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Mathematical Techniques for Wave Interaction with Flexible by Trilochan Sahoo

By Trilochan Sahoo

Mathematical innovations for Wave interplay with versatile Structures is a considerate compilation of many of the mathematical concepts used to house wave constitution interplay difficulties. The ebook emphasizes particular selection of the answer for a category of actual difficulties linked to Laplace- or Helmholtz-type equations fulfilling larger order boundary stipulations with the purposes of the idea of normal and partial differential equations, Fourier research, and more.

Features:

  • Provides a targeted mathematical remedy for gravity wave interplay with floating and submerged versatile structures
  • Highlights answer equipment for a different category of boundary worth difficulties in wave constitution interaction
  • Introduces and expands upon differential equations and the basics of wave constitution interplay problems

This is a perfect guide for naval architects, ocean engineers, and geophysicists facing the layout of floating and/or versatile marine constructions. The book’s underlying mathematical instruments may be simply prolonged to house actual difficulties within the quarter of acoustics, electromagnetic waves, wave propagation in elastic media, and solid‐state physics.

Designed for either the study room and autonomous research, Mathematical thoughts for Wave interplay with versatile Structures allows readers to understand and follow the mathematical instruments of wave constitution interplay examine to their very own work.

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Additional resources for Mathematical Techniques for Wave Interaction with Flexible Structures

Sample text

Hereafter, we will use the subscript 1 for the upper fluid, and 2 for the lower fluid, respectively. 34) in the respective fluid regions. 43) with Φ2 (x, y, z, t) satisfying the linearised bottom condition as in Eq. 45) at y = H and Eq. 46) as y → ∞. 50) ∂y ∂y ∂t where y = h + ζ2 (x, z, t) is the interface and ζ2 (x, z, t) is called the interfacial elevation. In addition, from Bernoulli’s equation as in Eq. 40), the fluid pressures P1 and P2 at the interface of y = h + ζ2 are given by 1 ∂Φj + ∂t 2 ∂Φj ∂x 2 + ∂Φj ∂y 2 + ∂Φj ∂z 2 − gy = − Pj .

Thus, the shear force and the bending moment at the connecting edge satisfy the edge conditions as given by ([21], [150]) EI −EI ∂2 ∂ζ − ∂2 ∂ζ + − , + ν 2 ζ ± = k55 2 ∂x ∂z ∂x ∂x ∂ ∂2 ∇2xz + (1 − ν) 2 ζ ± = k33 ζ + − ζ − . 82) General introduction 25 However, the limiting case k33 = k55 = 0 corresponds to the case of free-edge conditions, which physically implies that the bending moment and the shear force vanish at that edge of the structure. This case occurs when the study involves the wave scattering by a freely floating elastic plate.

110), the condition on the submerged plate at y = h for 0 < x < ∞, −∞ < z < ∞ is obtained as E2 I2 ∇4xz + N2 ∇2xz + ρp2 d2 ∂2 ∂t2 ∂Φ2 ∂ 2 Φ2 ∂Φ2 = ρ2 −g ∂y ∂t2 ∂y 2 ∂ Φ1 ∂Φ1 . 112) − ρ1 −g ∂t2 ∂y Assuming that the motion is simple harmonic in time with angular frequency ω, the fluid motion is described by the velocity potentials Φj (x, y, z, t) = Re{φj (x, y, z)e−iωt } for j = 1, 2. Further, it is assumed that the deflection of the floating and submerged plates are of the forms 34 Mathematical techniques for wave interaction with flexible structures ζj (x, z, t) = Re{ζj (x, z)e−iωt) }.

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