The scattering of a scalar wave by a semi-infinite rod of circular cross section

1955 ◽  
Vol 247 (934) ◽  
pp. 499-528 ◽  
Keyword(s):  
Boundary Condition ◽  
Circular Cross Section ◽  
Scattering Coefficient ◽  
Wave Number ◽  
Infinite Cylinder ◽  
Angle Of Incidence ◽  
Pressure Amplitude ◽  
Average Pressure ◽  
Scalar Wave ◽  
Harmonic Scalar

The form of the exact solution for the scattering of a plane harmonic scalar wave by a semi-infinite circular cylindrical rod of diameter 2 a is found when the boundary condition is u — 0 or du/dv — 0,where u represents the scalar field and v is the normal to the rod. When the angle of incidence is n, i.e. the angle between the direction of propagation of the incident wave and the normal (out of the rod) to the end is , the average pressure amplitude on the end of the rod and the scattering coefficient are found for the boundary condition 0. Graphs are given showing the behaviour of these quantities for the range 0 < < 10, where k is the wave-number. When ka reaches 10, the quantities have almost become constant. For small values of ka the scattering coefficient is shown to be {ka)2', it appears from the numerical results that this is, in fact, a fairly close approximation for ka <2. It is further shown that the average pressure amplitude on the end for other angles of incidence is approximately the product of the average pressure amplitude for an angle of incidence of n and the amplitude of the symmetric mode (ka < 3-83) which the incident field would produce inside a hollow semi-infinite cylinder occupying the same position as the rod. When the boundary condition is u — 0 and ka is small it is proved that the scattered field is the same as that due to a semi-infinite hollow cylinder longer by an amount 0-la approximately. A similar result does not hold for the boundary condition 0. The theory is extended to the case when a pressure pulse falls on a circular rod. It is found that the pressure on the end drops almost to its final value in the time taken for a wave to travel the diameter of the rod, and that the average pressure during this process is given, at time by (0-9154-0-745(2— a0tja)2}I approximately, where a0 is the speed of sound. Tables, in the range 0(0-25)10 of ka, of the ‘split’ functions which arise in connexion with a semi-infinite cylinder are given.

scholarly journals Dynamic Stress and Displacement in an Elastic Half-Space with a Cylindrical Cavity

2011 ◽  
Vol 18 (6) ◽  
pp. 827-838 ◽  
Author(s):  
İ. Coşkun ◽  
H. Engin ◽  
A. Özmutlu
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Half Space ◽  
Cylindrical Cavity ◽  
Circular Cross Section ◽  
Elastic Half Space ◽  
Least Square ◽  
Polar Coordinates ◽  
Wave Number ◽  
Forcing Function

The dynamic response of an elastic half-space with a cylindrical cavity in a circular cross-section is analyzed. The cavity is assumed to be infinitely long, lying parallel to the plane-free surface of the medium at a finite depth and subjected to a uniformly distributed harmonic pressure at the inner surface. The problem considered is one of plain strain, in which it is assumed that the geometry and material properties of the medium and the forcing function are constant along the axis of the cavity. The equations of motion are reduced to two wave equations in polar coordinates with the use of Helmholtz potentials. The method of wave function expansion is used to construct the displacement fields in terms of the potentials. The boundary conditions at the surface of the cavity are satisfied exactly, and they are satisfied approximately at the free surface of the half-space. Thus, the unknown coefficients in the expansions are obtained from the treatment of boundary conditions using a collocation least-square scheme. Numerical results, which are presented in the figures, show that the wave number (i.e., the frequency) and depth of the cavity significantly affect the displacement and stress.

An approximate method in high-frequency scattering

1962 ◽  
Vol 58 (4) ◽  
pp. 662-670
Author(s):  
A. Sharples
Keyword(s):  
Boundary Condition ◽  
Circular Cylinder ◽  
Integral Representation ◽  
Sound Wave ◽  
High Frequency ◽  
Correction Term ◽  
Scattering Coefficient ◽  
Extended Form ◽  
Geometrical Acoustics ◽  
Plane Sound

ABSTRACTThe diffraction of a high-frequency plane sound wave by a circular cylinder is investigated when the boundary condition on the cylinder is expressed by means of an equation of the form The special feature of this investigation is that an extended form of the Kirchhoff-Fresnel theory of diffraction is used to find an integral representation for the scattering coefficient. In order to avoid the complicated analysis which would be necessary to evaluate the integrals concerned, the more natural geometrical acoustics approach is used to find the first correction term in the scattering coefficient. Numerical results are given for large and small values of the impedance Z.

THE SCATTERING OF A PLANE WAVE BY A ROW OF SMALL CYLINDERS

1960 ◽  
Vol 38 (2) ◽  
pp. 272-289 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Boundary Condition ◽  
Plane Wave ◽  
Integral Equations ◽  
Cross Sections ◽  
Simultaneous Equations ◽  
Wave Number ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Higher Order Terms ◽  
Scattering Cross Sections

Consideration is given to the scattering of a plane wave by N cylinders equispaced in a row. The problems associated with scatterers, both "soft" and "hard" in the acoustical sense, are treated. An application of Green's theorem together with the appropriate boundary condition on the cylinders leads to a set of simultaneous integral equations in the unknown function on the cylinders.Solutions in the form of series in powers of a small parameter δ (essentially the ratio of cylinder dimension to wavelength) are assumed. In the case of elliptic cylinders, the integral equations are reduced to sets of linear algebraic equations. Only for the first term in the solution for "soft" cylinders is it necessary to solve N simultaneous equations in N unknowns; all other equations involve essentially only one unknown. Far-fields and scattering cross sections are calculated. The case of two "soft" cylinders is given particular attention.Conditions for justification of the neglect of higher-order terms are discussed. It is found that all terms but the first (in either problem) may be neglected if [Formula: see text] and (N–1)/(ka) is sufficiently small. (Here a is the spacing between centers of adjacent cylinders, and k is the wave number.) For this reason these solutions are most useful when the number of cylinders is small.

Scattering of surface waves obliquely incident on a fixed half immersed circular cylinder

1984 ◽  
Vol 96 (2) ◽  
pp. 359-369 ◽  
Author(s):  
B. N. Mandal ◽  
S. K. Goswami
Keyword(s):  
Integral Equation ◽  
Surface Water ◽  
Circular Cylinder ◽  
Water Waves ◽  
Wave Number ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Obliquely Incident ◽  
Surface Water Waves

AbstractThe problem of scattering of surface water waves obliquely incident on a fixed half immersed circular cylinder is solved approximately by reducing it to the solution of an integral equation and also by the method of multipoles. For different values of the angle of incidence and the wave number the reflection and transmission coefficients obtained by both methods are evaluated numerically and represented graphically to compare the results obtained by the respective methods.

scholarly journals Logarithmic correlation functions for critical dense polymers on the cylinder

2019 ◽  
Vol 7 (3) ◽  
Author(s):  
Alexi Morin-Duchesne ◽  
Jesper Jacobsen
Keyword(s):  
Boundary Condition ◽  
Correlation Functions ◽  
Central Charge ◽  
Infinite Cylinder ◽  
Operator Product ◽  
Partition Functions ◽  
Loop Model ◽  
Perfect Agreement ◽  
Single Node ◽  
Highest Weight

We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter nn. In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity \alpha \in (0,\infty)α∈(0,∞). These correlators are defined as ratios Z(x)/Z_0Z(x)/Z0 of partition functions, where Z_0Z0 is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For Z(x)Z(x), the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions 11 and xx where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite nn using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n\to \inftyn→∞ are expressed as simple integrals that depend on the scaling parameter \tau = \frac {x-1} n \in (0,1)τ=x−1n∈(0,1). For small \tauτ, the leading behaviours are proportional to \tau^{1/4}τ1/4, \tau^{1/4}\log \tauτ1/4logτ, \log \taulogτ and \log^2 \taulog2τ. We interpret the lattice results in terms of ratios of conformal correlation functions. We assume that the corresponding boundary changing fields are highest weight states in irreducible, Kac or staggered Virasoro modules, with central charge c=-2c=−2 and conformal dimensions \Delta = -\frac18Δ=−18 or \Delta = 0Δ=0. With these assumptions, we obtain differential equations of order two and three satisfied by the conformal correlation functions, solve these equations in terms of hypergeometric functions, and find a perfect agreement with the lattice results. We use the lattice results to compute structure constants and ratios thereof which appear in the operator product expansions of the boundary condition changing fields. The fusion of these fields is found to be non-abelian.

Approximate methods in high-frequency scattering

1957 ◽  
Vol 239 (1218) ◽  
pp. 338-348 ◽  
Keyword(s):  
Boundary Condition ◽  
Circular Cylinder ◽  
High Frequency ◽  
Fourier Transforms ◽  
Scattering Coefficient ◽  
Complete Agreement ◽  
Approximate Methods ◽  
Exact Theory ◽  
Geometrical Acoustics

Two approximate methods for deriving the high-frequency scattering coefficient of twodimensional obstacles without edges are described. In the first method a simple field which satisfies approximately the boundary condition near the points of glancing incidence is found. Elsewhere the geometrical acoustics field is used. The scattering coefficient is about 7 % in error. In the second method Fourier transforms are employed to find a field which satisfies the boundary condition over a wider region. This leads to results which, for the circular cylinder, are in complete agreement with those of the exact theory.

On the Viscous Flow Through a Long Pipe With Non-Constant Pressure Boundary Condition

2018 ◽  
Vol 73 (7) ◽  
pp. 639-644 ◽  
Author(s):  
Eduard Marušić-Paloka ◽  
Igor Pažanin
Keyword(s):  
Boundary Condition ◽  
Asymptotic Analysis ◽  
Small Parameter ◽  
Cross Section ◽  
Constant Pressure ◽  
Viscous Incompressible Fluid ◽  
Fluid Velocity ◽  
Circular Cross Section ◽  
Pressure Boundary Condition ◽  
Flow Through

AbstractWe investigate the flow of a viscous incompressible fluid through a straight long pipe with a circular cross section. The flow is driven by the prescribed pressures at the pipe’s ends, where pressure p0 on the pipe’s entry is assumed to be non-constant. Using asymptotic analysis with respect to the small parameter (being the ratio between the pipe’s radius and its length), we replace the non-constant pressure boundary condition with the effective one governing the macroscopic flow. We also derive the optimal boundary pressure p0 such that the fluid velocity through a pipe is maximal.

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