The shape of ship wakes: The Kelvin wedge

1 1 o 2sin 3  38.9

Dr Andrew French. August 2013 1 Contents • Ship wakes and the Kelvin wedge • Theory of surface waves – Frequency – Wavenumber – Dispersion relationship – Phase and group velocity – Shallow and deep water waves – Minimum velocity of deep water ripples • Mathematical derivation of Kelvin wedge – Surf-riding condition – Stationary phase – Rabaud and Moisy’s model – Froude number • Minimum ship speed needed to generate a Kelvin wedge • Kelvin wedge via a geometrical method? – Mach’s construction • Further reading 2

A wake is an interference pattern of waves formed by the motion of a body through a fluid. Intriguingly, the angular width of the wake produced by ships (and ducks!) in deep water is the same (about 38.9o). A mathematical explanation for this phenomenon was first proposed by Lord Kelvin (1824-1907). The triangular envelope of the wake pattern has since been known as the Kelvin wedge. http://en.wikipedia.org/wiki/Wake 3 Venetian water-craft and their associated Kelvin wedges

Images from Google Maps (above) and Google Earth (right), (August 2013)

4 Still awake? 5 The Kelvin Wedge is clearly not the whole story, it merely describes the envelope of the wake. Other distinct features are highlighted below:

Turbulent Within the Kelvin wedge we flow from bow see waves inclined at a wave and slightly wider angle propeller from the direction of travel of wash. * These the ship. (It turns effects will be out this is about 55o) * addressed in this presentation. The others Waves disperse within a ‘few will not! degrees’ of the Kelvin wedge. Is the number of waves predictable? What about their Away from the ship the ‘Kelvin wavelengths? They all look wedge group’ of waves somewhat similar.... * appear to curve away from the 55o wavefronts. Can we model this? 6 Mathematical theory of surface waves

Key topics: Frequency Wavenumber Dispersion relationship Phase and group velocity Shallow and deep water waves Minimum velocity of deep water ripples

7 A means of characterizing a wave-like disturbance is via a dispersion relation. This is an equation which relates the frequency of the wave to its wavelength, plus other parameters such as surface tension, fluid density, depth etc.

Wavelength l Define angular frequency   2f

2

and wavenumber k 

l Amplitude

For deep water gravity waves, the dispersion relationship is  2  gk wave velocity c p i.e. we ignore effects of the density of air, surface tension and assume the depth D  l i.e. kD1

 The phase velocity (of individual wave crests) can c  fl  be found from the dispersion relationship p k 8 The dispersion relationship allows to compute how fast groups of disturbances will travel. This group velocity can be different from the phase velocity. Relative motion of the wave crests to the overall envelope causes dispersion.

d group wave velocity cg 

cg dk

For deep water gravity driven waves

1   gk 2

  1 c   gk 2 p k

d  1 c   1 gk 2 g dk 2

1  cg  2 c p 9 In general, for vorticity free waves on the interface of two incompressible, Newtonian fluids, the dispersion relationship can be shown to be, for waves with amplitude G<< D

surface tension k 3  g   k  2  1 2 2  1cotanhkD

fluid densities depth  For waves on a water, air interface 1 10002 , so ignore 2 3 2 k      gk tanhkD For shallow water waves tanhkD  kD  1  k 4D  2   gk 2D 1

For deep water waves tanhkD 1 k 3  2   gk 1 10 2 Since k  the higher powers of k will contribute less for longer wavelengths l

2 2 e.g. waves coming ashore Hence for shallow water gravity waves   gk D just before they break, or waves in a shallow river or canal

Similarly for deep water waves  2  gk

Note for deep water ripples (or ‘capillary waves’) this approximation is invalid.

We must use the full k 3 dispersion relation    gk 1 Water:  = 1000 kgm-3 -1 Ripple phase velocity is:  = 0.0728 Nm

 k g2  g l cp      kk11 l2 

4g minimum phase velocity = 23 cms-1 Which has a minima at cp  4 1 11 Mathematical theory of the Kelvin Wedge

1 1 o 2sin 3  38.9

1 1 Note 1 1 tan 4 2  sin 3 12 Consider a ship moving at velocity v through deep water. It creates wavelike disturbances with a variety of wavelengths. i.e. the resulting disturbance has a spectrum of Fourier components.

The waves of interest are those which are most pronounced. i.e. the Kelvin Wedge. Let these be travelling at angle a to the ship’s velocity.

c p Unlike all other waves, which will naturally disperse (and also attenuate due to the viscosity of water) these waves continue to be sustained by the motion of the ship.

To achieve this, the phase velocity of these waves must equal the component of the ship’s velocity parallel to the wavevector k a . This is know as a ‘surf riding’ condition.

2  1 cp  vsina   gk gk 2 Hence sina    1 c   gk 2 v p k g  k  2 2 v sin a 13 At point P the phase of the waves at angle a is   ka r  krsina   

Now g k  v2 sin 2 a gr sina        v2 sin 2 a  gr sin 2 a(cosa   )  sina   (2sina cosa)    2  4  a v  sin a  gr  sina cosa    2sina   cosa v2 sin3 a

 When this means the wave phase doesn’t change as a varies. In other words  0  a we expect constructive interference for angles a when  0 This is known as the principle of stationary phase. a   0  sina cosa    2sina   cosa  0 a

Hence 2tana     tana 14 Using the addition formula for tangents 2 tana     tana  tana  tan   2   tana 1 tana tan   2 tana  2 tan   tana  tan2 a tan  tana  tan2 a  2tan  tana  tan   tan2 a  2

2 2 2 2 d tan a  2sec a  tana2 tana sec a   sec   2 da tan2 a  2 2 2 2 2 d 2 tan a  2 2 tan a 2 2 tan a  sec   sec a 2  sec a 2 da tan2 a  2 tan2 a  2

d Hence  0 when tana   2 By a similar argument to ‘stationary phase’ da let the Kelvin Wedge be at a when Hence a  tan1 2  tan1 2  54.7o

 tana    2  and therefore   tan1  tan1   19.5o  2     tan a  2   4 

2 = 38.9o is the angular width of the Kelvin Wedge and a = 54.7o is the angle of the principle waves in the ship’s wake

a  tan1 2 54.7o

 2  1  o 2  2 tan    38.9  4 

16 What wavelengths do we expect for waves in the Kelvin wedge?

From our ‘surf-riding condition’

g 2 g 2v2 sin 2 a k     l  v2 sin2 a l v2 sin 2 a g sina Now 1  sin 2 a  cos2 a and tana  cosa 1 1 1  1  sin 2 a  2 2 1 sin a tan a 1 tan2 a 1 Hence if tana   2  sin 2 a   2 1 3 1 2

4v2 3gl This result could be used to  l  or v  infer ship velocities from their 3g 4 wake pattern 17 Wavelength of wake waves of the Isle of Wight Ferry (Image from Google maps August 2013) is approximately (3mm / 19mm) x 20m = 3.2m, although at this resolution it is somewhat difficult to measure accurately

3gl Hence v  = 2.7ms-1 4

Now 1ms-1 = 1.944 knots which means the model predicts the ferry is travelling at about 5.3 knots

According to the Red Funnel website the car ferry travels between 12 and 14 knots ..... However, it could have been slowing down as it enters Southampton water!

This sounds like an excellent experimental physics project, i.e. to test the validity of the above equation. 18 The idea of using Google maps to investigate the Kelvin wedge model has recently been posed by Rabaud and Moisy (arXiv: 1304.2653v1 9 April 2013)

Their thesis attempts to explain deviations to the Kelvin wedge model observed for higher ship speeds. They develop a model which characterizes the wave angle in terms of Froude number, although at ‘low’ Fr the fixed Kelvin wake angle of 38.9o is preserved to be consistent with observations.

Froude number is the ratio of ship’s speed to deep water wave speed. It assumes wavenumber k inversely scales with ship length L

1 v  2 1 Fr  cp  gk k  Rabaud & Moisy’s model cp L  v v Hence Fr    1  1  2 gL g    L 

19 Minimum ship speed needed to generate a Kelvin wedge

9g Water:  = 1000 kgm-3 v  4  = 0.0728 Nm-1 41

v > 20 cms-1

minimum phase velocity = 23 cms-1

20 What is the minimum ship speed needed to generate a wake? 4g This minimum phase velocity for deep water gravity waves is cp  4 1 2 gl 4v2 and for the Kelvin Wedge Now cp   l  l 1 2 3g

2 4 Hence 6g g 4v 4g 3g 2v 2 4g   4   v 2 2 3  4v 1 2 3g 1 1 1

2 3g 2v 4g 9g 4 2 4g   4  2v  3v 2 2  2v 1 3 1 1 1 2 3g 2v 4g 4 2 4g   9g  41v  61v 2  2v 1 3 1 1 4 2 9g  41v 12v g 1

4 2 41v 12v g 1 9g  0 21 4 2 41v 12v g 1  9g  0   9g 4 2 g v  4 41 v  3v   9g  0  1  41  2    -3  2 3 g 9 g  Water:  = 1000 kgm   -1 41  v     9g  0  = 0.0728 Nm  2   4   1  1  -1 2 v > 20 cms  g  4 v2  3   9g  9g  0 1 2   1  Compare to minimum 2 3 g v  2  0 phase velocity 1 4g 9g c  4  23 cms-1 v2  p 4 1 41 4  1.41 9g 9 v  4 4  1.22 41 4 22 Kelvin wedge via a geometrical method?

1 1 o 2tan 4 2  38.9

23 Is there an easier way to generate the Kelvin wedge pattern? Is a deep water wave wake pattern analogous to the shock waves caused by a supersonic aircraft?

To investigate the latter let us first consider Mach’s construction

shock front Object moving at speed v creating waves ‘Infinitesimally thin’ as it moves spherical shells of disturbance are created continuously as the object moves. They radiate out at the wave speed c

24 Mach’s construction

c is the wave speed v is the velocity of the object making the waves

Mach number v M  c

ct

 vtsin   ct vt  c     sin 1   sin 1 1  v  M

25 26 2 For the Kelvin Wedge we know tan   4

1 2 1 1 2 hence sin     sin   1 sin   1 3 1 1 9 1 2 2 tan  16

Hence Mach’s construction gives the Kelvin Wedge result if the corresponding Mach number is M = 3.

Is this just serendipity? Is it at all relevant? What does M = 3 mean for a ship moving at velocity v, generating deep water, gravity driven surface waves? (Which of course have wave speeds which vary with wavelength).

T.E. Faber attempts to discuss this possible connection in Fluid Dynamics for Physicists (p194). He makes a argument involving the fact that the group velocity of deep water gravity waves is half their phase velocity. However, he admits the whole discussion is on somewhat shaky foundations!

27 Further reading

Simulation of the Kelvin Wedge http://www.math.ubc.ca/~cass/courses/m309-01a/carmen/Mainpage.htm

Rabaud and Moisy’s model. “Ship wakes: Kelvin or Mach angle?” (arXiv: 1304.2653v1 9 April 2013) http://arxiv.org/pdf/1304.2653.pdf http://blog.physicsworld.com/2009/07/27/google-earth-physics/ http://physicsworld.com/cws/article/news/2013/may/30/physicists-rethink-celebrated-kelvin-wake-pattern-for-ships

Faber, T.E. Fluid Dynamics for Physicists. Cambridge University Press. 1995. p188-194 http://en.wikipedia.org/wiki/Wake

Perkinson, J. “Gravity waves: The Kevin wedge and related problems” http://web.mit.edu/joyp/Public/HarvardWork/AM201FinalPaper.pdf 28