Convergence of eigenfunction expansions for flexural gravity waves in infinite water depth

In the present paper, point-wise convergence of the eigenfunction expansion to the velocity potential associated with the flexural gravity waves problem in water wave theory is established for infinite water depth case. To take into account the hydroelastic boundary condition at the free surface, a flexible membrane is assumed to float in water waves. In this context, firstly the eigenfunction expansion for the velocity potentials is obtained. Thereafter, an appropriate Green’s function is constructed for the associated boundary value problem. Using suitable properties of the Green’s functions, the vertical components of the eigenfunction expansion is written in terms of the Dirac delta function. Finally, using the property of the Dirac delta function, the convergence of the eigenfunction expansion to the velocity potential is shown.

Modeling Fire Spread in a Real-Time Coupled Atmospheric-Vegetation Fire: An Analytical Approach

The ability to analyse the rate of fire spread outbreak in a real-time coupled Atmospheric-vegetation fire has become increasingly vital as forest fire fighters are building diverse kinds of models to combat the dangers/effects of fire spread across a given fire vicinity. This paper theoretically examines the analysis of fire spread in a real fire environment. A partial differential equations (PDE) governing the phenomenon is presented. The analytical solution of the model is obtained via direct integration and eigenfunction expansion technique, which displays the influence of the parameters involved in the system. The effect of change in parameters such as Frank-Kamenetskii number, Radiation number, Peclet energy number and Activation energy number are presented graphically and discussed. The results obtained show that Frank-Kamenetskii number, Radiation number, Peclet energy number, and Activation energy number all reduced transient state temperature.

A Feynman–Kac formula for magnetic monopoles

We consider the quantum mechanics of a charged particle in the presence of Dirac’s magnetic monopole. Wave functions are sections of a complex line bundle and the magnetic potential is a connection on the bundle. We use a continuum eigenfunction expansion to find an invariant domain of essential self-adjointness for the Hamiltonian. This leads to a proof of the Feynman–Kac formula expressing solutions of the imaginary time Schrödinger equation as stochastic integrals.

Couette Resistance Due to a Sliding Plate Over a Plate with Stripes of Non-Homogeneous Slip

Couette flow with non-homogeneous partial-slip stripes on one plate is studied. Drag and flow rate are found by an efficient eigenfunction expansion and point match method. Longitudinal motion (parallel to the stripes) experiences lower drag than transverse motion. As the gap width between the two plates approaches zero, the drag increases to a finite value if the stripes have partial slip, as comparison to the infinite value for no slip. Analysis of the region near the junction of a perfect stick-slip boundary shows a weak stress singularity while there is no singularity for partial slip junctions.

Wave Interaction With a Pair of Submerged Floating Tunnels in the Presence of An Array of Submerged Porous Breakwaters

In this article, a coupled model is proposed for wave interaction with a pair of submerged floating tunnels in the presence of an array of bottom-standing trapezoidal porous breakwaters. The theory of Sollitt and Cross is adopted to govern the fluid flow inside the porous medium. For constant water-depth, the eigenfunction expansion method is employed, whereas for varying water-depth, the eigenfunction expansion method along with the mild-slope approximation is employed. The solutions, thus derived, are matched at the shared boundaries under defined physical conditions. First, the performance of a single breakwater of impermeable and permeable type in reducing wave forces on tunnels is analyzed. Next, the performance of two and three submerged breakwaters is studied. The reflection and transmission coefficients of waves are high in the absence of the submerged breakwater and in the presence of an impermeable breakwater. These coefficients significantly reduce in the presence of the submerged porous breakwater. As a result, the horizontal and vertical forces acting on bridges and tunnels are substantially subsided. Wave forces on tunnels reduce with an increase in the angle of incidence. Multiple porous breakwaters show better performance in mitigating wave forces on tunnels. Higher wave force on tunnels is noticed in intermediate water-depth. The findings can enhance the knowledge of submerged porous breakwaters’ performance in reducing wave loads on bridges and tunnels.

Measurement of the Radius of Metallic Plates Based on a Novel Finite Region Eigenfunction Expansion (FREE) Method

Eddy current based approaches have been investigated for a wide range of inspection applications. Dodd-Deeds model and the truncated region eigenfunction expansion (TREE) method are widely applied in various occasions, mostly for the cases that the sample is relatively larger than the radius of the sensor coil. The TREE method converts the integral expressions to the summation of many terms in the truncated region. In a recent work, the impedance of the co-axial air-cored sensor due to a plate of finite radius was calculated by the modified Dodd-Deeds analytical approach proposed by authors. In this paper, combining the modified analytical solution and the TREE method, a new finite region eigenfunction expansion (FREE) method is proposed. This method involves modifying its initial summation point from the first zero of the Bessel function to a value related to the radius of the plate, therefore makes it suitable for plate with finite dimensions. Experiments and simulations have been carried out and compared for the verification of the proposed method. Further, the planar size measurements of the metallic circular plate can be achieved by utilising the measured peak frequency feature.

Measurement of the Radius of Metallic Plates Based on a Novel Finite Region Eigenfunction Expansion (FREE) Method

Eddy current based approaches have been investigated for a wide range of inspection applications. Dodd-Deeds model and the truncated region eigenfunction expansion (TREE) method are widely applied in various occasions, mostly for the cases that the sample is relatively larger than the radius of the sensor coil. The TREE method converts the integral expressions to the summation of many terms in the truncated region. In a recent work, the impedance of the co-axial air-cored sensor due to a plate of finite radius was calculated by the modified Dodd-Deeds analytical approach proposed by authors. In this paper, combining the modified analytical solution and the TREE method, a new finite region eigenfunction expansion (FREE) method is proposed. This method involves modifying its initial summation point from the first zero of the Bessel function to a value related to the radius of the plate, therefore makes it suitable for plate with finite dimensions. Experiments and simulations have been carried out and compared for the verification of the proposed method. Further, the planar size measurements of the metallic circular plate can be achieved by utilising the measured peak frequency feature.