The solitary wave of maximum amplitude

1966 ◽  
Vol 26 (2) ◽  
pp. 309-320 ◽  
Author(s):  
Charles W. Lenau
Keyword(s):  
Power Series ◽  
Solitary Wave ◽  
Velocity Potential ◽  
Maximum Amplitude ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Complex Velocity ◽  
Constant Form ◽  
Complex Velocity Potential ◽  
Infinite Power

The maximum amplitude of the solitary wave of constant form is determined to be 0·83b, where b is the depth far from the crest. In the analysis it is assumed that the crest is pointed and the motion is two-dimensional and irrotational. The complex velocity potential is expressed in terms of known singularities and an infinite power series with unknown coefficients. Approximate solutions are obtained by truncating the power series after N terms, where N = 1, 3, 5, 7, and 9. The amplitude, a measure of the error, and several other pertinent quantities are computed for each value of N.

scholarly journals The Two-Phase Hell-Shaw Flow: Construction of an Exact Solution

2013 ◽  
Vol 18 (1) ◽  
pp. 249-257
Author(s):  
K.R. Malaikah
Keyword(s):  
Exact Solution ◽  
Velocity Potential ◽  
Time Dependent ◽  
Computational Domain ◽  
Phase Problem ◽  
Complex Velocity ◽  
Two Phase ◽  
Interface Evolution ◽  
Complex Velocity Potential ◽  
Branch Cuts

We consider a two-phase Hele-Shaw cell whether or not the gap thickness is time-dependent. We construct an exact solution in terms of the Schwarz function of the interface for the two-phase Hele-Shaw flow. The derivation is based upon the single-valued complex velocity potential instead of the multiple-valued complex potential. As a result, the construction is applicable to the case of the time-dependent gap. In addition, there is no need to introduce branch cuts in the computational domain. Furthermore, the interface evolution in a two-phase problem is closely linked to its counterpart in a one-phase problem

scholarly journals Approximate Solutions of Two-Dimensional Burgers' and Coupled Burgers' Equations by Residual Power Series Method

2018 ◽  
Vol 22 ◽  
pp. 01044
Author(s):  
Selahattin Gulsen ◽  
Mustafa Inc ◽  
Harivan R. Nabi
Keyword(s):  
Power Series ◽  
Taylor Series Expansion ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Series Method ◽  
Power Series Method ◽  
Burgers Equations ◽  
Residual Power Series Method ◽  
Residual Power Series ◽  
Residual Power

In this study, two-dimensional Burgers' and coupled Burgers' equations are examined by the residual power series method. This method provides series solutions which are rapidly convergent and their components are easily calculable by Mathematica. When the solution is polynomial, the method gives the exact solution using Taylor series expansion. The results display that the method is more efficient, applicable and accuracy and the graphical consequences clearly present the reliability of the method.

The almost-highest wave: a simple approximation

1979 ◽  
Vol 94 (2) ◽  
pp. 269-273 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Free Surface ◽  
Gravity Wave ◽  
Velocity Potential ◽  
Vertical Plane ◽  
Simple Approximation ◽  
Complex Velocity ◽  
Complex Velocity Potential

The crest of a steep, symmetric gravity wave is shown to be closely approximated by the expression \[ x+iy = \frac{\alpha +\gamma i\chi}{(\beta + i\chi)^{\frac{1}{3}}}, \] where x, y are co-ordinates in the vertical plane, χ is the complex velocity potential and α, β, γ are certain constants. This expression is asymptotically correct both for small and for large values of |χ|; and the free surface agrees with the exact profile calculated by Longuet-Higgins & Fox (1977) everywhere to within 1·5 per cent. The pressure at the surface is constant to within 5 per cent.

scholarly journals The flow of a compressible liquid in the neighbourhood of the throat of a constriction in a circular wind channel

1932 ◽  
Vol 135 (827) ◽  
pp. 498-511 ◽  
Keyword(s):  
Power Series ◽  
Elastic Property ◽  
Compressible Fluid ◽  
Velocity Potential ◽  
Complete Solution ◽  
Maximum Velocity ◽  
Equation Of Motion ◽  
Compressible Liquid ◽  
Two Dimensional ◽  
Irrotational Motion

As a result of earlier work by G. I. Taylor on the two-dimensional motion of a compressible fluid, it appears evident that the elastic property of a fluid places a limitation upon the maximum velocity which can exist in a field in order that a certain type of irrotational motion may continue to be possible. So far a complete solution of the equation of motion of a compressible fluid in any particular problem has eluded the workers on this subject, and the greatest theoretical advance came when Taylor used the idea of expanding the velocity potential in a power series about the point of maximum velocity.

The Froude number for solitary waves

1984 ◽  
Vol 97 ◽  
pp. 193-197 ◽  
Author(s):  
J. B. McLeod
Keyword(s):  
Solitary Wave ◽  
Froude Number ◽  
Solitary Waves ◽  
Constant Velocity ◽  
Inviscid Fluid ◽  
Two Dimensional ◽  
Constant Form ◽  
Image Position ◽  
Horizontal Bottom

SynopsisThe paper is concerned with the problem of a solitary wave moving with constant form and constant velocity c on the surface of an incompressible, inviscid fluid over a horizontal bottom. The motion is assumed to be two-dimensional and irrotational, and if h is the depth of the fluid at infinity and g the acceleration due to gravity, then the Froude number F is defined byThe result that F>1 has recently been proved by Amick and Toland by means of a long and complicated argument. Here we give a short and simple one.

scholarly journals Numerical solutions of nonlinear fractional model arising in the appearance of the strip patterns in two-dimensional systems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Sunil Kumar ◽  
Amit Kumar ◽  
Shaher Momani ◽  
Mujahed Aldhaifallah ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Power Series ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Analytical Techniques ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Power Series Method ◽  
Analytical Technique ◽  
Residual Power Series ◽  
Residual Power

Abstract The main aim of this paper is to present a comparative study of modified analytical technique based on auxiliary parameters and residual power series method (RPSM) for Newell–Whitehead–Segel (NWS) equations of arbitrary order. The NWS equation is well defined and a famous nonlinear physical model, which is characterized by the presence of the strip patterns in two-dimensional systems and application in many areas such as mechanics, chemistry, and bioengineering. In this paper, we implement a modified analytical method based on auxiliary parameters and residual power series techniques to obtain quick and accurate solutions of the time-fractional NWS equations. Comparison of the obtained solutions with the present solutions reveal that both powerful analytical techniques are productive, fruitful, and adequate in solving any kind of nonlinear partial differential equations arising in several physical phenomena. We addressed $L_{2}$ L 2 and $L_{\infty }$ L ∞ norms in both cases. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present aforesaid methods and noted excellent agreement. In this study, we use the fractional operators in Caputo sense.

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