A wavelet

Tsunami or maremoto waves occur in response to earthquakes or landslides on the seafloor of large bodies of water, as discussed in [

(a) Tsunami 52406 from DART, March 2011; (b) replica of tsunami-model profile adapted from [

In an apparently unrelated phenomena, rogue, freak, or monster waves are caused by small ripples or currents in layers near the water's surface [

(a)

Tsunami and rogue waves are perturbations of the water-surface elevation function

This work considers localized plane waves, or wavelets, on a flat sea propagating only in the

The forcings

It seems natural that wavelets should appear in the study of surface water waves. The wavelets presented here are particularly well suited for surface waves. In particular, we show that when the forcing

The first quantity that we introduce is the function

As part of our study, we employ the Jacobi theta function, defined for

For

From [

The appearance of theta functions is a consequence of the Laplace transform of (

The use of theta functions in frequency space provides decaying versions of

A tsunami wave is the consequence of a spontaneous change in elevation on the seafloor, which creates a variable pressure field throughout the volume of water. This sets up forces that extend to the surface of the ocean causing it to be moved up and down, locally. The perturbed wave height then propagates away from this disturbance. In still water, the surface wave speed

Suppose that the faultline on the seafloor is parallel to the

Consider the forced one-dimensional wave equation for the water-level function

For any

The expression in (

Similar results hold for height functions

Here, we model the Japan tsunami of March 11, 2011, using (

Figure

DART 21418: Data showing earthquake (

(a) Comparison of

Figure

Figure

Figure

DART 21413: Data showing

Figure

Wake Island: Data showing

A debate continues on the physical cause of rogue waves, [

In this section, we show that small amplitude forces, over long periods of time, can naturally produce large rogue waves. This demonstrates the existence of a resonance for the system externally forced by (

Let

The proof of this result is given in the last section. It is used here to show that small amplitude forces, over long periods of time, can naturally produce rogue waves. That is, for

Let

Without loss of generality, set

Let

From (

When a slight drift in the rogue-generating current is present, there may be a speed

Let

Applying the operator

For

There is a corresponding theorem for the moving

Note that, as is demonstrated in Figure

Parameters:

We now have a collection of solutions to differential equations that give the qualitative behavior of a physical phenomena. Next, to detect, analyze, store, and recover a tsunami waveform, it is common to use a wavelet analysis [

The results of a signal analysis and synthesis, briefly presented here, consist of

By [

On March 11, 2011, an earthquake of magnitude 9.0 occurred off the coast of Japan causing a tsunami no less than

(a) Tsunami 46411 from DART, March 2011; (b) relative magnitude of coefficients for 3 scales

On January 1, 1995, a rogue wave was detected on the Draupner platform in the North Sea. Surface wave heights were recorded using a laser-detection method [

(a) Rogue 1520 from Draupner, January 1995; (b) relative magnitude of coefficients for 3 scales

In this section, the estimates for the differences of

We first prove the estimate for the differences involving

The change of variables

From (

We now show that

Let

We next estimate the portion of the integral in (

Now, let

The proof for

The rogue data from the Draupner platform was graciously supplied by Dr. Sverre Haver from Statoil, Norway. The tsunami data came from DART buoys and was graciously provided by Dr. George Mongov at NOAA. The DART data is kept at the National Geophysical Data Center/World Data Center (NGDC/WDC) Historical Tsunami Database, Boulder, CO, USA. Figure