80
deterministic descriptions of offshore waves
By referring to Figures 3.9 and 3.10, we see that both the cnoidal and
free stream function théories are appropriate for describing this wave, for which
computer-aided solutions are needed.
2. Rather than generate our own numerical results, refer to results of Masch
and Wiegel (1961). That is,
Compute: T y/g/d = 15^32.2/20 = 19.03
Table 2:
c2/M = 0.9675
Compute: c = 24.96 ft/sec
3. As a check on this resuit, we find that the wave characteristics place it
near Cases 3D and 4D given by Dean (1974), but Dean’s tabulated values do
not encompass the numerical values of the présent problem.
4.
From linear theory the wave celerity is c = yfgd — 25.37 ft/sec.
5.
From solitary theory the wave celerity is c — y/g(d + H) = 26.62 ft/sec.
These last two values of c are reasonably close to the value 24.96 ft/sec
obtained from cnoidal theory. This does not imply, however, that the other flow
parameters are necessarily in close agreement, and those parameters need to be
calculated from cnoidal or free stream function theory.
Linear wave theory will be employed extensively in subséquent chapters to
calculate fluid loading on offshore structures.
PROBLEMS
3.1 For simple wave swells with periods T = 2m/u — X/c, 1 < T <30 sec,
where the depth to wave length ratio satisfies d/X > 0.5, the waves are defined
as deepwater waves. In this case show that the wave celerity and period hâve
the following approximate forms:
S
If T = 10 sec and X = 512 ft, calculate the wave celerity and the minimum
depth for which these relationships are valid.
3.2 A simple, shallow water wave is defined when the ratio of water depth
to wavelength satisfies d/X < 0.05. In this case, show that the wave celerity is
given approximately by c = yf^d. If À = 512 ft, what is the maximum water
depth for which this relationship holds?
3.3 For simple linear waves of length A at intermediate water depths d,
then 0.05 < d/X < 0.5. Deduce that the square of the wave celerity is given by
the following formula.
g X
2?rd
c = — tanh----2tt
A
deterministic descriptions of offshore waves
By referring to Figures 3.9 and 3.10, we see that both the cnoidal and
free stream function théories are appropriate for describing this wave, for which
computer-aided solutions are needed.
2. Rather than generate our own numerical results, refer to results of Masch
and Wiegel (1961). That is,
Compute: T y/g/d = 15^32.2/20 = 19.03
Table 2:
c2/M = 0.9675
Compute: c = 24.96 ft/sec
3. As a check on this resuit, we find that the wave characteristics place it
near Cases 3D and 4D given by Dean (1974), but Dean’s tabulated values do
not encompass the numerical values of the présent problem.
4.
From linear theory the wave celerity is c = yfgd — 25.37 ft/sec.
5.
From solitary theory the wave celerity is c — y/g(d + H) = 26.62 ft/sec.
These last two values of c are reasonably close to the value 24.96 ft/sec
obtained from cnoidal theory. This does not imply, however, that the other flow
parameters are necessarily in close agreement, and those parameters need to be
calculated from cnoidal or free stream function theory.
Linear wave theory will be employed extensively in subséquent chapters to
calculate fluid loading on offshore structures.
PROBLEMS
3.1 For simple wave swells with periods T = 2m/u — X/c, 1 < T <30 sec,
where the depth to wave length ratio satisfies d/X > 0.5, the waves are defined
as deepwater waves. In this case show that the wave celerity and period hâve
the following approximate forms:
S
If T = 10 sec and X = 512 ft, calculate the wave celerity and the minimum
depth for which these relationships are valid.
3.2 A simple, shallow water wave is defined when the ratio of water depth
to wavelength satisfies d/X < 0.05. In this case, show that the wave celerity is
given approximately by c = yf^d. If À = 512 ft, what is the maximum water
depth for which this relationship holds?
3.3 For simple linear waves of length A at intermediate water depths d,
then 0.05 < d/X < 0.5. Deduce that the square of the wave celerity is given by
the following formula.
g X
2?rd
c = — tanh----2tt
A