where *N*z and *N*c are the number of zero-upcrossings and crests in the wave record, respectively. We

emphasize that in Tucker's method of wave train analysis, crests are defined by zero-crossing. Note also by

definition of these periods that *T*c # Tz.

(h) The list of definitions stated above is not all-inclusive, and several other statistical quantities may

be obtained from a wave train analysis (Ochi 1973; IAHR 1986). For example, the rms surface elevation *η*rms

(described later in the short-term sea states section) (*σ *in IAHR list) defines the standard deviation of the

surface elevation, and the significant wave height *H*s is related to *η*rms by

(II-1-117)

(4) Significant wave height.

(a) The *significant wave height H*s (or *H*1/3) is the most important quantity used describing a sea state

and thus, is discussed further here for completeness. The concept of significant wave height was first

introduced by Sverdrup and Munk (1947). It may be determined directly from a wave record in a number

of ways. The most frequently used approach in wave-by-wave analysis is to rank waves in a wave record and

then choose the highest one-third waves. The average of the chosen waves defines the significant wave height

as

j *H*i

1

(II-1-118)

3

where *N *is the number of individual wave heights *H*i in a record ranked highest to lowest.

(b) Sverdrup and Munk (1947) defined significant wave height in this fashion because they were

attempting to correlate what sailors reported to what was measured. Hence, this is an empirically driven

definition. Today, when wave measuring is generally automated, some other parameter might be appropriate,

but significant wave height remains in recognition of its historical precedence and because it has a fairly

tangible connection to what observers report when they try to reduce the complexity of the sea surface to one

number. It is important to recognize that it is a statistical construct based only on the height distribution.

Knowing the significant height from a record tells us nothing about period or direction.

(5) Short-term random sea state parameters.

(a) It is well-known that any periodic signal *η(t) *with a zero mean value can be separated into its

frequency components using the standard *Fourier analysis*. Periodic wave records may generally be treated

as random processes governed by laws of the probability theory. If the wave record is a random signal, the

term used is *random waves*. For a great many purposes, ocean wave records may be considered random (Rice

1944-1945, Kinsman 1965, Phillips 1977, Price and Bishop 1974).

(b) The statistical properties of a random signal like the wave surface profile may be obtained from a

set of many simultaneous observations called an *ensemble *or set of signals {*η*1(t), η2(t), η3(t),...}, but not from

a single observation. A single observation even infinitely long may not be sufficient for determining the

spatial variability of wave statistics. An ensemble consists of different realizations or measurements of the

process *η(t) *at some known locations. To determine wave properties from the process *η(t)*, certain

assumptions related to its time and spatial variation must be made.

II-1-68

Water Wave Mechanics

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