Phase Jump Due to Partial Reflection of Irregular Water Waves at Steep Slopes

Phase Jump due to Partial Reflection of Irregular Water Waves at Steep Slopes

Supplement to: „Wave Resonances Detected in a Wave Tank and in the Field“, 5th Int. Symposium WAVES 2005, Madrid, Spain, Paper 134

Prof. Dr.-Ing. Fritz Büsching Bielefeld University of Applied Sciences Fluid Dynamics Laboratory Minden, Germany

urn:nbn:de:0066-201011165 http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 1/25 Scope of Presentation • Statement: The process of wavewave breakingbreaking at slopes ≤ n ≤ 2:1:13:1 is inextricably linked with the simultaneoussimultaneous formationformation of a reflectedreflected wavewave andand aa wavewave ofof transmissiontransmission. Superimposition of incident and reflected waves results in the formation of a partialpartial clapotisclapotis comprising of a phase jump. • Analogue: Fresnel´s Equations describing reflectionreflection andand transmissiontransmission of light at uniform planar interfaces.

• http://www.digibib.tu-bs.de/?docid=00047044Reflection coefficientsCoastlab 2010, BarcelonaCr = f(Hr/Hi, Δφ0/IV19/08/2014)< 2/250.

Partial standing waves breaking on a slope with reference to the point IP of the still water level intersecting the slope

Main topic: Influence of a phase shift Δϕ on breakers interacting with sloping structures

Slopes Damping ≤ n ≤ 2:1:13:1 Structure

SWL

IP d = 0.626m Wave Maker With and without re-Reflection

0.79m 1.79m Measurements of water level deflections quasi synchronously at 91 wave probe stations equally spaced 10 cm.

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 3/25 Tests on the stability of Hollow Cubes Δx Model scale: 1:5

Slopes: ≤ n ≤ 2:1:13:1 Wave heights up to H = 0.35m

Punging breaker on Collapsing breaker on Δ ≈ 15.0 mx quasi Smooth Slope Hollow Cubes

Cr = -0.32 Cr = -0.16

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 4/25 Maximum run up

Differences: Δy • extent • phase

Δ ≈ 40.0 my

Run down of collapsing Run down of plunging breaker on Hollow Cubes breaker on Smooth Slope http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 5/25 Energy Density Spectra of Water Level Deflections (Composite Power Spectra)

~IA ~IA

256 Components Δf = 0.00543 Hz

cut off cut off cut off cut off

Permeable revetment Plane revetment (measured synchronously)

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 6/25 Energy lines resulting from 256 frequency components total frequency range 0.0326 – 1.3997 Hz IA(p x Hz) (resultant clapotis length)

Emin

256 Components Δf = 0.00543 Hz

Slope 1:3

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 7/25 General properties of partial standing waves at a slope

Breakers Partial Clapotis Envelopes

Envelopes

Partial Clapotis Energy

Potential energy

− EE E = maximum energy at loop i Reflection max,i min,i max,i C ,ir = where and coefficient max,i + EE min,i Emin,i = minimum energy at node i

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 8/25 Energy lines of 12 definable partial clapotis waves frequency range 0.4015 – 0.8030 Hz (74 components)

Subfrequency ranges of 0.014 ≤ n.Δf ≤ 0.085 Hz

Slope 1:3 http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 9/25 12 partial standing waves at slope 1:m = 1:3

Minimum partial clapotis node distances from IP

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 10/25 Modified data procession and presentation for slopes 1:2

Hollow Cubes piled up to form a stepped face hollow seawall structure (2-layer-system). Slope: 1:2 Model scale: 1:10

Hollow Cubes with the stem C = +0.2 placed at one r bottom edge

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 11/25 Modified data procession and presentation for slopes 1:2

Comments: cut off noise frequencies • At steeper slope 1:2 measurements closer to IP.

• Set of partial clapotis IP waves identifiable, although re-reflection effect prevented.

• Energy contents of partial clapotis waves cut off noise piled up with reference frequencies to gauge station numbers

Close to IP • nodes at smooth slope and IP • loops at hollow slope.

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 12/25 Monochromatic waves at smooth slopes: NODE DISTANCE FROM STRUCTURE Relative node distances a/LSMOOTH with SLOPESreference to slope angle a/L 0,25-0,3 0.25 0,2-0,25 0,15-0,2 0.20 0,1-0,15 0,05-0,1 0.15 0-0,05 0.10 -0,05-0 -0,1--0,05 0.05 a/L 0.00 IP -0.05 0,30 .450 0,25 .529 0,20

0,15 .625 FREQUENCY [Hz] 0,10 0,05 .766 ≤ ≤ 22.217.1 sT IP 0,00 .800 -0,05

-0,10 .853 90,00 84,29 63,43 45,00 26,56 18,43 Joint trends: SLOPE ANGLE [o] Nodes closer to IP with tanα = 1:m 1:0.1 1:0.5 1:1 1:2 1:3 • slope angle decreasing • frequency increasing

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 13/25 Two kinds of standing transversal waves (without transmission) Free end reflection without phase jump (shift) (wave crest reflectet by wave crest) Known examples: • Rope waves • Electromagnetic waves • Water waves (Clapotis)

Fixed end reflection with phase jump Δϕ = 180o (wave crest reflectet by wave trough & vice versa) Known examples: • Rope waves • Electromagnetic waves • Water waves (at a slope?)

incident wave: red reflected wave: blue resultant wave: black Click on the figure to start animation Animations after Walter Fendt (2003) http://www.walter-fendt.de/ph14d/stwellerefl.htm http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 14/25 Wave breaking combined with partial reflection and transmission (wave moving to the left)

Partial Clapotis

Runup Node Loop Node 1 4 2 3 6 Runup = transmitted IP 5 wave pulse (phase 4) Breaking wave • The existence of a node close to the slope provides evidence of partial reflection with a phase jump . • Note opposite transmitted and reflected deflections around IP. • Conservation of momentum:

Smaller & slower transmitted wave pulse ct < ci combined with negative reflection (wave crest reflected by wave trough & vv) http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 15/25 Wave breaking combined with partial reflection and transmission (wave moving to the left)

Runup = transmitted wave pulse (phase 4) Breaking wave Click on the figure to start animation • The existence of a node close to the slope provides evidence of partial reflection with a phase jump . • Note opposite transmitted and reflected deflections around IP. • Conservation of momentum:

Smaller & slower transmitted wave pulse ct < ci combined with negative reflection (wave crest reflectet by wave trough & vv) http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 16/25 AnalogueAnalogue fromfrom ropelight waves: waves: Fresnel´s Equations describing reflection and transmission of light at uniform planar interfaces

Interface of two refractive index media Partial reflection and lower higher transmission of a pulse travelling to the right from low to high refractive index medium.

Superimposition of positive incident and negative reflected pulses not shown ! Animation after Oleg Alexandrow (2007) ci cr ct < ci http://de.wikipedia.org/wiki/Datei/:Partial_transmittance.gif breaking Click on the figure to start animation wave Incident and transmitted Analogue behaviour of water reflected pulse pulse = runup waves breaking on a slope respectively (except rundown).

negative partial reflection

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 17/25 Analogue from rope waves:

Partial reflection and transmission of a puls travelling to the right from thinner to thicker rope diameter.

Boundary of two ropes thinner thicker incident pulse: white reflected pulse: red transmitted pulse: blue

Superimposition of positive incident and negative reflected pulse not shown !

Animations after B. Surendranath Reddy (2004) http://www.surendranath.org/Apps.html

cr = ci ct < ci Click on the figure to start animation breaking incident and wave transmitted Analogue behaviour of water reflected pulse pulse = runup waves breaking on a slope respectively (except rundown). negative partial reflection http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 18/25

Further considerations on the positioning of partial standing waves with reference to a sloping structure

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 19/25 Consequences of phase shifts on reflection coefficients

Wave height ratio: Hr / Hi = 1.0

< 4.00

> 0

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 20/25 Consequences of phase shifts on reflection coefficients

Wave height ratio: Hr / Hi = 0.7

< 3.4

> 0.6

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 21/25 Consequences of a phase shift on reflection coefficients

Partial Clapotis Envelopes Loop Loop Node Node Node Wave crest envelope Wave trough envelope Hmin L/4 L/4 Hmax ? L

Reflection Coefficient (by Healy, 1953):

H r max − HH min Wave C == where max = + HHH ri and min = − HHH ri r height ratio H i max + HH min

Phase shift ⎛ H π ⎞ − HH − 42.114.3 C f ⎜ r ,7.0 ϕ =Δ== ⎟ = max min = = 377.0 considered r ⎜ ⎟ ⎝ H i 4 ⎠ max + HH min + 42.114.3

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 22/25 Reflection coefficients in the range of parameters

≤ HH ir ≤ 0.1/1.0 and phase distances 0 ° ≤ Δϕ ≤ 180 °

positive reflection negative reflection

0.0 -0.1

-0.3

-0.5

-0.7

-1.0

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 23/25 Found reflection coefficients attached to parameters

≤ HH ir ≤ 0.1/1.0 and phase distances 0 ° ≤ Δϕ ≤ 180 °

positive reflection negative reflection

Hollow Cubes piled up, Slope 1:2 o Cr (Hr /Hi = 0.26, Δϕ = 48 )= + 0.17 Wave packet 0.48 < f < 0.52 Hz Type of breaker: undifinable

Smooth slopes, plunging breakers

1:n = 1:3

Cr = - 0.41

1:n = 1:2

Hollow Revetment, Slope 1:3 Cr = - 0.80 o Cr (Hr /Hi = 0.3, Δϕ = 144 )= - 0.24 Wave packet 0.48 < f < 0.55 Hz Type of breaker: collapsing

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 24/25 Trivial suggestions on the joint effect of reflection, transmission and dissipation of breaking waves at a slope

• Verify current findings in a natural scale: Specify phase shifts for longer waves and less inclined slopes.

• Include the phase difference Δϕ in the presentations of reflection coefficients and of types of breakers in addition to the Iribarren nr IR = αξ/tan s

• Standardize the application of composite response spectra, in order to obtain spectral coefficients of reflection, transmission and absorption.

Extended version: http://nbn-resolving.de/urn:nbn:de:0066-201008270

http://www.digibib.tu-bs.de/?docid=00047044 Coastlab 2010, Barcelona 19/08/2014 25/25