Interactions Between Transient Long Waves and Random Swell: Laboratory Investigations and Preliminary Analysis

James M. Kaihatu1, Hoda M. El Safty2, Belynda Alonzo3, Kyle Outten3, Brianna Schilling3, and Ying-Po Liao2

1Assistant Professor, Zachry Department of Civil Engineering, Texas A&M University, College Station, TX. Email: jkaihatu@civil.tamu.edu2Graduate Research Assistant, Zachry Department of Civil Engineering, Texas A&M University, College Station, TX

3NSF-REU Undergraduate Research Assistant, Zachry Department of Civil Engineering, Texas A&M University, College Station, TX

Gage

1

Gage

8

Gage

16

Gage

20

Test Number

Hs (cm) Tp (s) kh δ=a/h Ur

1 5 2 1 0.033 0.033

2 5 4 0.45 0.033 0.163

3 10 2 1 0.067 0.067

4 10 4 0.45 0.067 0.331

Two views of the 2004 Boxing Day Tsunami from Koh Jum Island, Thailand

Introduction and Motivation:The solitary wave has been the prototypical iconic wave for tsunami studies, particularly those performed in the laboratory. In recent years, though, there have been some question as to the appropriateness of this representation. For example, Madsen et al. (2008) showed that this paradigm is more appropriate for distances away from the generation area which are far longer than the width of the world’s oceans.

In line with this questioning of the solitary wave paradigm, we wish to study any possible interactive effects between the overlying swell wave and the underlying tsunami. In fact, this possibility of non-negligible interactive effects has been raised before in tsunami conferences (anonymous NSF reviewer) but has not been investigated in detail. In storm surge events, it has been hypothesized (Ewing et al. 2009) that the presence of short period waves riding on top of a reduced surge may have been responsible for failure of roadbeds and building foundations on the area of coastal Texas to the west of Hurricane Ike. In this spirit, our primary motivation was to see whether some aspects of the destructive power of a tsunami may be affected by the presence of a random swell on the surface.

For the purposes of our analysis, we take the view of a random swell interacting with a transient long wave, rather than with a tsunami. In this manner, we focus on the interactions between short and long wave motions.

Figure from Madsen et al. (2008) showing disintegration of a non-dispersive long wave

Experiments:Laboratory experiments were conducted in the NEES Tsunami Wave Basin (TWB) at the O.H. Hinsdale Laboratory at Oregon State University. Preliminary tests were run January 2010, with the primary experiments were run 3-14 March 2010. The overall experimental domain was 43.75m. The first 10m had a constant water depth of 0.75m. The next 7.5m had a bottom slope of 1:15, and transitioned to 1:30 for the remainder of the water depth.

The wavemaker at the TWB is capable of generating a wave as high as 0.8m in 1m water depth, and has a maximum stroke of 2.6 m. We used TMA spectra to generate the random wave field (the parameters are included in the above table). Transient long waves, random short waves, and combined motions (long wave either in the middle or at the end of the short wave series) were run and measured at various locations in the tank. Both free surface measurements (via resistance gages) and near-bottom velocities (via ADV) were recorded, with instruments suspended from an observation bridge spanning the tank widthwise. Sampling rate was 50 Hz. In addition, two web cameras recorded overhead video of the experiments; a pair of rectified images from the video is shown below.

Study Methodology:We endeavored to conduct experiments on the interaction of transient long waves with short, random surface swell, and use various analysis techniques to discern the interactions between the two disparate scales of wave motion. One approach would be to separate (as far as possible) the combined short-wave / long-wave motion, and compare to the individual short or long motions.

In this regard, the most popular approach would be to apply a Fourier analysis to the data, separate the long and short scales and investigate each. However, due to the transient nature of the long wave, and thus the interaction, standard Fourier analysis is unsuitable. Methods which allow for local variations in frequency are useful for nonstationary phenomena such as seen here.

Details of laboratory experiment at Tsunami Wave Basin. Top left: View of wavemaker. Top right: View of wavetank. Bottom left: Tank layout and location of gages under study. Bottom right: Details of swell conditions.

Wavelet transforms are generally useful for this, since they are able to focus on a relatively short span of time rather than the entire time series; thus, it is possible to obtain frequency spectra which vary in time. Another potentially useful technique is the Hilbert-Huang transform (Huang et al. 1998), which uses Hilbert transforms to separate the time series data into a series of embedded functions, each of which represents some inherent time scale of motion represented in the time series. These can then be analyzed individually or combined.

Analysis of Free Surface Data - Wavelet Transforms:For detailed analysis of the free surface signal, we focused on four gage locations, as shown in the tank layout diagram. A Morlet wavelet shape was used as the “mother” wavelet basis function for the wavelet analysis; this is the most commonly used wavelet shape.

The breaking location of the long wave was qualitatively defined, and a snapshot taken. It is clear that the long wave breaking occurs further offshore in the presence of short swell than with no swell present. To identify the potential effect of short waves on the hazardous aspect of tsunamis (runup speed, etc.) more detailed processing is required.

Snapshots from overhead video. Left: Long wave breaking with swell. Right: Long wave breaking without swell.

Wavelet analysis, long wave only.Top of each figure: time series. Bottom of each figure: time dependent frequency spectra. Top left: Gage 1. Top right: Gage 8. Bottom left: Gage 16. Bottom right: Gage 20. Frequency in rad/s.

As a qualitative measure, we note the distinctive triangular shape of the long-wave wavelet spectrum, and note how it changes when swell waves are present.

Wavelet analysis, long wave with swell 2. Top of each figure: time series. Bottom of each figure: time dependent frequency spectra. Top left: Gage 1. Top right: Gage 8. Bottom left: Gage 16. Bottom right: Gage 20. Frequency in rad/s.

Wavelet analysis, long wave with swell 4. Top of each figure: time series. Bottom of each figure: time dependent frequency spectra. Top left: Gage 1. Top right: Gage 8. Bottom left: Gage 16. Bottom right: Gage 20. Frequency in rad/s

It is evident that the presence of the swell, in this case, has a strong effect on the wavelet spectrum of the long wave. At Gage 16, for example, the predominance of the energy is pushed into frequencies higher than the peak frequency of the incident swell (shown with the white line). However, since the wavelet analysis still uses a basis function specified apriori, it is not clear whether the high frequency energy is an artifact of the analysis or not.

Analysis of Free Surface Data - Hilbert-Huang Transforms:The Hilbert-Huang transform (HHT) was developed to help separate out wave motions of disparate scales. Unlike apriori methods, the modal shapes come from the data. Huang et al. (1998) refer to each mode from the analysis as an “intrinsic mode function” or IMF, which represents a particular embedded scale of motion in the signal.

Application of HHT to long wave signal. Left: original signal at each analyzed gage (top to bottom: Gage 1, Gage 8, Gage 16 and Gage 20). Right: signal reconstructed from eight IMF modes at each gage.

Individual IMF modes from HHT analysis of long wave signal. Left eight plots: Modes 1 through 4. Right eight plots: Modes 5 through 8.

Blue: Gage 1Green: Gage 8

Blue: Gage 16Green: Gage 20

Blue: Gage 1Green: Gage 8

Blue: Gage 16Green: Gage 20

Mode 1 at Gages 1 and 8 appear to be better correlated to the short reflected waves in the lee of the long wave than to the long wave itself. This is not the case for Mode 1 at Gages 16 and 20. Additionally, the modal representation of the long wave appears to have a distinctive shape. We now apply the HHT to the combined signal (long wave with Swell 3). We show only the separated signals, and not the IMFs.

HHT analysis on combined signal; long wave with Swell 3. For each: Gages 1, 8, 16 and 20, top to bottom. Left: Original signal; note long wave appearing near t=170 seconds. Middle: Result of addition of IMF modes 1-3. Right: Result of addition of IMF modes 4-8.

It is clear that the HHT has the potential of separating the two scales of motion for individual analysis. This is the focus of future work.Acknowledgments: This work was funded by NSF CMMI Grant 0936579. We would like to thank the staff of the NEES Tsunami Wave Basin at Oregon State University, in particular Drs. Sungwon Shin, Tim Maddux and Dan Cox; Ms. Melora Park, Ms. Linda Frayler. Mr. Jason Killian and Mr. Adam Ryan.

References:Ewing, L.C., et al. 2009. Shore and Beach, 77, 9-23.Huang, N., et al. 1998. Proc. Royal Soc. London, Ser. A, 454, 903-995.Madsen, P.A., et al. 2008. J. Geophys. Res., 113, doi: 10.1029/2008JC004932.