Scattering of internal waves in a two-layer fluid flowing through a channel with small undulations |
| |
| Authors: | Smrutiranjan Mohapatra Swaroop Nandan Bora |
| |
| Institution: | (1) Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, 781039, India |
| |
| Abstract: | Using two-dimensional linear water wave theory, we consider the problem of normal water wave (internal wave) propagation over
small undulations in a channel flow consisting of a two-layer fluid in which the upper layer is bounded by a fixed wall, an
approximation to the free surface, and the lower one is bounded by a bottom surface that has small undulations. The effects
of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed
to calculate the first-order corrections to the velocity potentials in the two-layer fluid by using Green’s integral theorem
in a suitable manner and the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions
for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having
the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient
of twice the interface wave number and the ripple wave number. When this quotient approaches one, the theory predicts a resonant
interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples.
High reflection of the incident wave energy occurs if this number is large. Again, when a patch of sinusoidal ripples having
two different wave numbers for two consecutive stretches is considered, the interaction between the bed and the interface
near resonance attains in the neighborhood of two (singular) points along the x-axis (when the ripple wave number of the bottom undulation become approximately twice as large as the interface wave number).
The theoretical observations are presented in graphical form. |
| |
| Keywords: | Two-layer fluid Wave scattering Reflection coefficient Transmission coefficient Linear water wave theory Green’ s function Perturbation technique |
| 本文献已被 SpringerLink 等数据库收录！ |
|
