Fourier spectrum and shape evolution of an internal Riemann wave of moderate amplitude

Abstract

The nonlinear deformation of long internal waves in the ocean is studied using the dispersionless Gardner equation. The process of nonlinear wave deformation is determined by the signs of the coefficients of the quadratic and cubic nonlinear terms; the breaking time depends only on their absolute values. The explicit formula for the Fourier spectrum of the deformed Riemann wave is derived and used to investigate the evolution of the spectrum of the initially pure sine wave. It is shown that the spectrum has exponential form for small times and a power asymptotic before breaking. The power asymptotic is universal for arbitrarily chosen coefficients of the nonlinear terms and has a slope close to –8/3

Strongly Nonlinear Transverse Perturbations in Phononic Crystals

“…Initially sinusoidal wave profile ( ) is transformed into a trapezoidal one. This agrees well with the results of the wave transformation in a cubic nonlinear medium obtained in [17,18]. The transformation of the profile appears faster and is more pronounced for large-amplitude waves.…”

“…It is worth mentioning that at the moment of the onset of the gradient catastrophe in the profile ( ) the singularity of type 1/3 is formed, as it was recently shown for hyperbolic equations of a relatively general form [18,19]. Since the function ( ) is the integral of ( ), then at the breaking point the singularity is very weak ( 4/3 ) and that is why it is not visible in Figures 4(a), 4(c), and 4(e) in contrast to the graphs in Figures 4(b), 4(d), and 4(f).…”

“…In a traveling wave due to (18), the equations of system (15) become identical; therefore, later either of them can be used. It is conveniently rewritten as…”

“…Then the value of the function ( , ) at the moment of breaking can be calculated by formula (31) provided by (41), (42), and (47): The plot of points of breaking br and br , normalized to the value of the amplitude and value of the amplitude multiplied wave number , respectively, depending on the values of the quantity is shown in Figure 3. Temporal evolution of the transverse perturbations ( , ), normalized to the amplitude of the initial perturbation, calculated by formulas (18), (20), (22), and (31) for the harmonic initial perturbation with the values = 0.1, = 1, and = 10, is represented in Figure 4. From Figure 4 it is seen that with the propagation of the harmonic perturbation the wave profile becomes asymmetric and the asymmetry for the functions ( ) and ( ) is shown differently.…”