4-Seakeeping - Wave Properties (2012)

Figure 2 Free surface for a regular seaway Bhattacharyya 1978 The relationship of simple harmonic motion can be illustrated in the case of wave motion by plotting the distance as a func

SEAKEEPING – WAVE PROPERTIES

Note: These notes are drawn from those issued by Dr Jonathan Duffy to students of JEE329 Seakeeping & Manoeuvring at

the Australian Maritime College Dr Duffy has used edited extracts from the main reference books for the subject and

which are listed at the end of the module

1 Introduction

Seakeeping is the study of motions imposed upon a ship by waves It is necessary to have an understanding of seakeeping to predict the motions of a ship in waves to determine factors such as:

● Is a ship going to survive in a given seaway?

● Can the ship carry out the specified task or mission in a given seaway?

● Decide which design will perform the best

● Are the ship motions acceptable with respect to factors such as: slamming, deck wetness, speed loss, human performance, ride control

In order to assess the seakeeping performance of a ship it is necessary to ascertain the wave environment in which the ship will be exposed to Then, the motions of the ship in the seaway can be predicted and assessed against suitable criteria This process is summarised in Figure 1

Expected sea conditions

Ship motions in waves

Compare

Seakeeping design criteria

Figure 1 Overview of seakeeping assessment

The prediction of ship motions in waves is addressed in this course based on the assumptions that the wave and ship motions are sufficiently small to linearise

MODULE 4

2 Regular waves

An ocean wave is irregular; no two waves have exactly the same height and they travel in different directions at different speeds However, as we shall see later, an irregular seaway may

be represented by superposing (adding together) multiple regular waves Hence, it is necessary to firstly study the properties of regular waves An idealised water wave is a sinusoidal curve, either a sine curve or cosine curve, as shown in Figure 2

Figure 2 Free surface for a regular seaway (Bhattacharyya 1978)

The relationship of simple harmonic motion can be illustrated in the case of wave motion by plotting the distance as a function of time This is shown in Figure 3 Point P1 is obtained by

plotting a horizontal line through P to the required value at t = 1 second As P travels around the

circle, a curve P0P1P2 P8P9P10 is obtained by plotting horizontal lines from different positions of

P, as illustrated in Figure 3 The waves in Figure 3(a) and Figure 3(b) are out of phase by 45

degrees The phase angle, which shifts the curve along the t axis may be accounted for as

follows:

Where

ζ𝑎= wave amplitude

𝜔𝑤= circular frequency of the wave

t = time

𝜖 = phase angle

Note that the equations above are for a stationary wave

Figure 3 Sinusoidal wave from a radius vector (Bhattacharyya 1978)

If we now consider a distance x on the horizontal axis the equation for a stationary wave is:

ζ= ζ𝑎𝑠𝑖𝑛𝑘𝑥 (2)

Where k is the wave number, which is defined as the number of waves per unit distance along

the x axis and is given by

𝑘 =𝜔𝑤

𝑉 𝑤 (5)

Or

𝑘 =2𝜋

So the equation for a stationary wave becomes

ζ= ζ𝑎𝑠𝑖𝑛2𝜋𝐿

Equation 4 is plotted in Figure 4

Figure 4 Sine representation of a wave (Bhattacharyya 1978)

Now let us consider a progressive wave as illustrated in Figure 5

Figure 5 Propagation of a sine wave after time t (Bhattacharyya 1978)

Assuming that the dotted curve is the progressive wave after t seconds the equation for the curve

can be defined as:

Where θ is a function of x

Now, θ = 0 at t = 0.Thus, when t = 0 kx must equal 0 Also, θ = 0 at x= V w t This is possible

if 𝜃 = 𝑘(𝑥 − 𝑉𝑤𝑡) Therefore:

Therefore the equation of a sinusoidal wave travelling at a velocity V w is given by:

ζ = ζ𝑎𝑠𝑖𝑛2𝜋

It is possible to express the wave equation in terms of wave frequency ωw After a period of time

equal to the wave period T the value of ζ is the same as that at x0, as given by:

At any other time:

From simple harmonic motion we know that 2𝜋𝑡𝑇 = 𝜔𝑤𝑡 Therefore the expression for a sinusoidal wave is:

If there is a phase angle the form of the equation is:

or:

3 Wave Velocity

In previous research the properties of regular waves have been studied and it has been found that,

without placing a restriction on the water depth (h), the wave velocity (wave celerity) can be

approximated by:

𝑉𝑤2=𝑔𝐿𝑤

2𝜋 𝑡𝑎𝑛ℎ2𝜋ℎ

𝐿𝑤 (13)

The wave velocity for deep water (i.e h is large, such that ℎ

𝐿 𝑤 → ∞)is:

𝑉𝑤 = 𝑔𝐿𝑤

The wave velocity for shallow water (i.e h is small, such that 𝐿ℎ

𝑤 → 0)is:

As can be seen, in shallow water waves have the same velocity regardless of their wavelength

4 Water Particle Motion in a Wave

Water particles in a wave experience orbital motion The velocity of a particle at the crest of the wave moves in the direction of the wave, whereas the velocity of a particle at the trough of a wave is in the opposite direction of the wave motion The water particles move either vertically upward or downward when crossing the mean still waterline, as illustrated in Figure 6

The water particles do move bodily forward with the wave at a velocity much less than the wave velocity This is due to the fact that the particles travel further when travelling in the same direction as the wave (i.e at the crest of the wave) than when travelling in the opposite direction

to the wave This is illustrated in Figure 7

Figure 6 Motion of water particles in a wave (Bhattacharyya 1978)

Figure 7 Water particles during wave motion

(a) water particles in a forward moving wave, (b) motion of water particles in a wave

The radius of the orbit of a particle in a wave decreases as the depth from the water surface is increased The wave amplitude at a known depth below the water surface can be expressed as a function of the wave amplitude at the water surface, as follows:

ζ𝑧 = ζ𝑎𝑒−𝑘𝑧 (16)

Where

ζ𝑧 = wave amplitude at a depth below the surface

ζ𝑎 = wave amplitude at the surface

𝑧 = mean depth of the particle below the free surface

𝑘 = wave number = 2𝜋 𝐿 𝑤

Using this equation it can be seen that the radius of orbit of a particle in deep water decreases

rapidly as the distance below the surface is increased

Generally, for the purpose of predicting ship behaviour in waves water is assumed to be deep when the water depth exceeds half the wave length This is because the orbital motion of the water in the wave decays exponentially with depth as a function of wave length

The velocity of the water particles in the horizontal and vertical directions in deep water are given by:

𝑢 = 𝑘ζ𝑎𝑉𝑤e−𝑘𝑧𝑐𝑜𝑠𝑘(𝑥 − 𝑉𝑤𝑡)(horizontal direction) (17)

𝑤 = 𝑘ζ𝑎𝑉𝑤e−𝑘𝑧𝑠𝑖𝑛𝑘(𝑥 − 𝑉𝑤𝑡) (vertical direction) (18)

Once again it can be seen that the disturbance of water particles diminishes as the depth from the free surface increases

5 Pressure in a Wave

In calm water the pressure at depth z is:

In calm water the constant pressure contour is a straight line However, underneath regular waves the constant pressure contour is distorted, as shown in Figure 8

Figure 8 Constant pressure contours beneath a 100m wave: depth 100m (Lloyd 1998)

The pressure beneath the crest of a wave from the still water line is given by:

Where 𝑧 is measured downwards from the still waterline and ζ is given by:

Where h is the water depth For large values of h (i.e deep water) the ratio

cosh 𝑘(−𝑧+ℎ)

cosh 𝑘ℎ approaches 𝑒−𝑘𝑧 Therefore:

Hence, the pressure beneath the crest of a regular wave from the still waterline is:

The first term is the hydrostatic component and the second term is the hydrodynamic component The second term is positive or negative depending upon whether the wave profile is in the crest

or trough

It can be seen that the pressure under a regular wave at constant depth beneath the still waterline oscillates around the steady hydrostatic pressure The pressure fluctuations decrease with depth and become negligible for depths greater than about half the wave length

6 Energy in a Wave

The propagation of waves is essentially brought about by two things:

1 The inertia of the fluid

2 Gravity, which tends to maintain the water surface as a horizontal plane

A wave has both potential energy and kinetic energy The potential energy is due to the elevation

of the water level and the kinetic energy is due to the fact that the water particles have an orbital motion

Consider a small element of the wave in Figure 9

Figure 9 Potential energy in a regular wave (Lloyd 1998)

The mass per unit width (width is defined as a distance perpendicular to the page) = −𝜌ζ ∂x

The potential energy (E p ) relative to the calm water = mgy

If we define w as the width of the regular wave perpendicular to the page and if we assume that the centre of gravity is approximately −ζ

2

above the undisturbed free surface (x axis in Figure 9) then the potential energy is:

1

2

Integrating over the entire wavelength we obtain:

𝐸𝑝=𝜌𝑔 ζ𝑎 w

or:

Now we will consider kinetic energy Recall that kinetic energy (E k) = 1 2 𝑚v2 Consider a small element of fluid with width w (perpendicular to the page) in Figure 10

Figure 10 Kinetic energy in a regular wave (Lloyd 1998)

The mass of the element of fluid is:

and it has a total velocity of:

𝒘

The kinetic energy of the element is:

1

2

Integrating to obtain the total kinetic energy of the fluid in one wavelength between the surface and the bottom:

𝐸𝑘 =𝜌w

It can be shown that the kinetic energy of a wave system can be expressed as:

𝐸𝑘 = 1 4 𝜌𝑔ζ𝑎2 per unit area of wave surface (34)

The total energy of a sinusoidal wave is:

𝐸 = 𝐸𝑝+ 𝐸𝑘 = 1 4 𝜌𝑔ζ𝑎𝐿𝑤w + 1 4 𝜌𝑔ζ𝑎𝐿𝑤w = 1 2 𝜌𝑔ζ𝑎𝐿𝑤w (35)

or :

The wave energy is independent of wave frequency and is dependent upon only on wave amplitude

6.1 Energy Transmission & Group Velocity in a Wave

In deep water each individual wave within a group of regular waves propagates forward at the wave velocity 𝑉𝑤 and the energy is transmitted in the direction of the wave propagation However, the energy of the wave group propagates at 12𝑉𝑤 So after one wave period each wave will have moved forward one wave length, taking half of its associated energy with it The other half of the energy is left behind to be added to the energy brought forward by the next wave Therefore, the total energy per square metre within the group is kept constant

Consider a regular wave in a towing tank At the leading edge of the group the first wave will be propagating into calm water, so the orderly exchange of energy from wave to wave does not happen after one wave period, hence the energy of the leading wave is halved The wave amplitude is reduced and this process continues as the leading edge of the wave train propagates down the tank at 𝑉𝑤

In summary, for deep water, the leading edge of the group proper (defined as the position of the first wave of full amplitude) propagates down the towing tank at the group velocity, which is equal to 12𝑉𝑤 The group velocity is important as it is identical to the rate of transmission of energy in the waves Individual waves within the group propagate at the wave velocity 𝑉𝑤, which

is twice the group velocity

For water of any depth the group velocity is given by:

𝑉𝐺 =1

2𝑉𝑤+ 𝑔ℎ

2𝑉𝑤𝑠𝑒𝑐ℎ2 2𝜋ℎ

For deep water 𝐿ℎ

𝑤 is large and so:

since 𝑠𝑒𝑐ℎ2 2𝜋ℎ

𝐿 𝑤 → 0

7 Effect of Water Depth on Regular Waves

Oscillatory waves may be classified by the water depth in which they travel A deep water wave

is one that satisfies;

w

L

h / >1/2

Finite depth waves may be transitional or shallow water waves A transitional wave is one where:

1/ 20  h Lw /  1/ 2

A shallow water wave is one that satisfies:

w

L

h / < 1/20

Note that the boundary between shallow and transitional waves can vary between researchers For example, this boundary is defined as:

w

L

h / < 1/25 by Shore Protection Manual (1984)

The period of a regular wave is independent of water depth Therefore, the period remains the same regardless of water depth However, the wave velocity in water of finite depth is different

to that in deep water Therefore, the length of a wave travelling in finite water depths is different

to that in deep water

A summary of the properties of regular waves in any water depth are given in Table 1.Some simplifications can be made for deep water regular waves as can be seen in Table 2, where properties specific to deep water waves are provided Wave properties specific to intermediate and shallow water waves are provided in Shore Protection Manual (1984)

Elevations of lines of equal pressure 𝜁 = 𝜁𝑎

sinh⁡(−z + ℎ) sinh 𝑘ℎ cos 𝑘(𝑥 − 𝑉𝑤 𝑡) Surface profile (i.e elevation of line of

equal pressure at z = 0) ζo= ζacos 𝑘(𝑥 − 𝑉𝑤 𝑡)

Horizontal water velocity

𝑢 = ζa𝑉𝑤𝑘cosh 𝑘(−z + ℎ)

sinh 𝑘ℎ cos 𝑘( 𝑥 − 𝑉𝑤 𝑡) Vertical water velocity

𝑤 = ζa𝑉𝑤𝑘sinh 𝑘(−z + ℎ)

sinh 𝑘ℎ sin 𝑘( 𝑥 − 𝑉𝑤 𝑡) Wave velocity or celerity

𝑉𝑤= g𝐿𝑤 2π tanh 𝑘ℎ

1/2

Pressure

𝑃 = 𝜌gz ± ζa𝜌gcosh 𝑘(−z + ℎ)

cosh 𝑘ℎ cos 𝑘(𝑥 − 𝑉𝑤𝑡) Where 𝜌gz is hydrostatic pressure and ℎ is depth of water The sign of the second term is dependent on whether the wave profile is in the crest or trough

Table 1 Properties of harmonic waves in water of any depth (Bhattacharyya 1978)

Note: For very shallow water (i.e for h<𝐿𝑤/20), Vw = 𝑔ℎ

and for deep water (i.e h>𝐿𝑤/2), Vw = 𝑔Lw/2𝜋 1/2

Elevations of lines of equal pressure

(at a depth z) 𝜁𝑧= 𝜁𝑎𝑒−𝑘zcos 𝑘(𝑥 − 𝑉𝑤 𝑡)

Surface Profile

(i.e elevation of line ofequal pressure at

z=0) (1st approx.)

ζ = ζo= ζacos 𝑘(𝑥 − 𝑉𝑤 𝑡)

or

ζ = ζo= ζacos (𝑘𝑥 − 𝜔𝑤 𝑡) Horizontal water velocity 𝑢 = 𝑘ζa𝑉𝑤e−𝑘zcos 𝑘( 𝑥 − 𝑉𝑤 𝑡)

Vertical water velocity w = 𝑘ζa𝑉𝑤e−𝑘zsin 𝑘( 𝑥 − 𝑉𝑤 𝑡)

𝑇𝑤 = g

𝜔𝑤= g𝐿𝑤 2π

1/2

2

2πg

𝜔𝑤2=g𝑇𝑤

2

2π

𝐿𝑤 =

𝜔𝑤2

g =

g

𝑉𝑤2= 4π2

g𝑇𝑤2

g

1/2

Pressure

𝑃 = 𝜌gz ± ζa𝜌ge−𝑘zcos(𝑘x − 𝜔𝑤𝑡) Where 𝜌gz is hydrostatic pressure and ℎ is depth of water The sign of the second term is dependent on whether the wave profile is in the crest

or trough

Maximum wave slope

𝐿𝑤 =

πℎ𝑤

𝐿𝑤

2𝜌gζa2

Table 2 Properties of harmonic waves in deep water (Bhattacharyya 1978)

7 A Vessel in Regular Waves

The absolute frequency of the waves (ω w) may not be the same as the frequency encountered by

a ship with forward speed For example, a ship heading directly into waves, in a head sea, will meet successive waves much more quickly and the waves will appear to have a much higher frequency On the other hand, a ship travelling in a following sea is moving away from the waves and so the frequency of the waves will be lower If the ship is travelling beam on to the waves there will be no difference between the absolute frequency of the waves and that encountered by the ship The frequency at which the ship meets the waves is known as the encounter frequency and is a function of the absolute frequency of the wave, the ship speed and the angle between the direction of wave travel and the direction in which the ship is heading

The encounter frequency (ω e) is the important consideration with respect to ship motions in waves since this tells how the ship meets the waves, which then influences the motion of the ship Thus, in all calculations it is the encounter frequency that is used instead of the absolute

frequency The encountering angle (μ) is the angle between the direction of wave travel and the

direction of the ship’s heading When the ship is heading into a train of regular waves the angle μ

is 180° In following seas the encountering angle is 0° and in beam seas it is 90° or 270°.This is shown for three specific cases in Figure 11 A more general description is given in Figure 12

Figure 11 Definition of heading angles relative to waves (Bhattacharyya 1978)

Figure 12 Definition of heading angles relative to waves

V – boat speed