This question arose out of an office discussion based on the recent
Indian Ocean tsunami, although it is more about general physics and
less about tsunamis.

Assumptions:
1.  There's a large body of water with a surface area that is
effectively infinite.  There are no shores or boundaries to worry
about.

2.  The body of water has a completely smooth, uniform bottom and is
100 meters deep throughout.

3.  A large, dense sphere is dropped into the water, creating a ripple
effect.  We?re basically throwing a large stone in a calm pool.

Question:
How does the energy of the wave (ripple) change as the distance from
the sphere (epicenter) increases?  Does the energy from the impact of
the sphere dissipate linearly, perhaps as a function of the diameter
of the ripple?  Or does the energy dissipate exponentially, perhaps as
a function of the volume or area of water displaced by the impact?

What is the correlation between wave height and energy?

To give a little background, I?ve read several papers about tsunamis
while researching this, but everything I?ve read tends to focus on run
up of the wave on a continental shelf and shoreline.  To further
complicate matters, tsunamis are usually generated by the movement of
a large fault line rather than a single point.  I?m looking for an
answer to a much simpler question.

---

Clarification of Question by woolly-ga on 06 Jan 2005 06:12 PST

Thanks for the comments so far.  Racecar-ga, can you direct me to a
published source (web-based or otherwise) that might have the actual
formulae for the long and short wavelength waves?

Thanks.

---

Request for Question Clarification by hedgie-ga on 07 Jan 2005 04:01 PST

Here is simpler version of equations stated in the last comment,
and some definitions
http://en.wikipedia.org/wiki/Wave

One wonders how deep your office discussion go (?) and much energy
and effort you want to give to which issue.

If you consider (more simple) steady state system - then Energy (~square
of amplitude) decreases as inverse of the distance from the source (place
where you repeatedly excite the surface) 
in perfect analogy to 3D case described here:
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

If it is not continuous (one stone) it is more complex:

1) Exponentional dissipation of total energy (as the whole disturbance
   dies-out) -- dpepends on viscosity
2) transport of the energy from the center, by the travelling wave 
 -- and that depends on speed of the wave 

  Here the decrease is not simply linear, as energy is not all in the
perimeter - but is spread non-uniformly through excited area --
so you get something betwee inverse square, inverse distance and it
all is time dependent.

Do you want actual solution, with movie and all?
http://www.utdallas.edu/~pca015000/NATS_1311/09_28_04/surface_waves_in_pond.htm
(just a start :-)

---

Request for Question Clarification by hedgie-ga on 07 Jan 2005 04:19 PST

Sorry

that link to hyperphysics got confused.
Here is that 3D case explained

http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html

In the case of surface waves - a 2D case - you get inverse distance
instead of the inverse square. In nD case it surface of a 'unit sphere'.

but that is true for a steady state - a stream of pebles going in.

For a single event - burst of 'radiation' - you get exp(-alpha * t) 
(dissipation) combined with transport of energy.

There is an educational film-loop around (usually forogotten in the stacks of
 physics department of any local college) which show in slow motion how
 waves spread - photographed is steel ball dropping to milk (to make
the liquid opaque) -- it is quite pretty and perhpas you can show it
during the
 lunch break (?)  -- but there is no simple formula ..

You may want to look at the question/answer

http://answers.google.com/answers/threadview?id=401213

The word 'dissipates' in thermodynamics means that higher form (kinetic
energy) is convereted to heat - entropy increases.

You perhaps are asking about how the amplitude of wave changes with
distance from the impact point (and time since impact?) ..

Your idealized wave is propogating out in a circle of ever-increasing
circumference.  Even if there is no attenuation of overall energy (due
to friction), there is still less energy per unit available as the
circle grows larger. I imagine it would fall off pretty rapidly, as
the square of the distance from the epicenter, since this is how the
circumference increases.

This is pretty much the same thing that happens when you toss a pebble
in a pond, at least in an idealized sense (the real world is, of
course, incredibly more complex).

The dynamics of a tsunami are pretty different, so I don't know how
applicable your thought-experiment is to the situation in Sumatra and
the surrounding areas in the Indian Ocean.


pafalafa-ga

>>...I imagine it would fall off pretty rapidly, as the square of the
distance from the epicenter, since this is how the circumference
increases...<<

Ooops...I got myself mixed up.  The circumference increases linearly
with the distance from the center, not as the square.

paf

The energy in the longer wavelength waves decreases linearly with
distance from the source.  There is very little dissipation at these
long wavelenghts.  The little ripples die out faster.  Wave energy is
proportional to the square of the height.

The following link goes through the math of surface gravity waves and
capillary waves.

http://snowball.millersville.edu/~adecaria/ESCI485/esci485_lesson05_surface_waves.html