On Monday, February 17, 2003, at 05:56 AM, ChrisAtUpw@....... wrote:
> In a message dated 17/02/03, hernlund@............ writes:
>
> Hey Chris,
> =A0=A0=A0In the ideal gas treatment (this is where the approximate =
comes=20
> in), the equation of state is just PV=3DnkT where P is pressure, V is=20=

> volume, n is number of molecules, k is Boltzman's constant, and T is=20=

> temperature. This is the same as P=3DrkT where r is the number =
density.=20
> While this is quite approximate, it does capture the gross features of=20=

> the atmosphere surprisingly well. The hydrostatic part of the=20
> atmospheric equilibrium (force balance) is determined by dP/dz
> proportional to -r*g, where g is the gravitational acceleration. At=20
> constant T this gives you: dP/dz proportional to -P*g/kT or upon=20
> integrating just P proportional
> to exp(-z*g/kT) where z is the height. Since P is proportional to r in=20=

> this case, then density also falls off exponentially with height.
>
> Hi John,
>
> =A0=A0=A0=A0=A0=A0This seems to be a rather convoluted and confusing =
way of=20
> expressing a simple velocity.

Well...sorry about that...I tend to be quite convoluted when writing=20
e-mails late at night, and my thinking was obviously not clear (see=20
below).

> =A0=A0=A0=A0=A0=A0I haven't a clue what a number density is supposed =
to represent.=20
> Would you care to define it please?

If you do the algebra:
PV=3DnkT
P=3DrkT
then r=3Dn/V or the number of molecules per unit volume.

> =A0=A0=A0=A0=A0=A0It is quite easy to show that the velocity of sound =
in a gas is=20
> given by c =3D Sqrt(gamma x P / rho), where gamma is the ratio of the=20=

> specific heats, P is the pressure and rho is the density =3D mass /=20
> volume V. Hence PV ---> RT and you don't have to bother with P and V.

Yes, the sound speed is Sqrt(dP/drho) and under the assumptions of=20
adiabatic fluctuations you do get this expression. You are correct=20
about the sound speed, and after going through some simple derivations,=20=

I see that the growth in amplitude with height is not due to slowing=20
down of waves, but rather is about the density only. To make this more=20=

explicit (and hopefully not so convoluted), I'll explain this briefly=20
below:

The sound wave equation for a wave going upward (z) can be written in=20
terms of density (rho) as:
d[ln(rho)]/dt=3D-c*d[ln(rho)]/dz
and the solution is just the equation ln(rho)=3Df(z-c*t), where f(z) is=20=

the function describing the wave's profile that depends only on the=20
initial perturbations at t=3D0.
You find the true origin of the amplitude increase with height by=20
separating the density into a hydrostatic part rho' (a function of z=20
only) times a factor measuring the fluctuation, that is:
rho=3D(q+1)*rho'
where q+1 is the fluctuation factor. For example, if q=3D0.10 then this=20=

represents a ten percent increase in density relative to the=20
hydrostatic density while q=3D0 represents no density fluctuation. You=20=

now get, since the time dependence of rho' goes away:
d[ln(q+1)]/dt=3D-c*d[ln(q+1)]/dz-c*d[ln(rho')]/dz
Since rho' decreases exponentially with height as=20
rho'(z)=3Drho'(z=3D0)*exp(-z/H), where H is the scale height, then the =
wave=20
equation says that:
d[ln(q+1)]/dt=3D-c*d[ln(q+1)]/dz+(c/H),
So looking just at the first term on the right side, it represents a=20
wave traveling upward with no change in fluctuation amplitude,
d[ln(q+1)]/dt=3D-c*d[ln(q+1)]/dz,
with a solution,
ln(q+1)=3Df(z-c*t) + a constant.
while the second term represents a source that increases ln(q+1) by an=20=

amount (c/H) per unit time,
d[ln(q+1)]/dt=3D(c/H),
which has the solution,
ln(q+1) =3D t*c/H + a constant
This second term is the origin of the growing amplitudes, it is not a=20
slowing down of waves that causes this. The surprising thing is the=20
exponential increase in fluctuation q with time. If we take H=3D9000=20
meters, as a typical parameter, and by noting that c*t is ~ the=20
distance the wave has traveled,
ln(q+1) =3D z/H + ln[q(z=3D0)+1],
or,
q+1 =3D [q(z=3D0)+1]*exp(z/H)
So at the scale height (~9 km), the fluctuation factor q+1 has=20
increased by ~2.7 (density due to wave fluctuation is 2.7 times more or=20=

less than its static value), at two scale heights it increases by ~7.3=20=

, at three it is ~19.7, four ~53.1, etc.. Of course, when you get up to=20=

the ionosphere you have gone about 10 scale heights so the fluctuation=20=

factor has increased by ~22,000 or more. It sounds reasonable then,=20
that this is how you get 100 m displacements from small EQ or tsunami=20
motions at the surface that aren't very large at all.

So the physics still holds up, all this stuff above simply means that a=20=

decreasing mass (density) means that for the same energy in a traveling=20=

wave that the amplitude must increase...simply due to inertia and the=20
the exponential decrease of density.

I recently saw an interesting related story (brief blurb on cnn) that=20
satellites are trying to measure the hissing sound produced by spray=20
paint cans in LA to prevent graffiti...don't know how well this will=20
work.

Cheers!
John
On Monday, February 17, 2003, at 05:56 AM, ChrisAtUpw@....... wrote:

ArialIn a message dated
17/02/03, hernlund@............ writes:


Hey Chris,

=A0=A0=A0In the ideal gas treatment (this is where the approximate comes
in), the equation of state is just PV=3DnkT where P is pressure, V is
volume, n is number of molecules, k is Boltzman's constant, and T is
temperature. This is the same as P=3DrkT where r is the number density.
While this is quite approximate, it does capture the gross features of
the atmosphere surprisingly well. The hydrostatic part of the
atmospheric equilibrium (force balance) is determined by dP/dz

proportional to -r*g, where g is the gravitational acceleration. At
constant T this gives you: dP/dz proportional to -P*g/kT or upon
integrating just P proportional

to exp(-z*g/kT) where z is the height. Since P is proportional to r in
this case, then density also falls off exponentially with height.


Hi John,


=A0=A0=A0=A0=A0=A0This seems to be a rather convoluted and confusing way =
of
expressing a simple velocity.



Well...sorry about that...I tend to be quite convoluted when writing
e-mails late at night, and my thinking was obviously not clear (see
below).


=A0=A0=A0=A0=A0=A0I haven't a clue what a number density is =
supposed to
represent. Would you care to define it please?



If you do the algebra:

PV=3DnkT

P=3DrkT

then r=3Dn/V or the number of molecules per unit volume.


=A0=A0=A0=A0=A0=A0It is quite easy to show that the velocity of =
sound in
a gas is given by c =3D Sqrt(gamma x P / rho), where gamma is the ratio
of the specific heats, P is the pressure and rho is the density =3D mass
/ volume V. Hence PV ---> RT and you don't have to bother with P and V.



Yes, the sound speed is Sqrt(dP/drho) and under the assumptions of
adiabatic fluctuations you do get this expression. You are correct
about the sound speed, and after going through some simple
derivations, I see that the growth in amplitude with height is not due
to slowing down of waves, but rather is about the density only. To
make this more explicit (and hopefully not so convoluted), I'll
explain this briefly below:


The sound wave equation for a wave going upward (z) can be written in
terms of density (rho) as:

d[ln(rho)]/dt=3D-c*d[ln(rho)]/dz

and the solution is just the equation ln(rho)=3Df(z-c*t), where f(z) is
the function describing the wave's profile that depends only on the
initial perturbations at t=3D0.

You find the true origin of the amplitude increase with height by
separating the density into a hydrostatic part rho' (a function of z
only) times a factor measuring the fluctuation, that is:

rho=3D(q+1)*rho'

where q+1 is the fluctuation factor. For example, if q=3D0.10 then this
represents a ten percent increase in density relative to the
hydrostatic density while q=3D0 represents no density fluctuation. You
now get, since the time dependence of rho' goes away:

d[ln(q+1)]/dt=3D-c*d[ln(q+1)]/dz-c*d[ln(rho')]/dz

Since rho' decreases exponentially with height as
rho'(z)=3Drho'(z=3D0)*exp(-z/H), where H is the scale height, then the
wave equation says that:

d[ln(q+1)]/dt=3D-c*d[ln(q+1)]/dz+(c/H),

So looking just at the first term on the right side, it represents a
wave traveling upward with no change in fluctuation amplitude,

d[ln(q+1)]/dt=3D-c*d[ln(q+1)]/dz,

with a solution,

ln(q+1)=3Df(z-c*t) + a constant.

while the second term represents a source that increases ln(q+1) by an
amount (c/H) per unit time,

d[ln(q+1)]/dt=3D(c/H),

which has the solution,

ln(q+1) =3D t*c/H + a constant

This second term is the origin of the growing amplitudes, it is not a
slowing down of waves that causes this. The surprising thing is the
exponential increase in fluctuation q with time. If we take H=3D9000
meters, as a typical parameter, and by noting that c*t is ~ the
distance the wave has traveled,

ln(q+1) =3D z/H + ln[q(z=3D0)+1],

or,

q+1 =3D [q(z=3D0)+1]*exp(z/H)

So at the scale height (~9 km), the fluctuation factor q+1 has
increased by ~2.7 (density due to wave fluctuation is 2.7 times more
or less than its static value), at two scale heights it increases by
~7.3 , at three it is ~19.7, four ~53.1, etc.. Of course, when you get
up to the ionosphere you have gone about 10 scale heights so the
fluctuation factor has increased by ~22,000 or more. It sounds
reasonable then, that this is how you get 100 m displacements from
small EQ or tsunami motions at the surface that aren't very large at
all.


So the physics still holds up, all this stuff above simply means that
a decreasing mass (density) means that for the same energy in a
traveling wave that the amplitude must increase...simply due to
inertia and the the exponential decrease of density.


I recently saw an interesting related story (brief blurb on cnn) that
satellites are trying to measure the hissing sound produced by spray
paint cans in LA to prevent graffiti...don't know how well this will
work.


Cheers!

John