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Waves in Fluids: Ship Waves, Tsunamis, and Solitary Waves, Study notes of Physics

Georgia Institute of Technology - Main CampusPhysics

Various wave phenomena in fluids, including ship waves, tsunamis, and solitary waves. It covers topics such as phase invariance, dispersion relations, and hamilton's equations of motion. The document also discusses the behavior of waves in different environments and the role of gravity and viscosity.

Typology: Study notes

Pre 2010

Uploaded on 09/17/2009

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Chapter 15 Waves
1.Exercise 15.3 Ship Waves
(a) Consider a plane wave with frequency !0and wave vector k0as measured in the water’s
frame, and !and kas measured in the boat’s frame. For an observer with position x0in the water’s
frame and xin the boat’s frame, the phase he measures is:
k0!x0!!0tin terms of water’s frame variables
and k!x!!tin terms of boat’s frame variables
Since the phase is invariant under the change of reference frame, we can equate the above two
expressions and then differentiate both sides with respect to t, noting that for an observer moving
together with the boat, dx0/dt !u,dx/dt !0, we get:
!!!0!k0!u
By looking at Fig 15.3 and use kto denote the wave vector as measured in the water’s frame(as
the text does), we get
!!!0"ukcos"
(b) #is the angle between Vg0!uand u, elementary trigonometry then gives(see the sketch on
the right),
tan#!Vg0sin"/"u"Vg0cos"#
For stationary wave pattern !!0, using the !"k#we got in part (a), we see that
!0"k#!!ukcos"
(c) For capillary waves, !0""$/%#k3,Vg0!#!0/#k!"3/2#"$/%#k
Plugging these and u!!!0/"kcos"#into the expression for tan #, we get
tan#!"3tan"#/"1!2tan2"#
Capillary wave pattern for a given #exists only when we can find some "$"&/2,&#(i.e. only
forward waves can contribute to the pattern) satisfying the above equation. And it’s easy to show that
indeed for any #we can find such a "given by:
tan"!"!3!9"8tan2##/"4tan##when ##&/2
and tan"!"!3"9"8tan2##/"4tan##when #$&/2
For gravity waves, !0"gek, and Vg0!"1/2#ge/k, and we get
tan#!"!tan"#/"1"2tan2"#
Only when ##arcsin"1/3#can we find some "$"&/2, &#satisfying this equation, which is:
tan"!"!1%1!8tan2##/"4tan##(both solutions are valid)
This means that the gravity-wave pattern is confined to a trailing wedge with an opening angle
#gw !2arcsin"1/3#.Q.E.D.
2.Tsunamis from Japan
We can treat this as a 2-dimensional problem, i.e., only consider the horizontal components of
the velocity, which are almost independent of z. In what follows, %is the 2-dimensional derivative
operator.
(a) The mass per unit area is %"D"'#, and the mass flux per unit length is %"D"'#v"%Dv,to
the first order in perturbation. Then by mass conservation,
#$%"D"'#%/#t"%!"%Dv#!0
&#'/#t"%!"Dv#!0 (i) ( assuming constant %)
pf3

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Chapter 15 Waves

1.Exercise 15.3 Ship Waves (a) Consider a plane wave with frequency_!_ 0 and wave vector k 0 as measured in the water’s

frame, and_!_ and k as measured in the boat’s frame. For an observer with position x 0 in the water’s frame and x in the boat’s frame, the phase he measures is:

k 0! x 0!! 0 t in terms of water’s frame variables and k! x! !t in terms of boat’s frame variables Since the phase is invariant under the change of reference frame, we can equate the above two expressions and then differentiate both sides with respect to t, noting that for an observer moving together with the boat, dx 0 / dt! u , d x / dt! 0, we get:

!!! 0! k 0! u By looking at Fig 15.3 and use k to denote the wave vector as measured in the water’s frame(as the text does), we get !!! 0 " uk cos " (b) # is the angle between V (^) g 0! u and u , elementary trigonometry then gives(see the sketch on the right), tan #! V (^) g 0 sin " /" u " V (^) g 0 cos " # For stationary wave pattern_!! 0, using the!_ " k # we got in part (a), we see that ! 0 " k #!! uk cos " (c) For capillary waves,! 0 " " $ / % # k^3 , V (^) g 0! #! 0 /# k! "3/2# " $ / % # k Plugging these and u!!! 0 /" k cos " # into the expression for tan # , we get tan #! "3 tan " #/" 1! 2 tan 2 " # Capillary wave pattern for a given # exists only when we can find some " $ " & /2, & # (i.e. only forward waves can contribute to the pattern) satisfying the above equation. And it’s easy to show that indeed for any # we can find such a " given by:

tan "! "! 3! 9 " 8 tan 2 # #/"4 tan # # when # # & / and tan "! "! 3 " 9 " 8 tan 2 # #/"4 tan # # when # $ & / For gravity waves,! 0 " g (^) e k , and V (^) g 0! "1/2# g (^) e / k , and we get tan #! "! tan " #/" 1 " 2 tan 2 " # Only when # # arcsin"1/3# can we find some " $ " & /2, & # satisfying this equation, which is: tan "! "! 1 % 1! 8 tan 2 # #/"4 tan # # (both solutions are valid) This means that the gravity-wave pattern is confined to a trailing wedge with an opening angle #gw! 2 arcsin"1/3#. Q.E.D.

2.Tsunamis from Japan We can treat this as a 2-dimensional problem, i.e., only consider the horizontal components of the velocity, which are almost independent of z. In what follows, % is the 2-dimensional derivative operator. (a) The mass per unit area is % " D " ' #, and the mass flux per unit length is % " D " ' # v " %D v , to the first order in perturbation. Then by mass conservation, #$ % " D " ' #%/# t " %! " %D v #! 0 & # ' /# t " %! " D v #! 0 (i) ( assuming constant % )

The Navier-Stokes equation in this case is: # v /# t! !% P / % " g , whose vertical component tells us P! %g " '! z #, and whose horizontal components then tell us # v /# t!! g % ' (ii) Applying # t to both sides of eqn. (i) and then plugging in eqn (ii), we get #^2 ' /#^2 t! g %! " D % ' # (b) By plugging in a plane wave solution to the wave equation, we find the dispersion relation: !! k gD 1! i "% D / D #! " k / k^2 # " k gD $ 1! " i /2#"% D / D #! " k / k^2 #% The imaginary part of_!_ only affects the decaying(or growing) of the the wave amplitude but not it’s propagation direction, and furthermore it’s smaller than the real part by a factor of ( / l & 1 (where ( is the wave length and l is the scale over which D varies). Thus we take_!_ " k gD.

Using the Hamilton’s equations of motion introduced in Chapter 6 Geometrical Optics, we get d x / dt! % (^) k!! gD " k / k # and d k / dt! !% (^) x!! !" k /2# g / D % (^) x D Thus we see the direction of wave propagation is always deflected towards the shollower part of the ocean. (c) Create in the bottom of the Pacific Ocean a ridge going from Japan to LA(with equi-depth contours being elliptical curves and LA being at the focus)! This ridge will act as a lens, focusing those Tsunamis towards LA. Note that only very slight deflection(ver slight difference in ocean depth) is sufficient: Assume Japan extends 500km long, and the distance between Japan and LA is about 10,000km. Then only about 500km/10000km ' 5% change in the ocean depth is needed.. Q.E.D.

3.Solitary Waves in a Deformable Conduit (a) Assuming that a vary slowly with height, the solution for this part can be found in Section 12.4.5 Blood Flow B&T. The only difference between there and here is that here we no longer neglect gravity. By adding a % 1 g term to the driving force, we get [see eqn.(12.72) Poiseuille’s Law B&T], Q!! &a^4 "# p 1 /# z " % 1 g #/" 8 ) 1 # (b) The force per unit area on the conduit wall applied by fluid-2 consists of two parts: the pressure contribution p 2 and the viscous contribution f (^) viscous. Since fluid-2 has no vertical motion, we have p 2 !constant! % 2 gz. Now let’s calculate f (^) viscous in cylindrical coordinates. The velocity field due to the change in a is: v! v (^) r e (^) r , with v (^) r! " a / r #"# a /# t # f (^) viscous! T (^) rr!! 2 ) 2 *rr!! 2 ) 2 $"2/3## v (^) r /# r! "1/3# v (^) r / r % !# 2 ) 2 " a / r^2 #"# a /# t # where we have used formulas analogous to those in Box 10.2 B&T f (^) viscous " r! a #! 2 ) 2 "1/ a #"# a /# t # Thus the force per unit area applied by fluid-2 on the wall is constant! % 2 gz " 2 ) 2 "1/ a #"# a /# t #, and similarly the force per unit area applied by fluid-1 on the wall is p 1 " 2 ) 1 "1/ a #"# a /# t #. Equating these two forces and noting that ) 2 ( ) 1 , we get p 1!! % 2 gz " 2 ) 2 "1/ a #"# a /# t # "constant Thus the PDE relating Q and a is:

Q! " &a^4 /8 ) 1 #& % 2 g! % 1 g! #$ 2 " ) 2 / a #"# a /# t #%/# z ' The other PDE relating Q and a is given by mass conservation(i.e. volume conservation, assuming constant % ) for fluid-1 as follows, d

dt ) z 1

z 2 &a^2 dz! Q " z 1 #! Q " z 2 # & 2 &a "# a /# t #! !# Q /# z (c) 2 &a "# a /# t #! #" &a^2 #/# t! !# Q /# z

let Q! Q 0 " Q 1 f " z! ct #, then &a^2! " Q 1 / c # f " z! ct # , where we’ve set the additive constant to zero without loss of generality.