Summary Coastal Engineering, Robert M. Sorensen Third Edition, complete coastal engineering study material

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Basic Coastal Engineering Robert M Sorensen Third Edition
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Coastal Engineering

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Two-Dimensional Wave Equations and Wave Characteristics / 27


Example 2.5-1
Consider the wave from Example 2.3–1 when it has propagated into a water
depth of 10 m without refracting and assuming energy gains and losses can be
ignored. Determine the wave height and the water particle velocity and pressure
at a point 1 m below the still water level under the wave crest. (Assume fresh
water.)
Solution:
From Example 2.3–1 we have Lo ¼ 156 m and Eq. (2.14) gives

9:81(10)2 2p(10)
L¼ tanh
2p L
"1 m
which can be solved by trial to yield L ¼ 93:3 m. Then, k ¼ 2p=93:3 ¼ 0:0673
and from Eq. (2.38)
! "
1 2(0:0673)(10)
n¼ 1þ ¼ 0:874
2 sinh (2(0:0673)(10)

With K r ¼ 1, Eq. (2.42) yields
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
156
H¼2 ¼ 1:97 m
2(0:874)(93:3)

At the crest of the wave cos (kx " st) ¼ 1, and z ¼ "1, so Eq. (2.21) gives
$ %
p(1:97) cosh (0:0673)(9)
u¼ ¼ 1:01 m=s
10 sinh (0:0673)(10)

which is the total particle velocity since w ¼ 0 under the wave crest.Equation
(2.32) gives
$ %
1000(9:81)(1:97) cosh (0:0673)9
P ¼ "1000(9:81)( " 1) þ
2 cosh (0:0673)10
2
¼ 19; 113 N=m



Remember,Eqs. (2.40) to (2.42) neglect energy transfer to and from waves by
surface and bottom eVects. The nature of these eVects is discussed brieXy below.
Bottom eVects, of course, require that the water depth be suYciently shallow for
a strong interaction between the wave train and the bottom.