Coastal Hydraulics

Problems

 

 

Question 1

 

Consider the potential function below.

 

𝜙(x, z, t) = 0.05 cosh (π(z+0.1)) cos (πx - 6πt)

 

What are the values of the wave number, the wavelength, the wave speed and the period for this wave?

 

Find the equation of the surface profile function, and hence find its amplitude.

 

 

Question 2

 

State the necessary boundary conditions for the development of the 2-dimensional small amplitude wave solutions.

 

 

Question 3

 

Water in a rectangular channel flows with a steady and uniform current of 1m/s. The channel is 5m wide and the flow rate is 15m³/s. If wind is blowing steadily along the rectangular channel generating long waves propagating parallel upstream having wavelength of 80m, determine the wave period and the wave speed of the long waves.

 

 

Question 4

 

Water in a long wide river flows with a steady and uniform current of 1m/s. Wind is blowing steadily along the river generating long waves propagating parallel upstream with an average wave period of 12 sec. If the river is dredged to a uniform flow depth of 3 meters, determine the wavelength and intrinsic (free of the current effect) wave period of the long waves.

 

 

Question 5

 

Determine the accelerations in the horizontal and vertical directions in shallow water if the potential function is given as:

 

𝜙(x,z,t) = - Hg/2ơ * (cosh k(h + z)/cosh kh) sin( kx - ơt)

 

 

Question 6

 

Define group wave and derive an equation for describing it. Show the steps of the derivation.

 

 

Question 7

 

Given the following potential function for a linear wave as:

 

𝜙(x,z,t) = - Hg/2ơ * (cosh k(h + z)/cosh kh) sin( kx - ơt)

 

Derive the dispersion equation for the wave.

 

 

Question 8

 

Estimate the wave height at a structure to be constructed at the shore where the water depth is equal to 4m . A deep water wave height detected at a point about 0.5km from the shore is found to be 2m and the wave period is 6s. The water depth at this point is 50m and the slope of the seabed is 1 in 10. Use the potential function as below to estimate the maximum pressure at the structure.

(Hint: n = 1/2 (1 + 2kh/sinh(2kh)) and density of water is 1,000 kg/m³ )

 

𝜙(x,z,t) = - Hg/2ơ * (cosh k(h + z)/cosh kh) sin( kx - ơt)

 

 

Question 9

 

Given the velocity potential of a small amplitude wave as follows,

 

𝜙 = - ag/ơ * (cosh k(h + z)/cosh kh) cos( kx - ơt)

 

Obtain the dispersion equation of the wave

Find the horizontal velocity and vertical acceleration of the wave

Show that the horizontal displacement of the fluid particle (℥) can be expressed by the equation below.

 

℥ = a(cosh k(h + z)/sinh (kh) ) cos(kx - ơt)

 

 

Question 10

 

Show that the potential function given below is a solution for a linear standing wave in deep water, and determine the total pressure under this wave:

 

𝜙(x,z,t) = - (ag/ơ)exp(kz) cos(kx) sin(ơt)

 

 

Question 11

 

Show that the group velocity for waves on water of finite depth h is

 

Cg = c/2 (1 + 2kh cosesh 2kh )

 

If the wavelength is large compared to the depth, so that the shallow water wave theory applies, show that the group velocity is approximately equal to the wave speed.

 

Also, if deep water wave theory applies, show that the group velocity is approximately equal to half of the wave speed.

 

 

Question 12

 

If the dynamic free surface boundary condition can be simplified to the following unsteady Bernoulli equation

 

( gz - ∂𝜙/∂t + (u² + v²)/2 ) = C(t), for z = ƞ

 

Derive the free surface variation equation, ƞ, as shown below using the Bernoulli equation and the truncated Taylor series.

 

ƞ = 1/g *∂𝜙/∂t, for z = 0

 

Show all the steps and assumptions used in this derivation.

 

 

Question 13

 

Assuming steady state and the wave height (H) is linearly related to the water depth (h) in a surf zone in which H = Kh (K is a constant) and h is a function of x (the cross-shore distance), show that the equilibrium beach profile has the following relationship.

 

h ∝ x^2/3

 

 

Question 14

 

Given that the velocity potential function, 𝜙, of a two-dimensional small amplitude wave as follows, show that the horizontal acceleration in the water column is almost constant in shallow water conditions.

 

𝜙 = gH/2ơ * (cosh k(d + z)/cosh kd) sin( kx - ơt)

 

Here H is the wave height, d is the mean water depth, k is the wave number, g is the acceleration due to gravity and  ơ is the wave angular frequency.

 

Question 15

 

Start from the unsteady Bernoulli equation for an ideal fluid given below,

 

P/𝜌 + gz + 1/2( u² + w²) - ∂𝜙/∂t = C(t)

 

where C(t) is the unsteady Bernoulli constant, 𝜌 is the density of fluid, P is the pressure, and u and w are the horizontal and vertical water particle velocities, respectively. Determine the expression for the pressure field under a linear small amplitude waves. State all the assumptions used.

 

 

Question 16

 

A 15m long yacht is sailing in the deep water condition heading in the same direction of the wave motion. Determine the sailing speed of the yacht, if a sailor reports that the surface waves pass the yacht every 5 seconds with wavelength the same as the yacht.

 

 

Question 17

 

A deep-water wave with a wave height of 1m and period of 15s is propagating towards the coast normally. Calculate the maximum vertical velocity component and the maximum vertical displacement from the mean position for a particle 2m below the still water level where the water depth is 5m.

 

 

Question 18

 

The velocity potential 𝜙 for the fluid motion under a small-amplitude periodic progressive wave is given by

𝜙 = - Hg/2ơ * (cosh k(h + z)/cosh kh) sin( kx - ơt)

Establish the expression for the pressure fluctuation under the wave and sketch the variation of pressure with depth under the crest and trough of the wave.

 

A pressure gauge fixed 1m above the sea bed at a position where the mean water depth is 15m records a maximum pressure of 145kPa and period 9s. Determine the wavelength and height. Assume density of sea water = 1,025kg/m³.

 

Sketch the velocity components of the water particles under a small-amplitude periodic progressive wave.

 

 

Question 19

 

Linear wave theory is extensively used in practice. What are the governing assumptions that this theory is based on?

 

 

Question 20

 

A tank that is 200m long contains water up to 3m high. If the water oscillates inside the water tank in the longitudinal direction, the water surface variation (ƞ) can be determined by summing the two waves traveling in the opposite directions expressed as follows:

 

ƞ(x, t) = 0.3sin(kx - ơt) - 0.3sin(kx + ơt)

 

If the wave number of the oscillating wave inside the tank is 0.01571, find

the wave celerity

the wave period

the location of zero amplitude

the maximum water surface elevation, and

the maximum pressure under the lowest water surface elevation