Louis N. Howard

Note on a paper of John W. Miles

The theorem X established by Miles is given a simpler and more general proof. Some further theoretical results concerning the stability of heterogeneous shear flows are also presented, in particular a demonstration that the complex wave velocity of any unstable mode must lie in a certain semicircle. (auth)

Note on a heterogeneous shear flow

Goldstein (1931) has considered the stability of a shear layer within which the velocity and the density vary linearly and outside which they are constant. Rayleigh (1880, 1887) had found that the corresponding, homogeneous shear flow is unstable in and only in a finite band of wave-numbers. Goldstein concluded that a small density gradient renders the flow unstable for all wave-numbers. This conclusion appears to depend on the acceptance of all possible branches of a multi-valued eigenvalue equation, and it is shown that the principal branch of this eigenvalue equation yields one and only one unstable mode if and only if the wave-number lies in a band that decreases from Rayleighs band to zero as the Richardson number increases from 0 to ¼.

Neutral curves and stability boundaries in stratified flow

An example is presented which shows that the boundary of stability in an antisymmetric stratified shear flow is not necessarily marked by steady neutral waves with c = 0. The stability characteristics of stratified shear flow in the neighbourhood of the neutral curve are also discussed.

Normal modes of a rectangular tank with corrugated bottom

We study some effects of regular bottom corrugations on water waves in a long rectangular tank with vertical endwalls and open top. In particular, we consider motions which are normal modes of oscillation in such a tank. Attention is focused on the modes whose internodal spacing, in the absence of corrugations, would be near the wavelength of the corrugations. In these cases, the perturbation of the eigen-functions (though not of their frequencies) can be significant, e.g. the amplitude of the eigenfunction can be greater by a factor of ten or more near one end of the tank than at the other end. This is due to a cooperative effect of the corrugations, called Bragg resonance. We first study these effects using an asymptotic theory, which assumes that the bottom corrugations are of small amplitude and that the motions are slowly varying everywhere. We then present an exact theory, utilizing continued fractions. This allows us to deal with the rapidly varying components of the flow. The exact theory confirms the essential correctness of the asymptotic results for the slowly varying aspects of the motions. The rapidly varying parts (evanescent waves) are, however, needed to satisfy accurately the true boundary conditions, hence of importance to the flow near the endwalls.

Bifurcations Under Nongeneric Conditions

Much interest has been focused recently on bifurcation theorems (cf. [I] for bibliography), which state, for example, that if the linearization around a critical point of a differential equation satisfies some conditions, then the (nonlinear) equation has some special solutions near the critical point, e.g., other critical points or periodic solutions. A well known bifurcation theorem (e.g., [l, 2]), can be stated, in its simplest form, as follows: THEOREM 1. Let Xt = F(X, ,u) be a one-parameter family of autono- mous dazerential equations on Rn, F depending smoothly (e.g., Cz) on all of its n + 1 arguments, such that X = 0 is a critical point for each TV with 1 p 1 suficiently small (i.e., F(0, p) = 0). Let A, be the linearization (with respect to X) around X = 0 of F(X, t.~). Suppose that for y = 0, h = 0 is a simple eigenvalue of A, and let X(p) be the eigenvalue of A, which reduces to h = 0 for t.~ = 0. Suppose further that (dh/dp)(O) # 0. Then in every suficiently small neighborhood of (0,O) there is a curve in the n + 1 dimensional (X, p) space passing through X = 0, TV = 0, and distinct from the p axis, each of whose X components is a critical point for X, = F(X, p). Th ere are no other critical points in a neighborhood of x = 0, p = 0. This particular bifurcation theorem really just concerns the zeroes of a family of vector valued-functions. However, its main interest is in connection with solutions to differential equations. This form of 274

On higher order Bragg resonance of water waves by bottom corrugations

The exact theory of linearized water waves in a channel of indefinite length with bottom corrugations of finite amplitude (Howard & Yu, J. Fluid Mech., vol. 593, 2007, pp. 209-234) is extended to study the higher order Bragg resonances of water waves occurring when the corrugation wavelength is close to an integer multiple of half a water wavelength. The resonance tongues (ranges of water-wave frequencies) are given for these higher order cases. Within a resonance tongue, the wave amplitude exhibits slow exponential modulation over the corrugations, and slow sinusoidal modulation occurs outside it. The spatial rate of wave amplitude modulation is analysed, showing its quantitative dependence on the corrugation height, water-wave frequency and water depth. The effects of these higher order Bragg resonances are illustrated using the normal modes of a rectangular tank.

Bifurcations in Reaction–Diffusion Problems

Publisher Summary This chapter discusses bifurcations in reaction-diffusion problems. The general structure related bifurcation theorems is that from some hypotheses about the linearized version of the problem, conclusions are drawn about the full nonlinear one, valid in an appropriate neighborhood. This is analogous to the implicit function theorem, and indeed proofs of such bifurcation theorems typically involve transformations that convert the problem into a form in which the implicit function theorem can be applied. Another consequence of the hypotheses in the theorem stated above relates to the stability properties of the new family of critical points. When this is adjoined to the mere existence of the critical points, the bifurcation theorem becomes genuinely a result about differential equations. Bifurcations are said to exhibit an exchange of stabilities. To obtain a more satisfactory understanding of the shocks and to derive the conditions characterizing their propagation, one should return to the full reaction-diffusion equations and, at least in certain idealized cases, show that they do have solutions with a shock structure, a region of transition between two different plane wave solutions. This can be done for the case of a shock structure joining two plane-wave solutions with nearly equal wave numbers and frequencies by bifurcation methods used for the weak gas dynamic shock structure problem.

Time-Periodic and Spatially Irregular Patterns

Publisher Summary This chapter focuses on time-periodic and spatially irregular patterns. Slowly-varying waves cannot always remain slowly-varying. Eventually regions of increasingly rapid variation in local wave number can develop. It is the group velocities, not the phase velocities, which should be directed toward the transition zone; however, in the experimental situation, the phase and group velocities are in the same direction. Plane waves are the analogs of the constant states. Plane waves are not constant; however, are among the simplest special solutions. In the Zaiken–Zhabotinskii patterns, the patches of slowly-varying waves are frequently filled with a pattern of concentric circular wave crests propagating outward. The centers of these patterns resemble the boundaries where they meet other patches in being regions, which cannot be described as slowly varying waves.

Bragg Reflection of Water Waves in a Rectangular Channel

A series of corrugations on the bottom of a layer of water of otherwise uniform depth can have a cooperative effect on incident water waves. The phenomenon is well-known, called Bragg reflection or Bragg resonance. These effects on the normal modes of oscillation in a rectangular tank with corrugated bottom are studied using an asymptotic theory, and by developing an exact theory. The exact theory confirms the essential correctness of the asymptotic results for the slowly varying aspects of the motions. The rapidly varying components are, however, important to the flow near the boundaries. Higher order Bragg resonance, i.e. when the spacing of corrugations is an integer (greater than 1) multiple of half the water wavelength, is examined and the solution regimes (resonance tongues) are constructed using the exact theory.Copyright