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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. |
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Subject:
Re: Progression speed of water waves?
Answered By: hedgie-ga on 04 Oct 2004 04:12 PDT Rated: |
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:
and gave an additional tip of:
$6.00
If taken together, the answers and links provided did completely answer my question - thanks a lot! |
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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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