théories. The first finite amplitude theory, known as the trochoidal
theory, was developed by Gerstner (1802). It is so called because the
free surface or wave profile is a trochoid. This theory is mentioned
only because of its classical interest. It is not recommended for application, since the water particle motion predicted is not that observed in
nature. The trochoidal theory does, however, predict wave profiles quite
accurately. Stokes (1880) developed a finite-amplitude theory which is
more satisfactory than the trochodial theory. Only the second-order Stokes
équations will be presented, but the use of higher order approximations is
sometimes justified for the solution of practical problems.
For shallow-water régions, cnoidal wave theory, originally developed
by Korteweg and De Vries (1895), predicts rather well the waveform and
associated motions for some conditions. However, cnoidal wave theory has
received little attention with respect to actual application in the solution of engineering problems. This may be due to the difficulties in
making computations. Recently, the work involved in using cnoidal wave
theory has been substantially reduced by introduction of graphical and
tabular forms of functions. (Wiegel, 1960), (Masch and Wiegel, 1961.)
Application of the theory is still quite involved. At the limit of cnoidal
wave theory, certain aspects of wave behavior may be described satisfactorily by solitary wave theory. Unlike cnoidal wave theory, the solitary
wave theory is easy to use since it reduces to functions which may be
evaluated without recourse to spécial tables.
Development of individual wave théories is omitted, and only the
results are presented since the purpose is to présent only that information which may be useful for the solution of practical engineering
problems. Many publications are available such as Wiegel (1964), Kinsman
(1965), and Ippen (1966a), which cover in detail the development of some of
the théories mentioned above as well as others. The mathematics used here
generally will be restricted to element'ary arithmetic and algebraic operations. Emphasis is placed on sélection of an appropriate theory in accordance with its application and limitations.
Numerous example problems are provided to illustrate the theory
involved and to provide some practice in using the appropriate équations
or graphical and tabular functions. Some of the sample computations give
more significant digits than are warranted for practical applications.
For instance, a wave height could be determined to be 10.243 feet for
certain conditions based on purely theoretical considérations. This
accuracy is unwarranted because of the uncertainty in the basic data used
and the assumption that the theory is représentative of real waves. A
practical estimate of the wave height given above would be 10 feet. When
calculating real waves, the final answer should be rounded off.
2.22 WAVE FUNDAMENTALS AND CLASSIFICATION OF WAVES
Any adéquate physical description of a water wave involves both its
surface form and the fluid motion beneath the wave. A wave which can be
described in simple mathematical ternis is called a simple wave. Waves
2-3
theory, was developed by Gerstner (1802). It is so called because the
free surface or wave profile is a trochoid. This theory is mentioned
only because of its classical interest. It is not recommended for application, since the water particle motion predicted is not that observed in
nature. The trochoidal theory does, however, predict wave profiles quite
accurately. Stokes (1880) developed a finite-amplitude theory which is
more satisfactory than the trochodial theory. Only the second-order Stokes
équations will be presented, but the use of higher order approximations is
sometimes justified for the solution of practical problems.
For shallow-water régions, cnoidal wave theory, originally developed
by Korteweg and De Vries (1895), predicts rather well the waveform and
associated motions for some conditions. However, cnoidal wave theory has
received little attention with respect to actual application in the solution of engineering problems. This may be due to the difficulties in
making computations. Recently, the work involved in using cnoidal wave
theory has been substantially reduced by introduction of graphical and
tabular forms of functions. (Wiegel, 1960), (Masch and Wiegel, 1961.)
Application of the theory is still quite involved. At the limit of cnoidal
wave theory, certain aspects of wave behavior may be described satisfactorily by solitary wave theory. Unlike cnoidal wave theory, the solitary
wave theory is easy to use since it reduces to functions which may be
evaluated without recourse to spécial tables.
Development of individual wave théories is omitted, and only the
results are presented since the purpose is to présent only that information which may be useful for the solution of practical engineering
problems. Many publications are available such as Wiegel (1964), Kinsman
(1965), and Ippen (1966a), which cover in detail the development of some of
the théories mentioned above as well as others. The mathematics used here
generally will be restricted to element'ary arithmetic and algebraic operations. Emphasis is placed on sélection of an appropriate theory in accordance with its application and limitations.
Numerous example problems are provided to illustrate the theory
involved and to provide some practice in using the appropriate équations
or graphical and tabular functions. Some of the sample computations give
more significant digits than are warranted for practical applications.
For instance, a wave height could be determined to be 10.243 feet for
certain conditions based on purely theoretical considérations. This
accuracy is unwarranted because of the uncertainty in the basic data used
and the assumption that the theory is représentative of real waves. A
practical estimate of the wave height given above would be 10 feet. When
calculating real waves, the final answer should be rounded off.
2.22 WAVE FUNDAMENTALS AND CLASSIFICATION OF WAVES
Any adéquate physical description of a water wave involves both its
surface form and the fluid motion beneath the wave. A wave which can be
described in simple mathematical ternis is called a simple wave. Waves
2-3