Estimation and Analysis of Horizontal Bottom Velocities Due to Waves

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26L.29

Estimation

and Analysis

of

\

Horizontal

BoHom Velocities

Due to Waves

J. B. HERBICH end S. B. BRAHME Ocean Engineering Program

TAMU-SG-77-208 COE Report No.202 Aug. 1977

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OF

HORIZONTAL BOTTOM VELOCITIES DUE TO WAVES

by

John B. Herbich and

Shashikant B. Brahme Ocean Engineering Program

"

August 1977

TAMU-SG-77-208 COE Report No. 202

Partially supported through Institutional Grant 04-6-158-44012 to Texas A&M University

by the National Oceanic and Atmospheric Administration's Office of Sea Grants

Department of Commerce

and the United Nations Development Programme, United Nations

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2

G- J.r -

2 I

ABSTRACT

Maximum bot tom velocities caused by waves were calculated using di-gital computers. Four wave theories, Airy, Stokes third order, Cnoidal and Solitary,were applied in the computation.

Results of the study were tabulated and presented graphically to highlight the importance of various parameters affecting the maximum bottom velocity.

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The study described in this report was conducted as part of continu-ing research within the Ocean Engineercontinu-ing Program at Texas A&M University.

The Texas Engineering Experiment Station provided funds to defray part of the computer costs.

The manuscript was edited by Dr. Gisela Mahoney and typed for publi-cation by Joyce McCabe.

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I. INTRODUCTION

.

11.

PRESENT STUDY.

.

. .

lIl.

DATA ANALYSIS.

IV. CONCLUDING REMARKS

V. REFERENCES

VI. APPENDIX .

.

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1.

Tables .

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.

2.

Figures.

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Page

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1 3 4 6 7

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8 9

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27

Abstract

Preface.

Table of Contents.

v

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BOTTOM VELOCITIES DUE TO WAVES

I. INTRODUCTION

A storm or a hurricane generated in the,sea produces waves which move towards the shoreline. Water particles under such waves move in orbits. When the wave is in deep water the orbital velocity of the particles along the seabed is negligibly small and as such creates no problems. However, when the wave approaches the shallow water it under-goes considerable change. The height of the wave (H) increases, the length (L) decreases and thus the wave becomes more and more steep. A dimensionless parameter commonly used is the wave steepness. The orbital velocity of the particle also undergoes change. The wave finally breaks in the surf zone. The bottom velocity generated by waves is a function of the wave height, the wave period (T) and the water depth (h). Con-siderable bottom velocities have been observed particularly when a hurri-cane passes a certain area of the sea. These bottom velocities cause sediment movement of immense magnitude. At times the bottom velocities were found to be so high as to cause considerable damage to marine struc-tures and offshore pipelines. Considerable research was therefore done in this field to improve the understanding of wave phenomena. Many scientists and engineers have developed theories to describe the various parameters of waves in the best possible manner. Notable among these are Airy, Stokes, Keulegan, Le Méhaute; Dean, and others. Attempts were also made to compare the theories with the actual waves generated in nature. In the present report one important aspect of the waves, i.e., the horizontal bottom velocity is examined in view of its importance

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11. PRESENT STUDY

Four wave theories, i.e., Airy, Stokes 111, Cnoidal and Solitary Wave have been examined in the present studies. The important para-meters considered are the wave height, wave period and water depth. The following ranges of values are selected for these parameters in order to cover a wide range of possibilities.

i. Wave Heights 2 feet to 24 feet (significant) 3.56 feet to 42.72 feet (Hmax) ii. Wave Periods 2.9 seconds to 15 seconds iii. Water Depths 8 feet to 150 feet

Computations for the bottom velocities were made on the digital computer. Bottom horizontal velocities (maximum) under wave crest and wave trough were computed for different wave theories indicated .above. The validi-ties of the various wave theories were taken into account while preparing the computer programs. The data obtained from the computer were tabulated (Tables 1 to 18) and plotted to show the effects of the various parameters on the bottom velocities (Figures 1 to 14).

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The resu1ts obtained were found to be quite interesting and empha-sized the importance of various parameters being considered.

Figures 1 to 9 present plots of maximum bottom horizontal ve10cities under wave crest as a function of the water depth for different signifi-cant and corresponding maximum wave heights for two different periods. The resu1ts indicated the fo11owing:

1. Some variation in the maximum horizontal bottom ve10cities was ~ noticed for different wave theories as expected. The variation was mar-gina1 for low wave heights but increased considerab1y for higher wave heights.

2. The bottom ve10cities increase with increase in wave height and wave period and decrease with increase in water depths.

3. The bottom velocity was quite large for a significant wave of 24 feet (Hs) and 42.72 feet (Hmax) even in a depth of 150 feet of water. The magnitude of bottom velocity for a 15-second wave period was of the order of 4 feet per second and 7.5 feet per second, respective1y. A velocity of this magnitude is expected to cause considerable sediment movement a10ng the bed.

4. The maximum bottom velocity for any wave occurs at the breaking region where the sediment movement would be maximum.

5. The bottom velocity drops rapid1y initially from the breaker zone towards the sea up to some depth and then drops gradually till it becomes zero in the deep sea.

In order to assess the effects of wave heights and wave periods on the maximum bottom velocities, plots were made of maximum bottom velocities

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versus the wave heights. These are presented in Figures 10 to 12. These plots indicated a linear relationship between maximum bottom velocities and wave heights. The velocities also showed increase with increase in wave period as would be evident from the figures. A notable feature noticed was the scatter of data for depth of water of 80 feet for 15-second waves. Two lines have been drawn to indicate the upper and lower limits of velocities for this periode The scatter however reduces when the depth increases and for a 150 foot depth of water there is negligible scatter of data (see Figure 12).

There was very little difference between the maximum horizontal bottom velocities under a wave crest and that under a wave trough for most of the wave theories except the Stokes 111. This wave theory in-dicated considerable variation in the bottom velocities for crest and trough. In order to examine the exact nature of the variation, plots were made for maximum bottom velocities versus wave height for two typi-cal depths of water, i.e. 40 feet and 80 feet. The variations were very high for 40-foot depths compared to that for 80-foot depths of water. The difference in the velocities increased with wave heights and wave periods and decreased with depth. The results are presented in Figures 13 and 14.

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Maximum bot tom velocity under a wave is one of the important aspects of wave phenomena that needs thorough study since it causes sediment move-ment of immense magnitude along the bed. This affects the pipelines laid along the seabed and the scour around objects placed along the bottom. Bottom velocities were determined with the aid of a digital computer on the basis of four wave theories. Plots have been made to highlight the importance of various parameters affecting the maximum bottom velocity. It is suggested that maximum bottom velocities under a wave crest be con-sidered while estimating the sediment movement and designing offshore pipe-lines, platforms, and underwater objects.

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REFERENCES

1. Wiegel, R.L., "Oceanographic Engineering", Eng1ewood C1iffs, Prentice-Ha11, I nc ., 1964.

2. Skje1breia, L., "Gravity Waves. Stokes' Third Order Approximation Tables of Functions", Counci1 on Wave Research, The Engineering Foundation, 1959.

3. Masch, F.D., and Wiegel, R.L., "Cnoida1 Wave Tables of Function", Coun-ci1 on Wave Research, The Engineering Foundation, 1961.

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BOT TOM VELOCITIES FOR DIFFERENT WAVE THEORIES

SL WAVE HT. WAVE PERIOD WATER SOLITARY WAVE CNOIDAL STOKES 111 AIRY REMARKS

NO DEPTH THEORY

CREST TROUGH CREST TROUGH CREST TROUGH CREST TROUGH 2.3 2.7112 2.6366 Bottom veloc-1 2 ft 2.9 secs 8 ft 1.2720 1.2121 1.2201 1.2159 ities in ft/sec 24 ft - 0.l384 0.l384 0.l301 0.1297 40 ft 0.0141 0.0141 0.0127 0.0126 64 ft 0.0004 0.0004 0.0004 0.0004 80 ft 0.0000 0.0000 0.0000 0.0000 100 ft 0.0000 0.0000 0.0000 0.0000 150 ft 0.0000 0.0000 0.0000 0.0000 3.81 4.5765 4.3935 2 2 ft 10 secs 8 ft 1.9407 1.9404 24 ft 1.0440 1.0439 40 ft 0.711 0.724 0.7665 0.7302 0.7489 0.7489 64 ft 0.5247 0.5176 0.5213 0.5212 80 ft 0.4260 0.4230 0.4245 0.4245 100 ft 0.3347 0.3337 0.3342 0.3341 150 ft 0.1884 0.1883 0.1882 0.1882 I.D

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TABLE NO. 2

BOTTOM VELOCITIES FOR VARIOUS WAVE THEORIES _,

o

---SL WAVE HT. WAVE PERIOD WATER SOLITARY WAVE CNOIDAL STOKES III AIRY REMARKS

NO DEPTH THEORY

CREST TROUGH CREST TROUGH CREST TROUGH CREST TROUGH

4.15 4.7125 4.5842 3 3.56 ft 2.9 secs 8 ft 2.3738 2.1559 24 ft 0.2743 0.2743 40 ft 0.0304 0.0304 64 ft 0.0011 0.0011 80 ft 0.0001 0.0001 100 ft 0.000 0.000 150 ft 0.000 0.000 4.06 ft 4.7872 4.6569 4 3.56 ft 10 secs 8 ft 3.4545 3.4538 24 ft 1.8584 1.8582 40 ft 1.3881 1.2773 1.3331 1.3331 64 ft 0.9391 0.9165 0.9278 0.9278 80 ft 0.7604 0.7512 0.7556 0.7556 100 ft 0.5968 0.5935 0.5948 0.5948 150 ft 0.3355 0.3352 0.3350 0.3350

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BOTTOM VELOCITIES FOR DIFFERENT WAVE THEORIES

NO DEPTH THEORY

CREST TROUGH CREST TROUGH CREST TROUGH CREST TROUGH 4.83 5.2638 5.7205 5 4 ft 3.8 secs 8 ft 3.5122 2.5794 3.0919 3.0867 24 ft 0.8612 0.8573 0.8218 0.8210 40 ft 0.2379 0.2379 0.2197 0.2195 64 ft 0.0329 0.0329 0.0288 0.0287 80 ft 0.0087 0.0087 0.0074 0.0074 100 ft 0.0000 0.0000 0.0000 0.0000 150 ft 0.0000 0.0000 0.0000 0.0000 6.6 6.5868 6.4074 6 4 ft 10 secs 8 ft 3.8815 3.8807 24 ft 2.1750 1.8170 2.0883 2.0882 40 ft 1.4140 1.4340 1.5671 1.4222 1.4980 1.4979 64 ft 1.0568 1.0283 1.0426 1.0425 80 ft 0.8552 0.8435 0.8490 0.8490 100 ft 150 ft 0.3771 0.3767 0.3764 0.3746 _, ..J

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-TABLE NO. 4

BorTOM VELOCITIES FOR DIFFERENT WAVE THEORIES _.

N

SL WAVE HT. WAVE PERIOD WATER SOLITARY WAVE CNOIDAL STOKES III AIRY REMARKS

NO DEPTH THEORY

CREST TROUGH CREST TROUGH CREST TROJGH CREST TROJGH

7 7.12 ft 3.8 secs 8 ft '6.7331 3.7769 5.5036 5.4943 8.90 ft 24 ft _1.6482 _1.6316 _1.4643 _1.4629 40 ft _0.48₂₁ _0.4820 _0.3918 _0.3914 64 ft _0.0729 _0.0729 _0.0_'_{509 0.0508} 80 ft _0.0₂₀₆ ₀_.₀₂₀₆ _0.0₁₃₀ ₀_.₀₁₃₀ 100 ft 150 ft _0.0000 _0.0000 8 7.12 ft 10 secs 8 ft _6.9094 _6.9077 0.41 ft 8.5407 8.3642

.

24 ft 3.8520 2.8060 3.7172 3.7169 40 ft 2.4410 2.5160 2.8787 2.4207 2.6665 2.6663 64 ft _1.90₁₆ _1.8108 _1.8557 _1.8557 80 ft _1.₅₃₁₈ _1.4947 _1.5112 _1.5112 100 ft 150 ft _0.6729 _0.6716 _0.6700 _0.6700

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-BOT TOM VELOCITIES FOR DIFFERENT WAVE THEORIES

NO DEPTH THEORY

7.20 ft 6.9656 6.6691 9 6 ft 4.6 secs 8 ft 6.4739 3.0103 5.0804 5.0750 24 ft 1.9520 1.8942 1.8656 1.8647 40 ft 0.8242 0.8224 0.7805 0.7801 64 ft 0.2190 0.2190 0.1998 0.1997 80 ft 0.0891 0.0891 0.0790 0.0789 100 ft 0.0289 0.0289 0.0247 0.0247 150 ft 0.0017 0.0017 0.0014 0.0014 10 6 ft 10 secs 8 ft 5.8222 5.8211 9.18 ft 8.0191 7.8608 24 ft 3.2630 2.5050 3.1325 3.1323 40 ft 2.3995 2.0739 2.2470 2.2469 64 ft 1.5962 1.5319 1.5638 1.5638 80 ft 1.2879 1.2615 1.2735 1.2735 100 ft 1.0087 0.9992 1.0024 1.0024 150 ft 0.5665 0.5656 0.5646 0.5646 ... w

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TABLE NO. 6

-~

--SL WAVE HT. WAVE PERIOD WATER SOLITARY WAVE CNOIDAL STOKES III AIRY REMARKS

NO DEPTH THEORY

11 10.68 ft 4.6 secs 8 ft 9.0432 9.0335 12.35 ft 8.8421 8.7145 24 ft 3.6816 3.4580 3.3209 3.3192 40 ft 1.5987 1.5908 1.3892 1.3886 64 ft 0.4519 0.4519 0.3537 0.3537 80 ft 0.1919 0.1919 0.1406 0.1406 100 ft 0.0655 0.0655 0.0440 0.0440 150 ft 0.0044 0.0044 0.0024 0.0024 12 10.68 ft 10 secs 8 ft 10.3636 10.361' 14.48 ft 10.0396 9.9318 24 ft 5.3470 3.653 5.5758 5.5754 40 ft 3.6120 3.6940 4.4592 3.4344 3.9997 3.9995 64 ft 2.8886 2.6823 2.7836 2.7835 80 ft 2.3158 2.2314 2.2668 2.2668 100 ft 1.8075 1.7770 1.7843 1.7843 150 ft 1.0134 1.0106 1.0050 1.0050

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NO DEPTH THEORY

13 8 ft 5.3 secs P, ft 9.6 8.0532 7.8481 24 ft 3.1873 2.9237 2.9978 2.9968 40 ft 1.6207 1.6032 1.5495 1.5491 64 ft 0.6098 0.6095 0.5703 0.5701 80 ft 0.3126 0.3125 0.2868 0.2868 100 ft 0.1343 0.1343 0.1199 0.1199 150 ft 0.0161 0.0161 0.0135 0.0135 14 8 ft 10 secs 8 ft 11.34 ft 8.9128 8.7574 24 ft 4/1766 4.1763 40 ft 2.7800 2.7840 3.2615 2.6838 2.9960 ·2.9959 64 ft 2.1433 2.0284 2.0851 2.0850 80 ft 1.7244 1.6774 1.6980 1.6980 100 ft 1.3485 1.3316 1.3366 1.3365 150 ft 0.7567 0.7551 0.7528 0.7528 _, U1

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TABLE NO. 8

BOT TOM VELOCITIES FOR DIFFERENT WAVE THEORIES __,0'1

----SL WAVE HT. WAVE PERIOD WATER SOLITARY WAVE CNOIDAL STOKES III AIRY REMARKS

NO DEPTH THEORY

15 14.24 ft 5.3 secs 8 ft 18.2 ft 10.5784 10.4490 24 ft 6.0145 5.0483 5.3361 5.3361 40 ft 3.0807 3.0095 2.7581 2.7573 64 ft 1.2081 1.2064 1.0151 1.0480 80 ft 0.6392 0.6391 0.5106 0.5164 100 ft 0.2859 0.2859 0.2135 0.2134 150 ft 0.0378 0.0378 0.0240 0.0240 16 18.24 ft 11.2357 11.1593 24 ft 7.4344 7.4339 40 ft 4.6700 4.7180 6.1181 4.3138 5.3329 5.3327 64 ft 3.9008 3.5296 3.7115 3.7114 80 ft 3.1140 2.9617 3.0225 3.0224 100 ft 2.4248 2.3698 2.3791 2.3791 150 ft 1.3585 1.3533 1.3400 1.3399

.

- -# -_. '_"'-"-.

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--BOT TOM VELOCITIES FOR DIFFERENT WAVE THEORIES

NO DEPTH THEORY

17 10 ft 6 secs 8 ft 12.1 ft 9.0368 8.8984 24 ft 4.5908 3.8199 4.1924 4.1914 40 ft 2.5592 2.4789 2.4480 2.4476 64 ft 1.1955 1.1918 1.1374 1.1372 80 ft 0.7168 0.7163 0.6732 0.6731 100 ft 0.3736 0.3736 0.3450 0.3449 150 ft 0.0716 0.0716 0.0629 0.0629 18 10 ft 10 secs 8 ft 14 ft 9.8271 9.7157 24 ft 5.2288 5.2204 40 ft 3.423 3.470 4.1510 3.2513 3.7450 3.7449 64 ft 2.6981 2.5176 2.6064 2.6063 80 ft 2.1650 2.0911 2.1225 2.1224 100 ft 1.6906 1.6640 1.6707 1.6707 150 ft 0.9487 0.9456 0.9410 0.9410 _, ...

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TABLE NO. 10

BOTTOM VELOCITIES FOR DIFFERENT WAVE THEORIES ...

00

NO DEPTH THEORY

19 17.80 ft 6.0 secs 8 ft 20.41 ft 11.7362 11.6726 24 ft 8.7231 6.0856 7.4624 7.4624 40 ft 4.8187 4.5106 4.3575 4.3567 64 ft 2.3012 2.2853 2.0245 2.0245 80 ft 1.4107 1.4085 1.1983 1.1981 100 ft 0.7576 0.7574 0.6141 0.6140 150 ft 0.1565 0.1565 0.1120 0.1120 20 17.80 ft 10 secs 8 ft 22.16 ft 12.3341 12.2776 24 ft 9.2930 9.2940 40 ft 5.5650 5.6310 7.8445 5.0658 6.6661 6.6658 64 ft 4.9388 4.3507 4.6394 4.6392 80 ft 3.9276 3.6857 3.7781 3.7780 100 ft 3.0520 2.9644 2.9739 2.9738 150 ft 1.7093 1.7010 1.6750 1.6749

.

.-- . _-'.- .---

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--TABLE NO. 11

NO DEPTH THEORY

21 14 ft 7.4 secs 8 ft 16.9 ft 10.7119 10.6279 24 ft 7.9546 4.7827 6.6385 6.6375 40 ft 4.700 4.1404 4.3812 4.3807 64 ft 2.7015 2.6387 2.6008 2.6007 80 ft 1.9334 1.9172 1.8621 1.8261 100 ft 1.2753 1.2723 1.2203 1.2202 150 ft 0.4391 0.4390 0.4101 0.4101 22 14 ft 15 secs 8 ft 21.1 ft 12.1584 12.0967 24 ft 7.567 7.7564 40 ft 5.8233 ·5.8232 64 ft 4.4200 3.8120 4.3838 4.3838 80 ft 3.6770 3.5030 4.1385 3.3822 3.7896 3.7896 100 ft 2.9640 3.0460 3.4193 3.0453 3.2422 3.2422 150 ft 2.3930 2.3040 2.3377 2.3374 _. \0

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TABLE NO. 12

BOTTOM VELOCITIES FOR DIFFERENT WAVE THEORIES N

0

I

NO DEPTH THEORY

CREST TROUGH CREST TROUGH CREST TROOGH CREST TROUGH

23 24.92 ft 7.4 secs 8 ft 24 ft 31.26 ft 14.0555 14.0242 40 ft 8.8650 6.9025 7.7985 7.7977 64 ft _5.0633 _4.8250 _4.6295 _4.6292 80 ft _3.6478 _3.5839 _3.3145 _3.3143 100 ft ₂_.₄₃₉₁ _2.4265 _2.1722 _2.1720 150 ft _0.8759 _0.8757 ₀_.₇₂₉₉ _0.7299 24 24.92 ft 15.0 secs 8 ft 24 ft 32.1 ft 14.9468 14.9088 40 ft ₁_{0.3655 10.365} 64 ft 7.4690 5.8520 7.8032 7.8031 80 ft 6.2630 5.8780 7.7722 5.4298 6.7455 6.7455 100 ft 5.1300 5.3360 6.3130 5.1260 5.7711 5.7711 150 ft _4.3267 _4.0416 ₄_.₁₇₈₈ _4.1788 -I.

.

.._,--_. .~,~.~'

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SL WAVE HT. WAVE PERIOD WATER SOLITARY WAVE CNOIDAL STOKES III AIRY REMARKS

NO DEPTH THEORY

CREST TROUGH CREST TROUGH CREST TROUGH CREST TROUGH 25 18 ft 8.5 secs 8 ft 21.6 ft 12.1186 12.0602 24 ft 8.9962 8.9953 40 ft 6.9910 5.3088 6.2240 6.2236 64 ft 4.2530 3.9837 4.0495 4.0493 80 ft 3.2445 3.1534 3.1241 3.1240 100 ft 2.3579 2.3329 2.2756 2.2755 150 ft 1.0678 1.0667 1.0179 1.0179 26 18 ft 15 secs 8 ft 24 ft 25.5 ft 13.3686 13.3227 40 ft 7.4871 ·7.4870 64 ft 5.5980 4.6790 5.4341 4.1921 5.6363 5.6362 80 ft 4.680 4.3940 4.4577 3.8389 4.8724 4.8723 100 ft 3.7070 3.9560 3.0942 2.9467 4.1685 4.1685 150 ft 3.0184 3.0184 N

...

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TABLE NO. 14

BOTTOM VELOCITIES FOR DIFFERENT WAVE THEORIES N

N

~-~

-SL WAVE HT. WAVE PERIOD WATER SOLI TARY WAVE CNOIDAL STOKES III AIRY REMARKS

NO DEPTH THEORY

CREST TROUGH CREST TROUGH CREST TRaJGH CREST TROUGH

27 32.04 ft 8.5 secs 8 ft 24 ft 39.9 ft 15.9030 15.8809 40 ft 13.3515 7.9584 11.078~ 11.0780 64 ft 7.9682 6.9905 7 .210~ 7.2100 80 ft 6.0677 5.7265 5.560~ 5.5606 100 ft 4.4304 4.3334 4.0506 4.0505 150 ft 2.0583 2.0539 1.8118 1.8117 28 32.04 ft 15 secs 8 ft 24 ft 39.0 ft 16.4557 16.4312 40 ft 13.3270 3.3268 64 ft 8.8850 8.8580 10.0358 0.0356 80 ft 7.7560 7.1230 0.2874 6.5012 8.6728 8.6727 100 ft 6.3140 6.8630 8.2943 6.3335 7.4200 7.4199 150 ft 5.6216 5.1450 5.3728 5.3727

.

- -~_. ~~-~--'---

.

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NO DEPTH THEORY

~9 22 ft 9.5 secs 24 ft 26.7 ft 13.4640 13.4299 40 ft 5.9090 6.7560 9.5674 5.9656 8.0612 8.0608 64 ft 5.9139 5.1844 5.5112 5.5110 80 ft 4.6409 4.3546 4.4252 4.4250 100 ft 3.5405 3.4438 3.4135 3.4134 150 ft 1.8769 1.8695 1.8077 1.8077 30 22 ft 15 secs 24 ft 30.3 ft 14.5343 14.4988 40 ft 9.1509 9.1507 64 ft 6.8888 .6.8887 80 ft 5.6280 8.2780 6.7716 4.9323 5.9551 5.9551 100 ft 4.5520 4.7570 5.0949 5.0948 150 ft I'\: c....

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TABLE NO. 16

BOTTOM VELOCITEIS FOR DIFFERENT WAVE THEORIES N

+=-SL WAVE HT. WAVE PERIOD WATER SOLITARY WAVE CNOIDAL STOKES II I AIRY REMARKS

NO DEPTH THEORY

31 39.16 ft 9.5 secs 40 ft 48.6 ft 17.7956 17.3177 64 ft 11.1442 8.6099 9.8098 9.8095 80 ft 8.6820 7.6467 7.8768 7.8766 100 ft 6.126 6.2579 6.0761 6.0759 150ft 3.5508 3.5212 3.2177 3.2177 32 39.16 ft 15 secs 40 ft 6.2886 16.2883 48.5 ft 18.2747 18.2574 64 ft 2.2621 12.2620 80 ft 8.9120 8.2330 12.8958 7.3944 0.6001 ~0.6000 100 ft 7.3950 8.1880 10.3429 7.4219 9.0689 9.0688 150 ft 6.9443 6.2230 6.5'667 6.5667

'

.

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SL WAVE HT. WAVE PERIOD WATER SOLI TARY WAVE CNOIDAL STOKES III AIRY REMARKS

NO DEPTH THEORY

33 24 ft 10 secs 24 ft 29.1 ft 14.0622 14.0251 40 ft 10.9866 6.1131 8.9881 8.9876 64 ft 6.8081 5.7105 6.2553 6.2551 80 ft 5.3844 4.9291 5.0940 5.0939 100 ft 4.1713 4.0056 4.0097 4.0096 150ft 2.3369 2.3211 2.2584 2.2583 34 24 ft 15 secs 24 ft 32.5 ft 15.0544 15.0229 40 ft 9.9828 9.9826 64 ft 7.2640 5.6980 7.5151 7.5150 80 ft 6.0390 5.702 7.4548 5.2768 6.4965 6.4964 100 ft 4.9580 5.1710 6.0622 4.9673 5.5580 5.5580 150 ft 4.1614 3.8937 4.0246 4.0245 N (,Tt

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TABLE NO. 18

N

'"

SL WAVE HT. WAVE PERIOD WATER SOLITARY WAVE CNOIDAL STOKES III AIRY REMARKS

NO DEPTH THEORY

I

! CREST TROUGH CREST TROUGH CREST TROUGH CREST TROUGH

i I 35 42.72 ft 10 secs 40 ft 52.2 ft 18.2652 18.2507 64 ft 12.8832 9.1711 11.1345 11.134 ! 80 ft 10.0923 8.4750 9.0673 9.0671 100 ft 7.7842 7.1766 7.1373 7.1372 150 ft 4.3926 4.3308 4.0199 4.0198 36 42.72 ft 15 secs 40 ft 52.5 ft 18.9980 18.9829 64 ft 13.3769 13.376, 80 ft 9.3490 8.7380 14.2303 7.7842 11.5638 11.563f , 100 ft 7.8420 8.8700 11.3904 7.9224 9.8933 9.8932 150 ft 7.6162 6.7518 7.1637 7.1637 I

.

,

.

- -

.

.-.'---

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--Hmax 3.56 ft. T 2.9 Sec.; 10 Sec. ~

"

i

16

J

I

SYMIOL IX ~ ~ ~ ~ 12

JL

----

~

->_~ U

9

~ > ~

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._

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~ N ""'-J ,.;-. 111 ~ U

3.56' Wave - 10 Sec. 'eriod

I

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2.00',Wave - 10

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1"~ Wnvi

Wave - 2.9 Sec. '-riod

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--

-

--

-

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8-+---+-- -~ ;::) ~ )( ~ ~ 4

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.. - . e - 2.9 Sec. 'eriod 10 40 60 DEPTH OF WATER (ft)

•

WAVE THEORY Solltar,. Wave Cnoidal Wave Stoke.

m

@ [J _Air, _Wave

-

...

-I"' 10 100 120

Figure 1. MaximumBot.tom Ve10cities under Wave Crest

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20 ? Cl.! til <,

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u

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." C ::. .__... >- 12 l-N U CO ₀ _, !oU > ~ 0 8

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I-0 U) ~ ':) ~ )( 4 c( ~ WAVE DATA: H. 4.00 ft Hmax 7.12 ft T 3.8 & 10.0 Sec.

SYMIOL WAVE THEORY

X Salitary Wave

•

Cnoidal Wave

I) _S_t_oke.

m

0 Airy Wave

1.12' Wave - 10 Sec. Period

l- 4' Wave ...; 10 Sec. Period \

\

\ 7.12, Wave - 3.8 Sec. Period I

)'

(

\

\.(

,

4' _{Wave - 3.8} _Sec_. _Period

I(~

<, ...

r.,

I , ... I

----

~""'I!)---•

~---/

... Il

-

-I ...

-

'='

---

_-

~

..._It-

--_

• o

20 40 60 80 100 120 140 110 DEPTH OF WATER (ft) ,'.i ..:e.;,\

Figure 2. Bottom Veloeities under Wave Crest

~.~.

-I. ~

(32)

? 111 111 <,

_...

...

..._... r.:'- 16

• _•

_..

u

_..

•

." IC ::l .._/ >- ₁₂ ~ N _U \.0 0_..J &AI

>

~ 0 8 ~ ~ 0 CID ~ ::l ~ 4 )( Cl: ~ Hmax 10.68 ft T _4.6& 10.0 Sec.

SYMIOL WAVE THEORY

X Solitory Wa.,.

•

Cnoidal Wave

@ Stoke.

m

0 Airy Wave

10.68'Wave - 10 Sec. Period

-

10 Sec. Period

"

~ ... 10.68' Wave - 4.64.6 SecSec...PeriodPeriod

<, <, <,

----,

_{--- --- ---.}

-

___ RL ___

o

20 40 60 80 100 120 140 IlO DEPTH OF WATER (ft)

(33)

20 ~ Q,) 011 <,

..

....

... -c- 16

•

• ..

v

_..

.,

r: ::::t -..J >- 12

....

w _U 0 0 ~ ...., >

s

0 8

....

0 ca ~ :l ~ 4 )(

•

~ WAVE DATA: H. 8.0 ft Hmax 14.24 ft T 5.3 & 10 Sec. SYM.Ol WAVE THEORY

X Solitor, Wave ,

•

Cnoidal Wave ~ Stoke.

m

0 Air, Wave ~\ I \ I

/ \ \ 14.24 Wave - 10 Sec. Period

\ \ I

Wave - 10 Sec. fleriod 8.0

I

14.24 Wave - 5.3 Sec.Period 8.0' Wave - 5.1 Sec.Period

---

---.

-

--o

20 40 60 10 100 121 140 _IlO

DEPTH OF WATER (ft)

Figure 4. Maximum Bottom ~locities under Wave Crest

(34)

20 ~ ilI VI <,

..

...

'-' .-=, ₁₆

•

• ..

u

..

•

~ e :::t ___, >- 12

....

w U Q ...,j w > ~ Q 8

....

Q ICO ~ ::l ~ )( ₄ c(

s

I i I i i I WAVE DATA: H, 10 ft

I

Hmax 17.8 ft I T 6 & 10 Sec

I

SYMIOL WAVE THEORY

X Soli.ary Wave

•

Cnoidal Wave @ Stok.,

m

I ₀ _Airy _Wave 11.8 Wave - 10 Sec. Period Wave - 6 Sec. Period I Wave - 10 Sec. Period Wave - 6 Sec. Period <, -, <, <, <, <,

-

--

----_ -

_

_-

----

---o

20 40 60 80 100 120 140 160 OEPTH OF WATER <ft)

(35)

20 ? 41 lil <, ~

...

..._., r." 16

•

ti

..

U

..

•

." C ::;) .._., >- ₁₂

....

w _U N 0 _,

""

>

~ 0 8

....

_....

0 ~ ~ ::l ~ )( 4

cr:

~ WAVE DATA: _H. ₁₄ ft Hma. 29.42 ft T 7.4& 15 Sec.

SYM.Ol WAVE THEORY

X Sali.ary Wa". ~

•

Cnoidal Wave @ _Stob.

m

/ \\ Airy Wave / _{\ \} 0 I v I I \ \ ~ 24.92' WQve- 15 Sec. Period

!:

I \ \ \

,

'r

24.92'Wave - 7.4Sec.Period

,,\

\ (

,

\

14 Wave - 15 Sec. Period

~j <, 14' Wave - 7.4 Sec. Period

-

... I o~

",

~-_

~ / _<,

.

-_

-

-(

---

'""'(!

I <, 4

~

-I ~ I P ---- __ 1--I

~),

·4

--

_

/ 1-- __ _j_ I". / ~ ~ ~ /

-_

/ --"'1

_p----

,.

r---_

-

--

_

/

-

_

_{--__-..}

~

o

20 40 60 10 100 120 140 ₁₁₀ DEPTH OF WATER (ft)

(36)

I

Hmax 32.04 ft ?

I

QI _T _S.5 _& _{15 Sec.} 011 <,

-....

<;»

-c- 16- SYMIOl WAVE THEORY

•

\

• ..

\ X Solitory Wave U

..

\

•

_Cnoidal _Wave

•

\ "a _\. _@ _{Stok ••}

m

c :) _\. '-" _-, 0 Airy Wave >- ₁₂

_{, I}

_I

~ ...

w U \ ...

r

32.04 Wave - 15 Se, Pedad

w ₀ \ ...

_ [ 32.04' Wave"- 8.5 Se, Pe,;ad

\. ... _, 0

-

....

-... \

>

\.

-- 1:

I

-, --- 18 Wave - 15 Sec Period

~

• ....

1-_

r-lS'Wave -8.5 Sec Period

0 8 <, ~ ~ 0 CID ~

I

//

I

,

~

~'}

...

_-,

r

I

l--~

::l ~ 4 )( Cl

I //

I

---

_"

I

,--~

-_

--

_~

-~

o

20 40 60 10 100 120 140 DEPTH OF WATER (ft)

Figure 7.

Maximum Bottom Veloeities under

'

Wave Crest

(37)

?

>1\

QI lil <,

..

_/ _\ _\ / _\

-

,

\

'-/ /

-..-

16

• •..

I V

N(

\

'j'

..

•

\

"-." c: _\

"

,

::) ..._" <, >- ₁₂

....

w _u

1 /

's

\

+:> 0

_....

[;]

"-I0&oI > -, <, ~ <, 0 8

I

....

_j

_/1

1

<,

I

~

ï

<,__

....

0 a:I ~ ~ ~ )( 4 C( ~ . _I

22 Wave - 9.5 Sec Period

20 WAVE DATA: H. 22 ft Hma. 39.16 ft T 9.5&15 Sec. WAVE THEORY SYMIOL X Solitary Wave Cnoidal Wave Stoke.

m

Airy Wave

•

® o I

..._

-

..r

39.16 Wave - 15 Sec Petiod

- _- I _{~ 22}I_{Wave -} ₁₅_{Sec Per}_i_od

- I

JI

~.~

\!)Wave -9.5 Sec Period o

-

_-

_{-_}

o

20 40 50 10 100 120 ₁₄₀ DEPTH OF WATER (ft)

Figu

re

8 .

Max

im

um Bottom Veloeities under Wave Crest

~

(38)

? GJ lil <, .t: '--" ~ 16

•

• ..

u

..

•

"'D c: ::l -::» >- ₁₂

....

-u w ₀ (.Tl _, !oU >

s

0 8

....

0 aa ~ :J

s

)( 4 Cl:

s

SYMBOl WAVE THEORY

•

Cnoidal Wav .. ® Stokes Wave 0 _{Alry Wave}

/

V

~ 10 Sec.Period

v;

//,

~ 5 Se' Perled

...

-:

_./

...

". ~~

...

,..

~0

0

".

".!r

,..

o

5 10 15 20 2S 30 1S 40 4S WAVE HEIGHT (ft)

Figure 10.

Maximum Bottom Veloeities under Wave Crest

in 24 Feet Depth of Water

(39)

20

I

SYMBOL WAVE THEORY

?

•

C"aldal Wave

"

'"

® Stake. Wave <,

..

-

I

0 Alry Wav. "-' r." 16

• _•

_..

u

..

•

'V c: ::;) -..J >- 12

....

I

I I ~ I U 0

w ₀ 15 Sec. Period (Upper Limit)

0'1 _,

'"

>

I

15 Sec Period (Lower Limit) ~ 0 8 ~ ~ 0 la ~ ::,:) ~ )( 4 C ~ WATER

I

DEPTH: 80'

o

5 10 15 20 25 30 lS 48 41 WAVE HEIGHT (ft)

Figure 11

.

Maximum Bottom Veloeities under Wave Crest

in 80 Feet Depth of Water

(40)

9 QI 411 <,

..

-'-.J

r.:--•

_•

_..

u

..

•

." C ::l ..__., >- I-W U -....j 0_~

'"

>

s

Q I- l-Q ca ~ :l ~ >< C ~ 12 +---~---_+---4_---~---_+---~---+_---~---_+ .~---+_---r_---~---_+---~ --4 ...I~ --WATER DEPTH: 150'

5se,·p~1

10 Sec.Period r.:. ~

o

5 10 15 20 25 30 3S 40 45 WAVE HEIGHT (ft)

Figure 12.

Maximum Bottom Ve10cities under Wave Crest

in 150 Feet Depth of Water

(41)

20 ? QI lil <, •

...

... ~ 16

•

• ..

u

..

•

"0 c: :::::» -..J >- 12

....

w U co 0_{_,} w

>

~ 0

• ....

_....

0 CID ~ :) ~ )( 4 c( ~ 10 Sec.Period

/

-:

~om V.lo';'yUnd., c- ...

/

-:

/'"

-..-V

--

--- ~ .o"om V.lod'y ""de' T'ough

_-~~

~-.--'"

r

-

-.-»> • ~~ ~~

...

A'"

....

'

./

o

5 10 15 20 25 30 35

,.

4S WAVE HEIGHT (h)

Figure 13. Maximum Bottom Veloeities for Stokes Wave Theory

in 40 Feet Depth of Water

(42)

20 ? QI lil <,

-....

~ r.:-. 16

• _•

_..

I///,@/

U

..

•

." r: tv ::;) ..._, /

Velocity Under Crest - I---- ,/

> 12 ./'

....

₍₁₅ _Sec.) ./'

U ./'

W ₀

....

<.0 ... _Velocity _Under _Crest

... > ₍₁₀ _Sec.) ~ 0

• ....

....

0 al ~

I

iir'~./' -> ~

""

"-- V.loclty Und., T'•• lh ::l ~ ₍₁₀ _5.c~ >< 4 C

I

1 _1- ~ 1

I

'-- Velocity Und., T'•• lh ~ I (15 Sec.)

o

5 10 15 20 25 30 35 40 WAVE HEIGHT (ft)

Figure 14. Maximum Bottom Veloeities for Stokes Wave Theory

in 80 Feet Depth of Water

(43)

(44)

Estimation and Analysis of Horizontal Bottom Velocities Due to Waves

Estimation

and Analysis

of

Horizontal

BoHom Velocities

Due to Waves

OF

"

G- J.r -

TABLE OF CONTENTS

I.

INTRODUCTION

.

.

.

.

PRESENT STUDY.

.

. .

. .

DATA ANALYSIS.

IV.

CONCLUDING REMARKS

V.

REFERENCES

VI.

APPENDIX .

.

·

·

· .

Tables .

· · ·

.

Figures.

.

•

· ·

Page

i;;

;v

v

. . . .

.

.

. .

.

Abstract

Preface.

Table of Contents.

.

.

.

.

.

.

.

.

.

...

.

.

.

'

.

.

.

.

'"

.

.

.

.

"

i

J

I

I

I

_·

_·

_{· .}

_{· · ·}

_•

_{· ·}

_..

_...

_•

_..

_..