University:
Massachusetts Institute of Technology
Course:
2.24 | Ocean Wave Interaction with Ships and Offshore Energy Systems (13.022)
Academic year:

2019
Views:

152

Pages:

14
Author:

Brent Cantu

PLANE PROGRESSIVE LINEAR WAVES AS ILLUSTRATED IN THE FIRST FEW SLIDES, REGULAR TIME-HARMONIC PLANE PROGRESSIVE WAVES ARE THE FUNDAMENTAL BUILDING BLOCK IN DESCRIBING THE PROPAGATION OF SURFACE WAVE DISTURBANCES IN A DETERMINISTIC 1 & STOCHASTIC SETTING AND IN PREDICTING THE RESPONSE OF FLOATING STRUCTURES IN A SEASTATE. THEY SATISFY THE BOUNDARY-VALUE PROBLEM! { ∂²φ/∂t² + g∂φ/∂z = 0, z≥0 { ∇²φ = 0, -H≤z≤0 { ∂φ/∂z = 0, z = -H THE LAST CONDITION IMPOSES A ZERO NORMAL FLUX CONDITION ACROSS A SEAFLOOR OF CONSTANT DEPTH, H. THE RELATION BETWEEN ω AND K IS KNOWN AS THE DISPERSION RELATION, OFTEN WRITTEN IN THE FORM: ω = f(k) f(k) DEPENDS ON THE PHYSICS OF THE WAVE PROPAGATION PROBLEM UNDER STUDY. IN THE CASE OF SOUND PROPAGATION: ω = Ck = f(k) WHERE C IS THE SPEED OF SOUND. THE ABOVE RELATION SUGGESTS THAT ALL SOUND WAVES IN A MEDIUM WITH ISOTROPIC PROPERTIES PROPAGATE WITH THE SAME PHASE SPEED REGARDLESS OF THEIR FREQUENCY OR WAVE LENGTH. THESE ARE KNOWN TO BE NON-DISPERSIVE WAVES. SURFACE WAVES ARE DISPERSIVE SINCE f(k) IS A NONLINEAR FUNCTION OF K. IT WILL BE DERIVED BELOW. DISPERSION RELATION IN DEEP & SHALLOW WATERS IN FINITE DEPTH: ω² = g k tanh kH OR Ω² = w²H/g = kH tanh kH => Ω² = tanh w THIS IS A NONLINEAR ALGEBRAIC EQUATION FOR w = kH WHICH HAS A UNIQUE SOLUTION AS CAN BE SHOWN GRAPHICALLY [GRAPH] tanh w: MONOTONICALLY INCREASING WITH AN ASYMPTOTIC VALUE OF 1 Ω²/w: HYPERBOLA MONOTONICALLY DECREASING UNIQUE REAL ROOT W(Ω²) CAN ONLY BE FOUND NUMERICALLY. YET IT ALWAYS EXISTS AND THE ITERATIVE METHODS THAT MAY BE IMPLEMENTED ALWAYS CONVERGE RAPIDLY.* 1 GIVEN (ω, H) → W*(Ω² = w²H/g) = W* = kH => 2π/λ = H * W*(Ω²) => λ = 2πH / W*(Ω²) SO GIVEN THE WAVE FREQUENCY ω → λ IN DEEP WATER ; H → ∞ tanh kH → 1 => ω² = gk ; DEEP WATER DISPERSION RELATION PHASE SPEED: C = ω/k = ω/ (ω²/g) = √g/ω OR C = ω/k = √g/ω = gT/2π SO THE SPEED OF THE CREST OF A WAVE WITH PERIOD T=10 secs IS APPROXIMATELY 15.6 m OR ABOUT 30 KNOTS!