Citation
Lau, Joseph P. (1968) Steady surface wave pattern in a shear flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PEETKY40. https://resolver.caltech.edu/CaltechTHESIS:12212015100519870
Abstract
The subject under investigation concerns the steady surface wave patterns created by small concentrated disturbances acting on a nonuniform flow of a heavy fluid. The initial value problem of a point disturbance in a primary flow having an arbitrary velocity distribution (U(y), 0, 0) in a direction parallel to the undisturbed free surface is formulated. A geometric optics method and the classical integral transformation method are employed as two different methods of solution for this problem. Whenever necessary, the special case of linear shear (i.e. U(y) = 1+ϵy)) is chosen for the purpose of facilitating the final integration of the solution.
The asymptotic form of the solution obtained by the method of integral transforms agrees with the leading terms of the solution obtained by geometric optics when the latter is expanded in powers of small ϵ r.
The overall effect of the shear is to confine the wave field on the downstream side of the disturbance to a region which is smaller than the wave region in the case of uniform flows. If U(y) vanishes, and changes sign at a critical plane y = y_{cr} (e.g. ϵy_{cr} = 1 for the case of linear shear), then the boundary of this asymmetric wave field approaches this critical vertical plane. On this boundary the wave crests are all perpendicular to the xaxis, indicating that waves are reflected at this boundary.
Inside the wave field, as in the case of a point disturbance in a uniform primary flow, there exist two wave systems. The loci of constant phases (such as the crests or troughs) of these wave systems are not symmetric with respect to the xaxis. The geometric optics method and the integral transform method yield the same result of these loci for the special case of U(y) = U_{o}(1 + ϵy) and for large Kr (ϵr ˂˂ 1 ˂˂ Kr).
An expression for the variation of the amplitude of the waves in the wave field is obtained by the integral transform method. This is in the form of an expansion in small ϵr. The zeroth order is identical to the expression for the uniform stream case and is thus not applicable near the boundary of the wave region because it becomes infinite in that neighborhood. Throughout this investigation the viscous terms in the equations of motion are neglected, a reasonable assumption which can be justified when the wavelengths of the resulting waves are sufficiently large.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Engineering 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Engineering and Applied Science 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  23 May 1968 
Record Number:  CaltechTHESIS:12212015100519870 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:12212015100519870 
DOI:  10.7907/PEETKY40 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  9337 
Collection:  CaltechTHESIS 
Deposited By:  INVALID USER 
Deposited On:  21 Dec 2015 19:06 
Last Modified:  20 Dec 2019 19:30 
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