = 5.3314..., k9/2,1 = 5.8684..., k3/2,2 = 6.0292..., k5,1 = 6.4156..., k2,2 = 6.7061..., k11/2,1 = 6.9597..., k0,2 = 7.0155..., k5/2,2 = 7.3670..., k6,1 = 7.5012..., k1/2,3 = 7.7898... (18) It should be noted that 𝑘 1,1 -the first value in (10) -coincides with k1,1 which is second here, whereas the eighth value in (10) is only fifteenth here. It is clear that every eigenvalue k2 𝑚/2,𝑠 with even 𝑚 is also an eigenvalue of problem ( 8), but for eigenvalues with odd 𝑚 this is not true.…”

Section: Vertical Circular Container With a Radial Bafflementioning

confidence: 99%

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Sloshing in vertical cylinders with circular walls: the effect of radial baffles

Kuznetsov¹,

Motygin²

2021

Preprint

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The behaviour of sloshing eigenvalues and eigenfunctions is studied for vertical cylindrical containers that have circular walls and constant (possibly infinite) depth. The effect of breaking the axial symmetry due to the presence of radial baffles is analysed. It occurs that the lowest eigenvalues are substantially smaller for containers with baffles going throughout the depth; moreover, all eigenvalues are simple in this case. On the other hand, the lowest eigenvalue has multiplicity two in the absence of baffle. It is shown how these properties affect the location of maxima and minima of the free surface elevation and the location of its nodes.

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Section: Vertical Circular Container With a Radial Bafflementioning

confidence: 99%

“…For the properties of the cross-product in the left-hand side, see, e.g., [17,18] and references therein. A set of values 𝑘 • 𝑚,𝑠 , computed for 𝜌 = 1/2, is shown in Fig.…”

Section: Vertical Annular Container Without Bafflementioning

confidence: 99%

Sloshing in vertical cylinders with circular walls: the effect of radial baffles

Kuznetsov¹,

Motygin²

2021

Preprint

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“…with 0 denoting differentiation with respect to r. These zeros are evaluated based on a method devised by Sorolla et al [22]. The coefficients A mn and the angular frequency x appearing in Eq.…”

Section: Using Boundary Conditions (3) and (4) The Velocity Potentiamentioning

confidence: 99%

Linear sloshing frequencies in the annular region of a circular cylindrical container in the presence of a rigid baffle

Choudhary

Bora

2017

Sādhanā

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Sloshing in any type of container may invite instability to it. If some part of the free liquid surface in the annular region of a specially designed circular cylindrical container is covered with an annular baffle, the natural frequencies and the response of the liquid in the container undergo a drastic change. A partly covered free surface shifts the natural frequency above and away from the control frequency of the vehicle, in which the liquid-filled container is placed, which results in the reduction of sloshing mass participating in the dynamic motion of the system. The fundamental natural frequency of an inviscid and incompressible liquid is determined for increasing width of the baffle that is attached to the outer tank wall on the free surface. It is observed that by increasing the width of the baffle, natural frequencies can be significantly increased. Investigations are also carried out for different values of Bond number, which depicts different states of surface tension, and for varying values of the part of the radius in the fluid region. It is also observed that by increasing the fluid height inside the container, the natural frequencies can be increased, which results in reduction of sloshing.

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“…The respective eigenvalues are tabulated (Bauer 1964 ; Bridge and Angrist 1962 ; Truell 1943 ) or they can be found by using numerical routines (Sorolla et al. 2013 ) and empirical approximations (Laslett and Lewish 1962 ). The computer algebra system MATHEMATICA ® (Wolfram Research, Inc., Champaign, IL, USA, Wolfram 1999 ) provides the command for numerical computation of the eigenvalues.…”

Section: Cylindrical Bessel Functionsmentioning

confidence: 99%

Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions

et al. 2015

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The cylindrical Bessel differential equation and the spherical Bessel differential equation in the interval with Neumann boundary conditions are considered. The eigenfunctions are linear combinations of the Bessel function or linear combinations of the spherical Bessel functions . The orthogonality relations with analytical expressions for the normalization constant are given. Explicit expressions for the Lommel integrals in terms of Lommel functions are derived. The cross product zeros and are considered in the complex plane for real as well as complex values of the index and approximations for the exceptional zero are obtained. A numerical scheme based on the discretization of the two-dimensional and three-dimensional Laplace operator with Neumann boundary conditions is presented. Explicit representations of the radial part of the Laplace operator in form of a tridiagonal matrix allow the simple computation of the cross product zeros.

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Algorithm to Calculate a Large Number of Roots of the Cross-Product of Bessel Functions

Cited by 9 publications

References 26 publications

Sloshing in vertical cylinders with circular walls: the effect of radial baffles

Sloshing in vertical cylinders with circular walls: the effect of radial baffles

Linear sloshing frequencies in the annular region of a circular cylindrical container in the presence of a rigid baffle

Orthogonality, Lommel integrals and cross product zeros of linear combinations of Bessel functions

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