Reference no: EM132528868
Question 1) The velocity potential describing a wave motion is given as:
Φ(x, z, t) = H/2.c.cosh[k(z + h)⌉/sinh(kh).sin(σt - kx)
a) Derive the linear dispersion relationship, explaining the consecutive steps. What type of waves, regarding the relative water depth, does it describe?
b) Determine the water particle trajectory.
c) Consider a wave with the following characteristics: L = 325m (wave length), h = 20m (local water depth). Find: the phase velocity, c; the wave period, T, and the wave number, k.
Question 2) A wave motion is represented by a velocity potential of the form:
Φ(x, z, t) = H/2/.g/σ. cosh⌈k(h + z)⌉/cosh(kh).cos(kx - σt)
a) Show that the potential satisfies the Laplace equation. What is the physical meaning of the Laplace Equation?
b) What types of waves does this potential represent, progressive or standing? If the waves are progressive, what is the direction of their propagation? Please, explain.
c) Determine the form of the water surface elevation, associated with this potential.
d) Determine and discuss the particle trajectories.
e) Derive a simplified expression for the potential in shallow water
Question 3) For the same velocity potential of the previous question (3), consider a wave with length L = 75 m and height H = 4m, which is propagating in water of 5 m depth. Calculate the maximum force due to the wave that acts on a surface of 3 m2 on the sea bottom. Consider that the density of sea water is ρ = 1025 kg/m3 and the acceleration of gravity is g = 9.81m/s2.
Question 4) Observations of the water particle motion in a small-amplitude wave system have resulted in the following data for a total water depth of 1 m:
Major semiaxis = 0.1m ; Minor semiaxis = 0.05m
These observations apply to a particle which mean position is at mid depth. Determine the height, the period and the length of the wave propagating on the free surface.
Question 5) A wave with period T = 7.5 s is registered by two pressure sensors, one at the sea bottom and another at 6.5 m above the bottom. The registered dynamic pressure amplitudes at the sensors are 1.3 × 104 N/m2 and 1.5 × 104 N/m2, respectively. Find the local water depth, d, the wave height, H, and the wave length, L. Consider: g = 9.81 m/s2; ρ = 992 kg/m3; the z-axis is positive upwards.
Question 6) The following potential describes a sinusoidal progressive wave:
Φ(x, z, t) = H/2.g/σ.cosh⌈k(h + z)⌉/cosh(kh).sin(kx - σt)
a) Find the form of the wave profile η(x, t);
b) Calculate the potential energy, EP, per unit surface area;
c) Calculate the kinetic energy, EK, per unit surface area.
Question 7) A wave with height H = 3 m is approaching a rectilinear coast with a uniform slope 0.05 and isobaths parallel to the coastline. The wave starts feeling the bottom at a distance of 1600 m offshore and at that time the wave front forms an angle of 30°with the coastline. Assume that the linear wave theory is valid and that there is no loss of energy through the orthogonals. Calculate for d = 15 m:
a) The wave period and wave length;
b) Wave celerity and wave group celerity;
c) Wave height and the angle between the wave fronts and the coastline (use fig 4.19 from Dean&Dalrymple);
d) The average wave energy per unit surface area, E, and the mean energy flux, f. Consider that the density of sea water is ρ = 1025kg/m3 and the acceleration of gravity is g = 9.81m/s2;
e) Assume that the wave height does not change during the wave propagation towards shore. At what depth approximately wave breaking is expected to start?
Question 8) The harbour entrance is designed for the following deep water wave conditions at station A: HA = 4 m and T = 10 s. The width at station A is bA = 120 m; the depth at station B is hB = 18 m. Assume that the wave height is uniform across sections A and B. Find the width at station B, such that the wave height at station B resulting from the design wave is 1.7 m.
Question 9) On 8 November 2017 a new world record of the biggest wave ever surfed was established by Rodrigo Koxa in Nazare/Portugal, with 24.38 m (see the link below). An approximate wave period was reported as 18 seconds. Fortunately, there are many good-quality photos and videos online to prove the great achievement as well as to support the analyses of the wave - where he was pulled by a jet-ski (tow-in surfing technique). Rodrigo is now curious if it could be possible, in the future, to catch and ride the same wave without the support of a jet-ski, i.e., paddling on a surfboard on his own. In order to do that, he would have to paddle fast enough to get to a speed close to the wave propagation and to the crest displacement at the moment of wave breaking. Looking and analyzing the images of that gigantic wave:
a) List and explain with equations all the wave effects and the unique combination that make Nazare the biggest wave in the world;
b) Considering a mildly sloped sea bottom and the McCowan approximation, what is the water depth at the moment the wave collapsed?
c) Assuming a slope of sea bottom of 0.05, what is the distance of the wave breaking line to the shore? How fast a jet-ski at the coast would have to travel to rescue him assuming he cannot survive more than three consecutive gigantic waves breaking on his body without any support?
d) What is the velocity Rodrigo would have to paddle to catch the wave?
e) What is the wave length and Energy at the moment the wave was breaking? Rodrigo has a Tesla electric car and he wants to celebrate the great achievement by driving away with his ModelS that consumes 181Wh/km. Considering the whole energy associated with that single wave, hypothetically converted into the car's battery, how far could he go?
f) If he had failed to drop such wave, his body would have been smashed against the sea bottom. What would be maximum pressure on his ears?
g) Moving back in time, when that wave started feeling the sea bottom: What was the water depth, the wave length, and wave period? What was the group and phase velocity? How far was this condition from the coast, and how long did it take from the beginning of the shoaling effect until the wave broke?
h) The previous world record had been obtained by Garrett McNamara, who rode a 23.77 m wave on 1 November 2011. What is the difference in phase velocity and wave Energy between Rodrigo´s and Garrett´s waves?
Attachment:- list problems.rar