(1) Introduction.

(a) The most elementary wave theory is the *small-amplitude *or *linear wave theory*. This theory,

developed by Airy (1845), is easy to apply, and gives a reasonable approximation of wave characteristics for

a wide range of wave parameters. A more complete theoretical description of waves may be obtained as the

sum of many successive approximations, where each additional term in the series is a correction to preceding

terms. For some situations, waves are better described by these higher-order theories, which are usually

referred to as *finite-amplitude wave theories *(Mei 1991, Dean and Dalrymple 1991). Although there are

limitations to its applicability, linear theory can still be useful provided the assumptions made in developing

this simple theory are not grossly violated.

(b) The assumptions made in developing the linear wave theory are:

! The fluid is homogeneous and incompressible; therefore, the density ρ is a constant.

! Surface tension can be neglected.

! Coriolis effect due to the earth's rotation can be neglected.

! Pressure at the free surface is uniform and constant.

! The fluid is ideal or inviscid (lacks viscosity).

! The particular wave being considered does not interact with any other water motions. The flow is

irrotational so that water particles do not rotate (only normal forces are important and shearing forces

are negligible).

! The bed is a horizontal, fixed, impermeable boundary, which implies that the vertical velocity at the

bed is zero.

! The wave amplitude is small and the waveform is invariant in time and space.

! Waves are plane or long-crested (two-dimensional).

(c) The first three assumptions are valid for virtually all coastal engineering problems. It is necessary

to relax the fourth, fifth, and sixth assumptions for some specialized problems not considered in this manual.

Relaxing the three final assumptions is essential in many problems, and is considered later in this chapter.

(d) The assumption of irrotationality stated as the sixth assumption above allows the use of a mathemati-

cal function termed the *velocity potential Φ*. The velocity potential is a scaler function whose gradient (i.e.,

the rate of change of *Φ *relative to the x-and z-coordinates in two dimensions where x = horizontal,

z = vertical) at any point in fluid is the velocity vector. Thus,

MΦ

(II-1-1)

M*x*

is the fluid velocity in the x-direction, and

MΦ

(II-1-2)

M*z*

is the fluid velocity in the z-direction. Φ has the units of length squared divided by time. Consequently, if

Φ(x, z, t) is known over the flow field, then fluid particle velocity components *u *and *w *can be found.

Water Wave Mechanics

II-1-5

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