Ship Kelvin Wake

Wave and Wave Body InteractionsCurrent ChapterShip Kelvin WakeNext ChapterLinear Wave-Body InteractionPrevious ChapterWavemaker Theory

This is an incomplete page

Contents

 [hide] 

• 1 Introduction
  • 1.1 Translating Coordinate System
  • 1.2 Kelvin wake
  • 1.3 Application of the Group velocity
  • 1.4 Solution of the equation for angle

Introduction

Wake created behind a ship

A ship moving over the surface of undisturbed water sets up waves emanating from the bow and stern of the ship. The waves created by the ship consist of divergent and transverse waves. The divergent wave are observed as the wake of a ship with a series of diagonal or oblique crests moving outwardly from the point of disturbance. These wave were first studied by Lord Kelvin

Translating Coordinate System

We have a the standard fixed coordinate system 

x,y,zx,y,z

 and a moving coordinate systems which is moving in the 

xx

 direction with speed 

UU

. We denote the moving coordinate systems in the 

xx

 direction by

x¯=x+Utx¯=x+Ut

Let 

Φ(x,t)Φ(x,t)

 be the velocity potential describing the potential flow generated by the ship relative to the earth frame. The same potential expressed relative to the ship frame is 

Φ¯(x¯,t)Φ¯(x¯,t)

. The relation between the two potentials is given by the identity

Φ(x,y,z,t)=Φ¯(x¯,y,z,t)=Φ¯(x−Ut,y,z,t)Φ(x,y,z,t)=Φ¯(x¯,y,z,t)=Φ¯(x−Ut,y,z,t)

where the relation between the coordinates of the two coordinate systems has been introduced. Note the time dependence occurs in two places in 

Φ¯Φ¯

 and in one place in 

ΦΦ

. The governing equations are always derived relative to the earth coordinate system and time derivatives are initially taken on 

ΦΦ

. Therefore

dΦdt=ddtϕ¯(x−Ut,y,z,t)=∂ϕ∂t−U∂ϕ∂xdΦdt=ddtϕ¯(x−Ut,y,z,t)=∂ϕ∂t−U∂ϕ∂x

All time derivatives of the earth fixed velocity potential 

ΦΦ

 which appear in the free surface condition and the Bernoulli equation can be expressed in terms of derivatives of 

Φ¯Φ¯

 using the Galilean transformation derived above.

If the flow is steady relative to the ship fixed coordinate system

∂Φ¯∂t=0∂Φ¯∂t=0

but

dΦdt=−U∂Φ¯∂xdΦdt=−U∂Φ¯∂x

or, the ship wake is stationary relative to the ship but not relative to an observed on the beach.

Kelvin wake

Diagram of the Kelvin Wake

Local view of Kelvin wake consists approximately of a plane progressive wave group propagating in direction 

θθ

. As noted above surface wave systems of general form always consist of combinations of plane progressive waves of different frequencies and directions. The same model will apply to the ship kelvin wake. Relative to the earth frame, the local plane wave in Infinite Depth takes the form

Φ=igAωekz−ik(xcosθ+ysinθ)+iωtΦ=igAωekz−ik(xcos⁡θ+ysin⁡θ)+iωt

Relative to the ship frame

barΦ=igAωekz−ik(xcosθ+ysinθ)−i(kUcosθ−ω)tbarΦ=igAωekz−ik(xcos⁡θ+ysin⁡θ)−i(kUcos⁡θ−ω)t

But relative to the ship frame waves are stationary, so we must have:

kUcosθ=ωkUcos⁡θ=ω

or

ωk=Cp=Ucosθωk=Cp=Ucos⁡θ

This implies the following

• The phase velocity of the waves in the kelvin wake propagating in direction 
  θθ
   must be equal to 
  UcosθUcos⁡θ
  , otherwise they cannot be stationary relative to the ship.
• Relative to the earth system the frequency of a local system propagating in direction 
  θθ
   is given by the relation 
  ω=kUcosθω=kUcos⁡θ

Relative to the earth system the Infinite Depth Dispersion Relation for a Free Surface states

ω2=gkω2=gk

so that

λ(θ)=2πU2cos2θgλ(θ)=2πU2cos2⁡θg

This is the wavelength of waves in a Kelvin wake propagating in direction 

θθ

 which are stationary relative to the ship.

Application of the Group velocity

An observer sitting on an earth fixed frame observes a local wave system propagating in direction 

θθ

 travelling at its group velocity 

dωdKdωdK

 by virtue of the Rayleigh device which states that we need to focus on the speed of the energy density (

∼∼

 wave amplitude) rather than the speed of wave crests. So, relative to the earth fixed inclined coordinate system 

(X′,Y′)(X′,Y′)

:

X′t=Vg=dωdKX′t=Vg=dωdK

Or

X′=dωdKt ⟹ ddK(KX′−ωt)=0X′=dωdKt ⟹ ddK(KX′−ωt)=0

X′=Xcosθ+Ysinθ=xcosθ+ysinθ+UtcosθX′=Xcos⁡θ+Ysin⁡θ=xcos⁡θ+ysin⁡θ+Utcos⁡θ

So:

KX′−ωt=K(xcosθ+ysinθ)+(KUcosθ−ω)tKX′−ωt=K(xcos⁡θ+ysin⁡θ)+(KUcos⁡θ−ω)t

However 

(KUcosθ−ω)t=0(KUcos⁡θ−ω)t=0

 so that the Rayleigh condition for the velocity of the group takes the form:

ddK[K(θ)(xcosθ+ysinθ)]=0ddK[K(θ)(xcos⁡θ+ysin⁡θ)]=0

By virtue of the dispersion relation derived above:

K(θ)=gU2cos2θK(θ)=gU2cos2⁡θ

It follows from the chain rule of differentiation that Rayleigh's condition is:

ddθ[gU2cos2θ(xcosθ+ysinθ)]]=0ddθ[gU2cos2⁡θ(xcos⁡θ+ysin⁡θ)]]=0

At the position of the Kelvin waves which are locally observed by an observer at the beach.

• So the "visible" waves in the wake of a ship are wave groups which must travel at the local group velocity. These conditions translate into the above equation which will be solved and discussed next. More discussion and a more mathematical derivation based on the principle of stationary phase can be found in Newman 1977.

Solution of the equation for angle

Graphical image of the equations

The solution of the above equation will produce a relation between 

yxyx

 and 

θθ

. So local waves in a Kelvin wake can only propagate in a certain direction 

θθ

, given 

yxyx

.

Simple algebra leads to:

yx=−cosθsinθ1+sin2θ=yx(θ)yx=−cos⁡θsin⁡θ1+sin2θ=yx(θ)

which implies that

• yx(θ)yx(θ)
   is anti-symmetric about 
  θ=0θ=0
   and each part corresponds to the Kelvin wake in the port and starboard sides of the vessel. The physics on

either side is identical due to symmetry.

• θ=0θ=0
  : waves propagating in the same direction as the ship. These waves can only exist at 
  Y=0Y=0
   as seen above.

• θ=π2θ=π2
  : waves propagating at a 
  90∘90∘
   angle relative to the ship direction of forward translation.

• θ=35∘16′θ=35∘16′
  : (or 35,26°) waves propagating at an angle 
  θ=35∘16′θ=35∘16′
   relative to the ship axis. These are waves seen at the caustic of the Kelvin wake.

Let the solution of 

yx(θ)yx(θ)

 be of the form, when inverted:

Region I:θ=f1(yx)Region I:θ=f1(yx)

Region II:θ=f2(yx)Region II:θ=f2(yx)

Note that observable waves cannot exist for values of 

yxyx

 that exceed the value shown in the figure or 

yx∣∣Max=2−3/2yx|Max=2−3/2

. This translates into a value for the corresponding angle equal to 

19∘28′19∘28′

 (or 19,47°) which is the angle of the caustic for any speed 

UU

.

"transverse" and "divergent" wave systems in the Kelvin wake

The crests of the wave system trailing a ship, the Kelvin wake, are curves of constant phase of:

xcosθ+ysinθcos2θ=Cxcos⁡θ+ysin⁡θcos2⁡θ=C

In 

Region IRegion I

:

C=xcosf1(yx)+ysinf1(yx)cos2f1(yx)≡G1(yx)C=xcos⁡f1(yx)+ysin⁡f1(yx)cos2⁡f1(yx)≡G1(yx)

In 

Region IIRegion II

:

C=xcosf2(yx)+ysinf2(yx)cos2f2(yx)≡G2(yx)C=xcos⁡f2(yx)+ysin⁡f2(yx)cos2⁡f2(yx)≡G2(yx)

Plotting these curves we obtain a visual graph of the "transverse" and "divergent" wave systems in the Kelvin wake.

---

This article is based on the MIT open course notes and the original article can be found here

Ocean Wave Interaction with Ships and Offshore Energy Systems

Categories: 

• Ocean Wave Interaction with Ships and Offshore Structures
• Incomplete Pages
• Linear Water-Wave Theory

Wave and Wave Body InteractionsCurrent ChapterShip Kelvin WakeNext ChapterLinear Wave-Body InteractionPrevious ChapterWavemaker Theory

This is an incomplete page

Contents

 [hide] 

• 1 Introduction
  • 1.1 Translating Coordinate System
  • 1.2 Kelvin wake
  • 1.3 Application of the Group velocity
  • 1.4 Solution of the equation for angle

Introduction

Wake created behind a ship

A ship moving over the surface of undisturbed water sets up waves emanating from the bow and stern of the ship. The waves created by the ship consist of divergent and transverse waves. The divergent wave are observed as the wake of a ship with a series of diagonal or oblique crests moving outwardly from the point of disturbance. These wave were first studied by Lord Kelvin

Translating Coordinate System

We have a the standard fixed coordinate system 

x,y,zx,y,z

 and a moving coordinate systems which is moving in the 

xx

 direction with speed 

UU

. We denote the moving coordinate systems in the 

xx

 direction by

x¯=x+Utx¯=x+Ut

Let 

Φ(x,t)Φ(x,t)

 be the velocity potential describing the potential flow generated by the ship relative to the earth frame. The same potential expressed relative to the ship frame is 

Φ¯(x¯,t)Φ¯(x¯,t)

. The relation between the two potentials is given by the identity

Φ(x,y,z,t)=Φ¯(x¯,y,z,t)=Φ¯(x−Ut,y,z,t)Φ(x,y,z,t)=Φ¯(x¯,y,z,t)=Φ¯(x−Ut,y,z,t)

where the relation between the coordinates of the two coordinate systems has been introduced. Note the time dependence occurs in two places in 

Φ¯Φ¯

 and in one place in 

ΦΦ

. The governing equations are always derived relative to the earth coordinate system and time derivatives are initially taken on 

ΦΦ

. Therefore

dΦdt=ddtϕ¯(x−Ut,y,z,t)=∂ϕ∂t−U∂ϕ∂xdΦdt=ddtϕ¯(x−Ut,y,z,t)=∂ϕ∂t−U∂ϕ∂x

All time derivatives of the earth fixed velocity potential 

ΦΦ

 which appear in the free surface condition and the Bernoulli equation can be expressed in terms of derivatives of 

Φ¯Φ¯

 using the Galilean transformation derived above.

If the flow is steady relative to the ship fixed coordinate system

∂Φ¯∂t=0∂Φ¯∂t=0

but

dΦdt=−U∂Φ¯∂xdΦdt=−U∂Φ¯∂x

or, the ship wake is stationary relative to the ship but not relative to an observed on the beach.

Kelvin wake

Diagram of the Kelvin Wake

Local view of Kelvin wake consists approximately of a plane progressive wave group propagating in direction 

θθ

. As noted above surface wave systems of general form always consist of combinations of plane progressive waves of different frequencies and directions. The same model will apply to the ship kelvin wake. Relative to the earth frame, the local plane wave in Infinite Depth takes the form

Φ=igAωekz−ik(xcosθ+ysinθ)+iωtΦ=igAωekz−ik(xcos⁡θ+ysin⁡θ)+iωt

Relative to the ship frame

barΦ=igAωekz−ik(xcosθ+ysinθ)−i(kUcosθ−ω)tbarΦ=igAωekz−ik(xcos⁡θ+ysin⁡θ)−i(kUcos⁡θ−ω)t

But relative to the ship frame waves are stationary, so we must have:

kUcosθ=ωkUcos⁡θ=ω

or

ωk=Cp=Ucosθωk=Cp=Ucos⁡θ

This implies the following

• The phase velocity of the waves in the kelvin wake propagating in direction 
  θθ
   must be equal to 
  UcosθUcos⁡θ
  , otherwise they cannot be stationary relative to the ship.
• Relative to the earth system the frequency of a local system propagating in direction 
  θθ
   is given by the relation 
  ω=kUcosθω=kUcos⁡θ

Relative to the earth system the Infinite Depth Dispersion Relation for a Free Surface states

ω2=gkω2=gk

so that

λ(θ)=2πU2cos2θgλ(θ)=2πU2cos2⁡θg

This is the wavelength of waves in a Kelvin wake propagating in direction 

θθ

 which are stationary relative to the ship.

Application of the Group velocity

An observer sitting on an earth fixed frame observes a local wave system propagating in direction 

θθ

 travelling at its group velocity 

dωdKdωdK

 by virtue of the Rayleigh device which states that we need to focus on the speed of the energy density (

∼∼

 wave amplitude) rather than the speed of wave crests. So, relative to the earth fixed inclined coordinate system 

(X′,Y′)(X′,Y′)

:

X′t=Vg=dωdKX′t=Vg=dωdK

Or

X′=dωdKt ⟹ ddK(KX′−ωt)=0X′=dωdKt ⟹ ddK(KX′−ωt)=0

X′=Xcosθ+Ysinθ=xcosθ+ysinθ+UtcosθX′=Xcos⁡θ+Ysin⁡θ=xcos⁡θ+ysin⁡θ+Utcos⁡θ

So:

KX′−ωt=K(xcosθ+ysinθ)+(KUcosθ−ω)tKX′−ωt=K(xcos⁡θ+ysin⁡θ)+(KUcos⁡θ−ω)t

However 

(KUcosθ−ω)t=0(KUcos⁡θ−ω)t=0

 so that the Rayleigh condition for the velocity of the group takes the form:

ddK[K(θ)(xcosθ+ysinθ)]=0ddK[K(θ)(xcos⁡θ+ysin⁡θ)]=0

By virtue of the dispersion relation derived above:

K(θ)=gU2cos2θK(θ)=gU2cos2⁡θ

It follows from the chain rule of differentiation that Rayleigh's condition is:

ddθ[gU2cos2θ(xcosθ+ysinθ)]]=0ddθ[gU2cos2⁡θ(xcos⁡θ+ysin⁡θ)]]=0

At the position of the Kelvin waves which are locally observed by an observer at the beach.

• So the "visible" waves in the wake of a ship are wave groups which must travel at the local group velocity. These conditions translate into the above equation which will be solved and discussed next. More discussion and a more mathematical derivation based on the principle of stationary phase can be found in Newman 1977.

Solution of the equation for angle

Graphical image of the equations

The solution of the above equation will produce a relation between 

yxyx

 and 

θθ

. So local waves in a Kelvin wake can only propagate in a certain direction 

θθ

, given 

yxyx

.

Simple algebra leads to:

yx=−cosθsinθ1+sin2θ=yx(θ)yx=−cos⁡θsin⁡θ1+sin2θ=yx(θ)

which implies that

• yx(θ)yx(θ)
   is anti-symmetric about 
  θ=0θ=0
   and each part corresponds to the Kelvin wake in the port and starboard sides of the vessel. The physics on

either side is identical due to symmetry.

• θ=0θ=0
  : waves propagating in the same direction as the ship. These waves can only exist at 
  Y=0Y=0
   as seen above.

• θ=π2θ=π2
  : waves propagating at a 
  90∘90∘
   angle relative to the ship direction of forward translation.

• θ=35∘16′θ=35∘16′
  : (or 35,26°) waves propagating at an angle 
  θ=35∘16′θ=35∘16′
   relative to the ship axis. These are waves seen at the caustic of the Kelvin wake.

Let the solution of 

yx(θ)yx(θ)

 be of the form, when inverted:

Region I:θ=f1(yx)Region I:θ=f1(yx)

Region II:θ=f2(yx)Region II:θ=f2(yx)

Note that observable waves cannot exist for values of 

yxyx

 that exceed the value shown in the figure or 

yx∣∣Max=2−3/2yx|Max=2−3/2

. This translates into a value for the corresponding angle equal to 

19∘28′19∘28′

 (or 19,47°) which is the angle of the caustic for any speed 

UU

.

"transverse" and "divergent" wave systems in the Kelvin wake

The crests of the wave system trailing a ship, the Kelvin wake, are curves of constant phase of:

xcosθ+ysinθcos2θ=Cxcos⁡θ+ysin⁡θcos2⁡θ=C

In 

Region IRegion I

:

C=xcosf1(yx)+ysinf1(yx)cos2f1(yx)≡G1(yx)C=xcos⁡f1(yx)+ysin⁡f1(yx)cos2⁡f1(yx)≡G1(yx)

In 

Region IIRegion II

:

C=xcosf2(yx)+ysinf2(yx)cos2f2(yx)≡G2(yx)C=xcos⁡f2(yx)+ysin⁡f2(yx)cos2⁡f2(yx)≡G2(yx)

Plotting these curves we obtain a visual graph of the "transverse" and "divergent" wave systems in the Kelvin wake.

---

This article is based on the MIT open course notes and the original article can be found here

Ocean Wave Interaction with Ships and Offshore Energy Systems

Categories: 

• Ocean Wave Interaction with Ships and Offshore Structures
• Incomplete Pages
• Linear Water-Wave Theory