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Q: Progression speed of water waves? ( Answered 5 out of 5 stars,   3 Comments )
Question  
Subject: Progression speed of water waves?
Category: Science > Physics
Asked by: justcurious157-ga
List Price: $30.00
Posted: 14 Sep 2004 15:26 PDT
Expires: 14 Oct 2004 15:26 PDT
Question ID: 401213
Speed of acustic waves in air or light waves are well known and pretty constant.
My question: What are the major parameters (energy input, water depth
etc.) and what is the basic equation to calculate the progression
speed of water waves in a lake generated by steamers, speed boats etc.
Background: As a passionate canoeist I wonder if it is (always)
possible to escape a wave front generated by a faster boat or ship
just by paddling. I sometimes succeeded but possibly just by chance.
My speed is about 8 to 10 km / hour max.
Answer  
Subject: Re: Progression speed of water waves?
Answered By: hedgie-ga on 04 Oct 2004 04:12 PDT
Rated:5 out of 5 stars
 
Speed of surface waves is given here:

http://www.physics.nmt.edu/~raymond/classes/ph13xbook/node7.html

 

speed goes up as square of depth -which leads to an observable effect
of crests 'falling over' as waves approach shore (and bottom is slowwing down).

Formula was derived and analysed by 
The Theory of Sound 1 by John William Strutt, 3rd Baron Rayleigh
http://www.measure.demon.co.uk/docs/Theory.html

and is reproduced in a classical book:
http://www.amazon.com/exec/obidos/tg/detail/-/0486602923/002-1451747-8990406?v=glance
justcurious157-ga rated this answer:5 out of 5 stars and gave an additional tip of: $6.00
If taken together, the answers and links provided did completely
answer my question - thanks a lot!

Comments  
Subject: Re: Progression speed of water waves?
From: hfshaw-ga on 14 Sep 2004 20:10 PDT
 
Waves on the surface or a body of water are an example of "surface 
waves".  For an incompressible fluid with zero viscosity (not a bad 
assumptions for the problem you have posed), the speed of propagation, 
c, of a surface wave of wavelength L in a body of water of depth (measured 
from the unperturbed surface to the bottom) H is given by:

  c = sqrt{[g * L * tanh((2*pi * H)/L)]/2*pi}

where g is the acceleration due to gravity (~9.8 m/s^2), pi = 3.14159...., 
and tanh is the hyperbolic tangent: 

    tanh(x) = {[exp(x) - exp(-x)]/[exp(x)+exp(-x)]}

For a given depth water, longer wavelengths travel faster.

A 1 m wavelength wave in a 3 m deep water body travels about 1.25 m/s 
or 4.5 km/hr.  A 2 m wavelength wave in the same depth water travels about 
6.5 km/hr, while a 5 m wavelength wave would travel about 10 km/hr.
Subject: Re: Progression speed of water waves?
From: racecar-ga on 15 Sep 2004 10:48 PDT
 
hfshaw has given a correct expression for the phase velocity of water
waves.  However, what you need to know to determine whether you can
outrun the wake of a bigger boat is the group velocity.  You might
have noticed that boat wakes are made up of not just a single V-shaped
crest, but a large number of crests, stacked up in a V-shaped pattern
behind the boat.  If you watch one of the crests, you see that it
starts at the inside edge of the V, propagates forward and outward to
the outside of the V, and dies out as other crests grow behind it.

If the water is deeper than half of the wavelength, the wave speed is
independent of depth.  This is likely to be the case for wake waves in
any body of water you'll be sharing with motor boats.  In this deep
water limit, the dispersion relation is w = sqrt(gk), where w is
frequency (radians per second), g is the acceleration of gravity, and
k is the wave number, which is 2*pi/wavelength.  The phase velocity,
which is the speed of each individual crest, is w/k = sqrt(g/k).  The
group velocity, which is the speed of the wake as a whole, is dw/dk =
(1/2)sqrt(g/k).  So the group velocity is just half the phase
velocity.  Using this expression, it turns out that you can outrun any
wave for which the wavelength L < 8 pi V^2 / g, where V is the speed
you paddle.  If you paddle at 10 km/h (2.78 m/s) you can outrun waves
shorter than 19.8 m.  If you paddle at 8 km/h (2.22 m/s) you can
outrun waves shorter than 12.7 m.  This is true even if the depth is
shallow enough to change wave speed, because waves get slower in
shallow water.  The actual wavelength of wake waves depends on the
geometry of the hull, mass of the boat, speed, etc.  But unless you're
talking about an oil tanker or something, I don't think the wake waves
will be as long as 19.8 m.  So you should in general be able to outrun
wakes, if you can maintain your top speed long enough.
Subject: Re: Progression speed of water waves?
From: racecar-ga on 04 Oct 2004 17:32 PDT
 
Wow--thirty bucks for an answer which consists entirely of:

- 3 links, 2 of which are irrelevant and unhelpful, as they relate to
sound waves and not surface waves.

- 1 piece of incorrect information (speed does not go up as depth
squared--in the shallow water limit it goes as the square root of the
depth, and in the deep water limit, it is independant of depth).

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