The Physics of Waves: A Guide to Wave Kinematics and Hydrodynamic Force Calculations

 Lecture 18

Wave Theory

Fig: progressive surface wave parameters

A monochromatic wave propagates at a phase velocity C in water with depth d, within an x, z coordinate system. The x-axis represents the still water level, and the bottom is located at z=−d. The wave's surface profile is described by z=η, where η is a function of xx and time t. The wave has a wavelength L and a height H, as illustrated in the figure. The wave travels a distance equal to its wavelength L in one period T.

C = L/T

The horizontal and vertical components of the water particle velocity at any instant are u and w, respectively. The horizontal and vertical coordinates of a water particle at any instant are given by ζ and ɛ, respectively. The coordinates are referenced to the center of the orbital path that the particle follows. At any instant, the water particle is located a distance d - (- z) =d + z above the bottom.

The following dimensionless parameters are often used:

k = 2π/L (wave number)

σ = 2π/T (wave angular frequency)

 

This equation indicates that for small-amplitude waves, the wave celerity is independent of the wave height. This Equation can also be written as:

These Equations collectively are commonly known as the dispersion equation. For a spectrum of waves having different periods (or lengths), the longer waves will propagate at a higher celerity and move ahead while the shorter waves will lag behind.

Surface waves can be classified based on the relative depth, defined as the ratio of water depth (d) to wavelength (L). As waves move from deep offshore waters into shallower nearshore waters, the wavelength decreases. However, this reduction in wavelength occurs more slowly than the decrease in water depth. Consequently, the relative depth (d/L) diminishes as waves approach the shore. When the relative depth exceeds approximately 0.5, the hyperbolic tangent function, tanh(2πd/L), becomes nearly equal to 1. Under these conditions, simplify above equations to more straightforward forms.

Example

A wave in water 100 m deep has a period of 10 s and a height of 2 m. Determine the wave celerity, length. What is the water particle speed at the wave crest?

Solution:

Assume that this is a deep-water wave

Lo = 9:81(10)² / 2π= 156 m

Since, d/L_o = 100/ 156 =0.64>0.5

C_o = gT/2π = (9.81 x 10) / (2x3.14) = 15.62 m/s

U_* = πHo/T = 3.14x2/10 = 0.63 m/s

When the relative depth is less than 0.5 the waves interact with the bottom. Wave characteristics depend on both the water depth and the wave period, and continually change as the depth decreases. The full dispersion equations must be used to calculate wave celerity or length for any given water depth and wave period.

Wave Kinematics

The following equations for water particle velocity in shallow water:

Pressure Field

The following equation for the pressure Weld in a wave:

The horizontal component of velocity from above equation leads to

Above equation becomes

P = nE/T

The value of n increases as a wave propagates toward the shore from 0.5 in deep water to 1.0 in shallow water. Above Equation indicates that n can be interpreted as the fraction of the mechanical energy in a wave that is transmitted forward each wave period.

Hydrodynamic Forces in Unsteady Flow

• Unsteady flow interacts with submerged solid bodies, exerting forces due to both flow velocity and acceleration.

·         Drag Force (F_d):

o    Caused by flow velocity.

o    Results from frictional shear stress and normal pressure.

o    Formula: F_d​=​C_d​Aρu²/2

§  ρ: Fluid density.

§  u: Flow velocity approaching the body.

§  Cd​: Drag coefficient (depends on body shape, orientation, surface roughness, and Reynolds number).

§  A:

§  For streamlined bodies (e.g., airfoil): Surface area.

§  For blunter bodies (e.g., circular/rectangular cross-sections): Projected cross-sectional area in flow direction.

 

• Accelerating Flow Past a Body:
• Flow velocity (u) and Reynolds number change continuously, causing the drag coefficient (C_d​) and drag force (F_d​) to vary significantly.
• Example: A wave passing a vertical cylindrical pile creates complex drag force patterns due to varying water particle velocities over time and along the pile.
• Inertial Force:
• Caused by flow acceleration, it has two components:

1.     Pressure Gradient Force: Results from the pressure gradient needed to accelerate the flow, creating a net force on the body.

2.     Added Mass Force: Occurs because the accelerating flow sets an additional mass of fluid into motion, requiring a force to accelerate it. This force depends on fluid density (ρ), acceleration, body shape, and volume.

Thus, when there is unsteady flow, the total instantaneous hydrodynamic force F on the body can be written

In potential flow, the added mass coefficient (k) varies with body shape and flow orientation:

·  Sphere: k=0.50

·  Cube (flow normal to a side): k=0.67

·  Circular cylinder (flow normal to axis): k=1.00

·  Square cylinder (flow normal to axis): k=1.20

Additional k values can be found in Sarpkaya and Isaacson (1981). In real fluids, k also depends on surface roughness, Reynolds number, and flow history. The term 1+k is commonly referred to as the coefficient of mass or inertia (C_m​), and the equation is often expressed using C_m​.

·         For potential flow past a circular cylinder, C_m​=2.0.

·         In real flow, C_m​ is often less than 2.0.

·         Above Equation, used for wave forces on submerged structures, is known as the Morison equation (Morison et al., 1950).

Piles, Pipelines, and Cables

• Marine Structures:
• Piles, pipelines, and cables are long cylindrical structures designed to withstand unsteady wave forces and steady current-induced drag.
• Cables:
• Similar to pipelines in stability but smaller in diameter (typically <15 cm).
• Used for mooring ships, buoys, and floating breakwaters.
• Pipelines:
• Can be much larger (up to 3 m in diameter, e.g., municipal waste outfall lines).
• Piles:
• Used in piers, offshore drilling structures, dolphins, and navigation aids.
• Typically, vertical or near-vertical, with diameters ranging from <1 m (piers) to several meters (deep-water oil drilling structures).
• May include horizontal or inclined cylindrical members for cross-bracing.

 

Piles

Total force on a vertical circular cylindrical pile

This wave-induced force causes a moment on the pile around the mudline given by

Based on laboratory and field experimental results, Hogben et al. (1977) and the U.S. Army Coastal Engineering Research Center (1984) have provided recommended design values for the drag coefficient (C_d​) and inertia coefficient  (C_m​). To determine C_m​, the Reynolds number (R) is calculated using the maximum water particle velocity in the wave (occurring at the wave crest and water surface). 

 

Example:

A vertical cylindrical pile having a diameter of 0.3 m is installed in water that is 8 m deep. For an incident wave having a height of 2 m and a period of 7 s, determine the horizontal force on the pile and the moment around the mudline when the pile is situated at the halfway point between the crest and still water line of the passing wave.

Solution:

Lo = 9:81(7)²/2π= 76:5 m

L/76.5 = tanh (2π8/L)

L = 55.2 m

K = 2π/L = 0.1138

The maximum horizontal particle velocity is

At crest cos (kx - σt) = 1

Recommended values for C_d and C_m given above yields

C_d = 0:72

C_m = 1:8

• For point along the wave σt=3π/4 [Represents the phase of the wave, σt=3π/4​ represents a specific point in the wave's cycle, where the wave is descending and below the still water line. It is useful for analyzing wave properties (e.g., velocity, acceleration, or forces) at that particular phase.]