Frequency Downshift in a Viscous Fluid John D. Carter - PowerPoint PPT Presentation

Frequency Downshift in a Viscous Fluid John D. Carter carterj1@seattleu.edu Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Major Collaborators ◮ Isabelle Butterfield (Seattle University) ◮ Alex Govan (Seattle University) ◮ Diane Henderson (Penn State University) ◮ Harvey Segur (University of Colorado at Boulder) Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Waves I’d Like to Model Photo from Shawn at Videezy.com. Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Modeling waves like those is too difficult for me because of: ◮ Wave breaking ◮ Air trapped in the fluid ◮ Vorticity ◮ Wind ◮ Interactions with the seafloor ◮ ... Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Waves I’m Going to Talk About Today Photo from http://teachersinstitute.yale.edu/curriculum/units/2008/5/08.05.06.x.html. Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Waves I’m Going to Talk About Today Two-dimensional modulated wave trains. Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Select Background Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Benjamin & Feir (1967) Theory and Experiments Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Benjamin-Feir Instability Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Benjamin-Feir Instability A time series that initially has the form Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Benjamin-Feir Instability A time series that initially has the form will evolve into due to the Benjamin-Feir (or modulational) instability. Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Yuen, Lake, Rungaldier, & Ferguson (1977) Experiments Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Frequency Downshifting Nonlinear deep-water waves 61 Wave 13 peaks height x=5 ft 13 peaks x= 10 ft 13 peaks x=15 ft 13 peaks x=20 ft ?? peaks x=25 ft 10 peaks x=30 ft FIGURE 5. Example of the long-time evolution of an initially uniform nonlinear wave train. Initial wave frequency is 3.6 Hz; oscillograph records shown on expanded time scale to display individual wave shapes ; wave shapes are not exact repetitions each modulation period because Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017 modulation period does not contain integral number of waves. components in the spectrum. The wave train appears to be in the process of losing its coherence and disintegrating. At a still later stage, however, as shown in the third spectrum, the energy has returned to the original frequency components (carrier, harmonics and side bands) of the initial wave train. The wave train has become almost fully demodulated, as can be seen in the corresponding wave form. This type of long-time behaviour of an unstable nonlinear system is unusual but not unknown. It was first discovered by Fermi, Pasta & Ulam (1940) during numerical Downloaded from http:/www.cambridge.org/core. Universitetsbiblioteket i Bergen, on 05 Jan 2017 at 10:21:46, subject to the Cambridge Core terms of use, available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S0022112077001037

Frequency Downshifting Nonlinear deep-water waves 61 Wave 13 peaks height x=5 ft 13 peaks x= 10 ft 13 peaks x=15 ft 13 peaks x=20 ft ?? peaks x=25 ft 10 peaks x=30 ft FIGURE 5. Example of the long-time evolution of an initially uniform nonlinear wave train. Initial wave frequency is 3.6 Hz; oscillograph records shown on expanded time scale to display individual wave shapes ; wave shapes are not exact repetitions each modulation period because Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017 modulation period does not contain integral number of waves. components in the spectrum. The wave train appears to be in the process of losing its coherence and disintegrating. At a still later stage, however, as shown in the third spectrum, the energy has returned to the original frequency components (carrier, harmonics and side bands) of the initial wave train. The wave train has become almost fully demodulated, as can be seen in the corresponding wave form. This type of long-time behaviour of an unstable nonlinear system is unusual but not unknown. It was first discovered by Fermi, Pasta & Ulam (1940) during numerical Downloaded from http:/www.cambridge.org/core. Universitetsbiblioteket i Bergen, on 05 Jan 2017 at 10:21:46, subject to the Cambridge Core terms of use, available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1017/S0022112077001037

Segur et al. (2005) Theory and Experiments Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Basic Experimental Setup Figure not to scale! Gauge 1 Gauge 2 Gauge 3 Gauge 12 Experiments conducted by Diane Henderson (Penn State University). Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Experimental Measurements x = 0 x = 0 0.3 0.1 0.15 0.01 0 0.001 -0.15 -0.3 0.0001 0 10 20 30 0 2 4 6 8 10 x = 50 x = 50 0.3 0.1 0.15 0.01 0 -0.15 0.001 -0.3 0.0001 0 10 20 30 0 2 4 6 8 10 x = 250 x = 250 0.3 0.1 0.15 0.01 0 0.001 -0.15 -0.3 0.0001 0 10 20 30 0 2 4 6 8 10 t (sec) frequency (Hz) Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Quantities of Interest ◮ The spectral peak, ω p ( x ), is defined as the frequency of the Fourier mode with largest magnitude at a location x Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Quantities of Interest ◮ The spectral peak, ω p ( x ), is defined as the frequency of the Fourier mode with largest magnitude at a location x ◮ The “mass” � L M ( x ) = 1 | B | 2 dt L 0 Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Quantities of Interest ◮ The spectral peak, ω p ( x ), is defined as the frequency of the Fourier mode with largest magnitude at a location x ◮ The “mass” � L M ( x ) = 1 | B | 2 dt L 0 ◮ The “linear momentum” � L P ( x ) = i BB ∗ t − B t B ∗ � � dt 2 L 0 Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Quantities of Interest ◮ The spectral peak, ω p ( x ), is defined as the frequency of the Fourier mode with largest magnitude at a location x ◮ The “mass” � L M ( x ) = 1 | B | 2 dt L 0 ◮ The “linear momentum” � L P ( x ) = i BB ∗ t − B t B ∗ � � dt 2 L 0 ◮ The spectral mean, ω m , is defined by ω m ( x ) = P ( x ) M ( x ) Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Quantities of Interest ◮ The spectral peak, ω p ( x ), is defined as the frequency of the Fourier mode with largest magnitude at a location x ◮ The “mass” � L M ( x ) = 1 | B | 2 dt L 0 ◮ The “linear momentum” � L P ( x ) = i BB ∗ t − B t B ∗ � � dt 2 L 0 ◮ The spectral mean, ω m , is defined by ω m ( x ) = P ( x ) M ( x ) A wave train is said to exhibit frequency downshifting if ω m or ω p decreases monotonically as it travels down the tank. Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Frequency Downshift | a ω | 2.5 F. Amps ω p 2.0 ω m 1.5 1.0 0.5 ω 2 4 6 8 10 Frequency downshift in both the spectral peak and spectral mean senses. Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

Frequency Downshift | a ω | 2.5 F. Amps 2.0 ω p 1.5 ω m 1.0 0.5 ω 2 4 6 8 10 Frequency downshift in the spectral peak sense. Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

More Experimental Background Segur et al. (2005) showed ◮ Frequency downshifting (FD) is not observed (in their tank) if the waves have “small or moderate” amplitudes Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

More Experimental Background Segur et al. (2005) showed ◮ Frequency downshifting (FD) is not observed (in their tank) if the waves have “small or moderate” amplitudes ◮ FD is observed if the amplitude of the carrier wave is “large” or if the sideband perturbations are “large enough” Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

More Experimental Background Segur et al. (2005) showed ◮ Frequency downshifting (FD) is not observed (in their tank) if the waves have “small or moderate” amplitudes ◮ FD is observed if the amplitude of the carrier wave is “large” or if the sideband perturbations are “large enough” ◮ If FD occurred, then ◮ ω m decreased monotonically ◮ FD occurred in the higher harmonics before in the fundamental Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017

More Experimental Background Segur et al. (2005) showed ◮ Frequency downshifting (FD) is not observed (in their tank) if the waves have “small or moderate” amplitudes ◮ FD is observed if the amplitude of the carrier wave is “large” or if the sideband perturbations are “large enough” ◮ If FD occurred, then ◮ ω m decreased monotonically ◮ FD occurred in the higher harmonics before in the fundamental Our goal is to provide a mathematical justification for these observations without relying on wind or wave breaking effects. Frequency Downshift in a Viscous Fluid John D. Carter April 26, 2017