Resonantly Forced Waves and the Effect of Tank Geometry on the Free Surface

This thesis examines the evolution of small-amplitude, resonantly forced oscillations of fluid in a shallow tank of finite extent.A rectangular tank and annular tank shape are both considered.The fluid in each tank is subject to periodic lengthwise or radial forcing, respectively.The resonant forcing of fluid in a rectangular tank has been examined in detail in prior work.How varying the geometry of the tank affects the evolution of the signal on the free surface is not yet fully understood.This work aims to contribute toward a fuller understanding of the interplay between geometric and resonant effects in axisymmetric geometries.The relevant governing equations for this type of fluid motion are first determined.Then, using perturbation methods and multiple scale techniques, asymptotic approximations are made to simplify the mathematical models for this problem.Various methods are then used to derive a solvability condition describing the periodic motion of the free surface of the fluid.The work done in the rectangular tank case is used to inform the work in the annular tank case.We will see there is a direct link between the two as the inner radius of the annular tank becomes large.Following this, undamped and damped numerical results are used to test the validity of the approximations for an annular tank with large inner radius and examine the qualitative effects the varying geometry of the tank has on the signal.The numerical results show the similarities and differences between the evolution of the signal in the two tanks.In some cases, the response in the annular tank is qualitatively distinct from that of the rectangular tank.The degree to which the underlying spectrum is commensurate along with geometric effects are shown to play a key role in determining how different these results are.√ λ 2 , ..., √ λ 6 to √ λ 1 plotted for various values of R 1 , and R 1 → ∞.The eigenvalues for R 1 ̸ = 0 are the zeros of (4.28), and for R 1 = 0, they are the zeros of J 1 .43 13 Evolution of the signal at r = 0 in the annular tank (R 1 = 50, R 1 = 2, R 1 = 1) compared against the signal at x = 0 in the rectangular tank (Red), no damping, ϵ = 0.00258, δ = 0.083 (κ = 0.1356).For the rectangular tank signal and R 1 = 50 and R 1 = 2, ω = 0.5.For R 1 = 1, ω = 0.51. . . . . . . . . . . . . . . . . . . . . . . .52 14 The steady state profiles of the rectangular tank signal and the annular tank signal (R 1 = 50, R 1 = 2, R 1 = 1).ϵ = 0.00258, δ = 0.083 (κ = 0.1356), ν = 0.0005.For R 1 = 50 and R 1 = 2, ω = 0.5.For R 1 = 1, ω = 0.51. . . . . . . . . . . . . . . . . . .53 15 The steady state profiles of the rectangular tank signal and the annular tank signal (R 1 = 2, R 1 = 1, R 1 = 0.2, R 1 = 0.1).ϵ = 0.00258, δ = 0.083 (κ = 0.1356), ν = 0.0005.For R 1 = 2, ω = 0.5.For R 1 = 1, ω = 0.51, For R 1 = 0.2, ω = 0.55.For R 1 = 0.1, ω = 0.57. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54 16 The resultant construction of η 0 (0, t) using the annular tank solvability condition (4.54) for γ = 0, γ = 0.089, γ = 0.177, γ = 0.266, γ = 1.109.ϵ = 0.00258, δ = 0.083 (κ = 0.1356), ω = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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