Whitham waves pdf

Figure 5 illus- Systems C and D, respectively. As for Fig. Figures 5 a and 5 b show the development of the instability Boussinesq systems 27 and 28 and the two Whitham—Boussinesq for System C as the jump height increases.

For this jump height, Fig. It should be 1. However, this result is for the stability of a gravity wave Stokes Phys.

Intermediate level 43 : thick blue line. However, given these restrictions, the instability wavenumbers are for the Whitham equation,65 but the instability for System D develops in reasonable agreement. The instability is the bore is becoming unstable. Further increase in the step height D stronger than that for System C in Fig.

The bore breaks up with a results in the instability becoming more pronounced, as expected, as series of solitary-like waves being shed ahead. The amplitudes of the illustrated in Fig. The intermediate level is well predicted, but waves of the bore itself are more uniform than in a stable bore, as for there is an extended wavetrain beyond the theoretical trailing edge, as the unstable bore of System C in Fig. This overall structure of the unsta- modulation equations.

The solution of Fig. It can be seen that there is a DSW solution of the water wave equations in a similar manner to modulation of the bore envelope, in contrast to the smooth envelope Systems C and D. Figure 5 e shows the water wave DSW for the of Fig.

There is then a distinct difference between the predictions instability. Solutions of Whitham—Boussinesq systems C 29 and D 30 and the water wave equations 2 — 5.

The wavenumber of the water wave DSW at the of Fig. The upstream Phys. System D then gives a better prediction of the they are much simpler to solve and analyze. There is a modulation of the bore envelope, which is similar the undular bore up until the onset of modulational instability. The noticeable difference with the System D is in better agreement with solutions of the water wave fully unstable System D bore of Fig.

However, the maximum difference between the lead wave instability. Overall, System D gives a better prediction of the develop- velocity as given by System D and the water wave equations is ment of the instability and gives a better prediction of the stability 0. This is not unexpected as the between these two Whitham—Boussinesq systems.

The amplitude Whitham—Boussinesq equations are being pushed beyond their region of the lead wave of the undular bore is not as well predicted as the of expected validity. The agreement between the amplitudes given by Systems VI. In addition, the these were standard Boussinesq systems for weakly nonlinear long Whitham—Boussinesq—Hamiltonian system, System D, gives a waves, one the standard Boussinesq system1 and the other the sys- good prediction for the initial jump height D for the onset of mod- tem arising from the Hamiltonian formulation of the water wave ulational instability of the undular bore, better than that of System equations.

In summary, equation1,44 extensions of these Boussinesq systems which are still the Whitham—Boussinesq systems are very accurate models of sur- weakly nonlinear, but include full linear water wave dispersion. Indeed, they can be The undular bore solutions of these four Boussinesq systems were concluded to provide a very accurate alternative to the full water compared with numerical solutions of the full water wave equa- wave equations for the study of water wave undular bores.

This tions. It was found that the two Whitham—Boussinesq systems give could prove useful as Whitham-type equations are much easier to improved agreement with solutions of the water wave equations, as solve and analyze than the full water wave equations. The agreement and Technology to develop a submitted research project, year for the amplitude of the lead wave of the undular bore is not as good and extension in In addition, the Whitham—Boussinesq systems of Edinburgh. El and M. D , 11—65 Whitham—Boussinesq systems and a somewhat slower transition to 46, — Clarke, R.

Smith, and D. Weather Rev. Porter and N. Fluid Mech. Smyth and P. Flaschka, M. Forest, and D. Yuan, R. Grimshaw, and E. Novotryasov, D. Stepanov, and I. Talipova, E. Pelinovsky, O. Kurkina, and A. Minzoni, and P. Esler and J. Ivanov and A. Fluids 32, V. Hur and L. Scott and D. Hur and A. Lowman and M.

London, Ser. A , 6—25 Marchant and N. Naumkin and A. Barsi, W. Wan, C. Sun, and J. Constantin and J. Wan, S. Jia, and J. Equations 22, — El, A. Gammal, E. Khamis, R. If the physical system is actually unstable, then prediction may not be possible. If these conditions are fulfilled, then the problem is said to be well posed, in the sense of Hadamard [Had].

Numerical schemes for particular PDE systems can be analyzed mathematically to determine if the solutions remain bounded. The consequence of this is that the initial condition propagates from left to right at constant speed c. If the initial condition is monotonically increasing with x, the characteristics will not overlap and the problem is well behaved.

In this situation we can only find a weak solution one where the problem is re-stated in integral form by appealing to entropy considerations and the Rankine-Hugoniot jump condition. PDEs other than equations 63 and 64 , such as those involving conservation laws, introduce additional com- plexity such as rarefaction or expansion waves. We will not discuss these aspects further here, and for additional discussion readers are referred to [Hir, chap.

The ODEs are solved along particular characteristics, using standard methods and the initial and boundary conditions of the problem. For more information refer to [Kno, Ost, Pol]. MOC is a quite general technique for solving PDE problems and has been particularly popular in the area of fluid dynamics for solving incompressible transient flow in pipelines. For an introduction refer to [Stre, chap.

Other invariant transformations are possible for many linear and nonlinear wave equations, for example the Lorentz transformation applied to the Maxwell equations, but these will not be discussed here. See figure 5. Wave refraction is caused by segments of the wave moving at different speeds resulting from local changes in characteristic speed, usually due to a change in medium properties.

Physically, the effect is that the overall direction of the wave changes, its wavelength either increases or decreases but its frequency remains unchanged. Wave diffraction is the effect whereby the direction of a wave changes as it interacts with objects in its path. The effect is greatest when the size of the object causing the wave to diffract is similar to the wavelength. A hard boundary is one that is fixed which causes the wave to be reflected with opposite polarity, e. A soft boundary is one that changes on contact with the wave, which causes the wave to be reflected with the same polarity, e.

If the propagating medium is not isotropic, i. The polarity of the partial reflection will depend upon the characteristics of the medium. In addition, for simplicity, consider the medium on both sides of the boundary to be isotropic and non- dispersive, which implies that all three waves will have the same frequency.

A striking example of this phenomena is the failure of the mile-long Tacoma Narrows Suspension Bridge. On 7 November the structure collapsed due to a nonlinear wave that grew in magnitude as a result of excitation by a 42 mph wind. Another less dramatic example of resonance that most people have experienced is the effect of sound feedback from loudspeaker to microphone.

A more complex form of resonance is autoresonance, a nonlinear phase-locking synchronizing phe- nomenon which occurs when a resonantly driven nonlinear system becomes phase-locked synchronized or in-step with a driving perturbation or wave.

For example, a sound wave will have a higher pitch and the spectrum of a light wave will exhibit a blueshift. They include: seismic S secondary waves, and electromagnetic waves, E electric field and H magnetic field , both of which oscillate perpendicularly to each other as well as to the direction of propagation of energy.

Light, an electromagnetic wave, can be polarized oriented in a specific direction by use of a polarizing filter. Longitudinal waves oscillate along the direction of wave propagation. They include sound waves pressure, particle displacement, or particle velocity propagated in an elastic medium and seismic P earthquake or explosion waves. Surface water waves however, are an example of waves that involve a combination of both longitudinal and transverse motion. Traveling-wave solutions are characterized by the fact that the profiles of these solutions at different time instants are obtained from one another by appropriate shifts translations along the x-axis.

Consequently, a Cartesian coordinate system moving with a constant speed can be introduced in which the profile of the desired quantity is stationary.

The term traveling-wave solution is also used in situations where the variable t plays the role of a spatial coordinate, y. The effect is that the wave amplitude varies with time but it does not move spatially. Clearly, the existence of nonlinear standing waves can be demonstrated by application of Fourier analysis. The path may be fixed or capable of being varied to suit a particular application.

The operation of a waveguide is analyzed by solving the appropriate wave equation, subject to the prevailing boundary conditions. There will be multiple solutions, or modes, which are determined by the eigenfunctions associated with the particular wave equations, and the velocity of the wave as it propagates along the waveguide will be determined by the eigenvalues of the solution.

Musical wind instruments, such as a flute, can also be thought of as acoustic waveguides. For detailed analysis and further discussion refer to [Lio, Oka].

One of the simplest form of wavefront to envisage is an expanding circle where its radius r, expands with velocity v, i. Simple circular sinusoidal wave-fronts propagating from a point source are shown in figure 7. Figure 7: Circular wave-fronts emanating from a point source. Depending upon the particular wave equation and medium in which the wave travels, the wavefront may not appear to be an expanding circle.

The path upon which any point on the wave front has traveled is called a ray, and this can be a straight line or, more likely, a curve in space. In general, the wavefront is perpendicular to the ray path, and the ray curvature will depend on the circumstances of the particular problem. For example, its curvature will be influenced by: an anisotropic medium, refraction, diffraction, etc.

This can result in waves having non-circular wave-fronts and hence curved rays. This situation, which occurs in many different applications, is illustrated in figure 8 where the curved wave-fronts are due to a combination of effects due to refraction, diffraction, reflection and a non-point disturbance. This idea was proposed by the Dutch mathematician, physicist, and astronomer, Christian Huygens, in , and is a powerful method for studying various optical phenomena [Enc].

The non-circular wave-fronts are clearly visible, which indicates curved rays. The points on a wave-front propagate from the wave source along so-called rays. This can occur when a solid object is forced through a fluid, for example in supersonic flight. As the cause of the disturbance subsides, the shock wave energy is dissipated within the fluid and it reduces to a normal, subsonic, pressure wave.

Note: A shock wave can result in local temperature increases of the fluid. This is a thermodynamic effect and should not be confused with heating due to friction. This is because the whip is tapered from handle to the tip and, when cracked, conservation of energy dictates that the wave speed increases as it progresses along the flexible cord.

As the wave speed increases it reaches a point where its velocity exceeds that of sound, and a sharp crack is heard. The expansion of the fluid, due to temperature and chemical changes force fluid velocities to reach supersonic speed, e. But perhaps the most striking example would be the shock wave produced by a thermonuclear explosion. This procedure uses a focused, high-intensity, acoustic shock wave to shatter the stones to the point where they are reduced in size such that they may be passed through the body in a natural way!

For further discussion relating to shock phenomena see [Ben, Whi]. We briefly introduce two topics below by way of example. He derived this result and was able to estimate, using only photographs of the blast released into the public domain in , that the yield of the bomb was equivalent to between Each of these photographs, crucially, contained a distance scale and precise time, see figure This result was classified secret but, five years later he published the details [Taya, Tayb], much to the consternation of the British government.

Sedov published similar independently derived results [Bet, Sed]. For further discussion relating to the theory refer to [Kam, Deb]. Times from instant of detonation are indicated in bottom left corner of each photograph Top first - left column: 0. The shock forms a high pressure, cone-shaped surface propagating with the aircraft.

This is the so-called N-wave - a pressure wave measured at sufficient distance such that it has lost its fine structure, see figure A sonic boom occurs when the abrupt changes in pressure are of sufficient magnitude. Thus, steady supersonic flight results in two booms: one resulting from the rapid rise in pressure at the nose, and another when the pressure returns to normal as the tail passes the point vacated by the nose.

This is the cause of the distinctive double boom from supersonic aircraft. The duration T varies from around ms for a fighter plane to ms for the Space Shuttle or Concorde. Another form of sonic boom is the focused boom. These can result from high speed aircraft flight ma- neuvering operations. These result in the so-called U-waves which have positive shocks at the front and rear of the boom, see figure Generally, U-waves result in higher peak over-pressures than N-waves [Nak]- typically between 2 and 5 times.

For further discussion related to sonic booms refer to [Kao]. Figure The U-wave sonic boom. As an aircraft passes through, or close to the sound barrier, water vapor in the air is compressed by the shock wave and becomes visible as a large cloud of condensation droplets formed as the air cools due to low pressure at the tail. A smaller shock wave can also form on top of the canopy. This phenomena is illustrated in figure However, a soliton is a solitary wave having the additional property that other solitons can pass through it without changing its shape.

But, in the literature it is customary to refer to the solitary wave as a soliton, although this is strictly incorrect [Tao]. Solitons are stable, nonlinear pulses which exhibit a fine balance between non-linearity and dispersion. They often result from real physical phenomena that can be described by PDEs that are completely integrable, i. Such PDEs describe: shallow water waves, nonlinear optics, electrical network pulses, and many other applications that arise in mathematical physics.

Where multiple solitons moving at different velocities occur within the same domain, collisions can take place with the unexpected phenomenon that, first they combine, then the faster soliton emerges to proceed on its way.

Both solitons then continue to proceed in the the same direction and eventually reach a situation where their speeds and shapes are unchanged. Thus, we have a situation where a faster soliton can overtake a slower soliton.

There are two effects that distinguishes this phenomena from that which occurs in a linear wave system. The first is that the maximum height of the combined solitons is not equal to the sum of the individual soliton heights. The second is that, following the collision, there is a phase shift between the two solitons, i. Figure Evolution of a two-soliton solution of the KdV equation.

Image illustrates the collision of two solitons that are both moving from left to right. The faster taller soliton overtakes the slower shorter soliton. Some additional discussion is given in section 4. More details may be found in Drazin and Johnson [Dra]. It is now generally accepted by the international scientific community to describe a series of traveling waves in water produced by the displacement of the sea floor associated with submarine earthquakes, volcanic eruptions, or landslides.

They are also known as tidal waves. Tsunami are usually preceded by a leading-depression N-wave LDN , one in which the trough reaches the shoreline first. Eyewitnesses in Banda Aceh who observed the effects of the December Sumatra Tsunami, see figure 8 , resulting from a magnitude 9.

Recent estimates indicate that this powerful tsunami resulted in excess of , deaths and extensive damage to property and infrastructure around the entire coast line of the Indian ocean [Kun].

Hence, tsunami waves are often modeled using the shallow water equations, the Boussinesq equation, or other suitable equations that bring out in sufficient detail the required wave characteristics. However, one of the major challenges is to model shoreline inundation realistically, i. As the wave approaches the shoreline, the water depth decreases sharply resulting in a greatly increased surge of water at the point where the wave strikes land.

This requires special modeling techniques to be used, such as robust Riemann solvers [Tor, Ran] or the level-set method [Set, Osh], which can handle situations where dry regions become flooded and vice versa.

Allan F. The goal of this paper is to prove the existence of an extreme solitary-wave solution of 1 and our plan is to use a global bifurcation theorem appearing in [ 11 ]; see also [ 10 , 15 , 16 ]. A key to our success is the fact that a lot of qualitative properties have been shown for the Whitham kernel, the Whitham symbol and the solutions of 1 , thanks to [ 9 , 21 , 22 ].

These guide us in choosing a convenient function space to study 1 and have been extremely useful in the application of the global bifurcation theorem.

In Sect. We also study how sequences of solutions converge and the Fredholm properties of important linear operators. Another key is the recently developed center manifold theorem for nonlocal equations in [ 24 ]. This result states that nonlocal equations with exponentially decaying convolution kernels are essentially local equations near an equilibrium.

It also provides a method to derive the local equation, which can then be studied using familiar ODE tools. Although the Whitham kernel has the required exponential decay, it fails a local integrability condition. Seeing that this condition is only for proving Fredholm properties of linear operators, we directly prove these properties instead. All necessary changes for the general center manifold theorem are listed in Appendix B. More specifically, we prove the following. Using Eq. We thus arrive at the first main result of this paper repeated as Theorem 3.

While both [ 19 ] and [ 31 ] contain existence results for supercritical solitary waves, the additional information provided by the center manifold approach concerning uniqueness is crucial in the subsequent analysis. To end Sect. This is in preparation for the global bifurcation theorem.

We rule out the loss of compactness alternative using qualitative properties of the solutions, how sequences of solutions converge and an integral identity for 1.

Then, we show that the blowup alternative happens as the Sobolev norm blows up and that an extreme solitary-wave solution is obtained in the limit.

More precisely, we have the following result repeated as Theorem 4. By demonstrating the use of recent spatial-dynamics tools, this paper serves as an example to studies of other nonlocal nonlinear evolution equations. In particular, these results will likely extend to a larger class of equation, such as in [ 5 ] and [ 20 ].

Finally, it is interesting to compare our results with the global bifurcation theory for the water wave problem. The existence of an unbounded, connected set of solitary water waves, including a highest wave in a certain limit, was proved by Amick and Toland [ 2 , 3 ] following several earlier small-amplitude results. Around the same time, Amick et al. The construction of the global solution continua in [ 2 , 3 ] is also different from ours.

While both proofs are based on nonlinear integral equations, the common approach in [ 2 , 3 ] is to first apply global bifurcation theory to a regularized problem and then pass to the limit.

On the other hand, we use global bifurcation theory directly on the solitary Whitham problem. A similar approach has in fact recently been used for solitary water waves with vorticity and stratification, but based on a PDE formulation [ 11 , 32 ]. For the water wave problem with vorticity and stratification, the limiting behavior of large-amplitude waves is more complex and there is numerical and some analytical evidence of overhanging waves; see for example [ 14 , 17 , 18 , 28 ] and references therein.

These spaces are equipped with the norms. We have. More specifically,. Since m is an even function, so is K. The fact that K is a positive function has been shown in Proposition 2.

When choosing appropriate function spaces for 1 , we will rely on the following qualitative properties of solutions. Item i is stated in Lemma 4. Items ii and iii are Proposition 3. Items iv , v and vi can be found in Theorem 4. The upper bound in viii comes from [ 22 ], Eq. The lower bound in vii comes from non-negativity and the following proposition. A proof of this can be found in [ 8 ], pp. It follows that. Modes of convergence of solution sequences will be important in ruling out alternatives from the global bifurcation theorem.

For each n , we have. We claim that the right-hand side forms an equicontinuous sequence. The square root of a non-negative equicontinuous sequence is an equicontinuous sequence. By Proposition 2. A standard result in the theory of paradifferential operators, for instance Theorem 2. Theorem 2. Another standard result in paradifferential calculus, for example Theorem 8. By iterating as many times as needed, the claim of the proposition is proved. Since Theorem 2. As a preparation for future bifurcation results, we study the operators.

Appendix A summarizes the relevant theorems from [ 27 ]. Noting that. The idea is to apply Proposition A. We define a positively homogeneous function A by.

In order to apply Proposition A. A calculation shows that. So, the value of A at 0, 1, 0, 1 is 1. According to Proposition A. The sign of the imaginary part equals the sign of. Proposition A. We have proved the first part of the main result of this section, which is the following. The application of Proposition A. The corresponding positively homogeneous function B is. In the notation of [ 24 ], Eq. The center-manifold reduction technique gives a reduced equation equivalent to the nonlocal Eq.

We consider Eq. For 10 , we use an adaptation of the center manifold theorem in [ 24 ]. The rest of the proof of the center manifold theorem in [ 24 ] remains the same.

We consider 10 together with the modified equation. More details are provided in Appendix B. It is hence reversible. The function f in v commutes with all translations and anticommutes with the reflection symmetry. We use Theorem B. Proposition 2. This means that in Hypothesis B. Statement vi concerning the reflection symmetry R follows directly from K being an even function. Hence, Theorem B. Equation 14 in v is given by Theorem B. According to Theorem B. Applying Theorem B. To prove Eq.

The linear equations are solved in Appendix C. This gives. Hence, all solutions to the linearized equation will be captured. The downside of this scheme is that our previous adaptation of the results in [ 24 ] cannot be applied directly.