Attenuation Coefficient for Sound

Thread view:

Enable EMail Alerts

Start New Thread

Thread rating:

Dr. Zaius - 01 Aug 2006 18:33 GMT

Hello everyone,

I have a question about attenuation of sound. From several sources,
I've found the following formula for the acoustic intensity of a plane
wave:
I(x) = I(0) exp [-2 alpha x]

where x is the distance in cm from the I(0) reference (which could be
the source of the sound)
alpha is called the amplitude attenuation coefficient in units of
[1/cm]

My first question is why is the factor of 2 put into the above acoustic
intensity formula? I've seen in other sources where it is instead
written:

I(x) = I(0) exp [- alpha x]

and also:

I(x) = I(0) 10 ^ [- alpha x / 10]
but in this case alpha is given in units of [dB / cm]

However, in the literature, I find most attenuation values are quoted
as 2alpha in units of [1/cm], so my second question is how do I convert
from 2alpha [1/cm] to units of [dB/cm]?

For example, for water at a frequency of 10MHz,
2alpha = 0.0479 [1/cm], which I'm told corresponds to 0.208 dB/cm
Apparently the conversion factor to go between these is a constant
value of approximately 4.3428, but where does this conversion value
come from?

I understand that there are also units of nepers, with which I see some
values quoted in the literature. I found this site:
http://www.microwaves101.com/encyclopedia/neper.cfm which explains that
conversion, so I'm fine with that. Just the above part I'm stuck with.

Thanks,

Dr. Zaius

Reply to this Message

robert bristow-johnson - 01 Aug 2006 19:33 GMT

> Hello everyone,
>
[quoted text clipped - 7 lines]
> alpha is called the amplitude attenuation coefficient in units of
> [1/cm]

> My first question is why is the factor of 2 put into the above acoustic
> intensity formula? I've seen in other sources where it is instead
> written:

i think that alpha is applied to amplitude ("voltage" or pressure)
attenuation and I(x), being intensity, is proportional to power which
is proportional to the square of amplitude. this alpha would be in
nepers/unit_length.

> I(x) = I(0) exp [- alpha x]

this alpha is twice the other alpha for the same situation.

> and also:
>
[quoted text clipped - 15 lines]
> http://www.microwaves101.com/encyclopedia/neper.cfm which explains that
> conversion, so I'm fine with that. Just the above part I'm stuck with.

nepers is the more mathematically natural unit of attenuation or gain
(as opposed to dB which is a human construct). assuming we could
communicate with them, aliens on the planet Zog would know what a neper
is. it's just like radians as the natural unit of angle (or phase
shift) rather than the arbitrarily defined degrees.

r b-j

Reply to this Message

Dr. Zaius - 01 Aug 2006 20:35 GMT

Ok, thanks for the info.

I'm guessing that in the literature, wherever they quote a value of
[1/cm], they really mean [nepers/cm].
If that is true, then this all seems to make sense now.

Also, I realized that the website I posted above actually does the
conversion wrong.
I used instead:

I_dB = 10 log (I/Io)

I_np = ln (I/Io)

I_dB / I_np = 10 / ln 10 = 4.3429444819 [dB/neper]

which works for converting from the published values of 2alpha [1/cm]
to [dB/cm]

robert bristow-johnson wote:

> > Hello everyone,
> >
[quoted text clipped - 48 lines]
>
> r b-j

Reply to this Message

robert bristow-johnson - 16 Aug 2006 02:34 GMT

i didn't realize that this was sitting around for 2 weeks (as often as
i look at this newgroup).

> Ok, thanks for the info.
>
> I'm guessing that in the literature, wherever they quote a value of
> [1/cm], they really mean [nepers/cm].
> If that is true, then this all seems to make sense now.

nepers and radians are perfectly dimensionless measure of
gain/attenuation and angle respectively. "1/cm" would mean "np/cm" for
gain/attenuation or "rad/cm" for phase shift.

> Also, I realized that the website I posted above actually does the
> conversion wrong.
> I used instead:
>
> I_dB = 10 log (I/Io)

this is correct because I and Io are both intensity which are *power*
quantities, not voltage (or pressure).

> I_np = ln (I/Io)

this is not correct. it is either

I_np = ln ( sqrt(I) / sqrt(Io) )

or simply
I_np = 1/2 * ln (I/Io)

> I_dB / I_np = 10 / ln 10 = 4.3429444819 [dB/neper]

no, the relationship is twice that.

1 neper = 8.6858889638 dB

> which works for converting from the published values of 2alpha [1/cm]
> to [dB/cm]

i think that is where your factor of 2 (which i also think is
incorrect) comes from.

r b-j

Reply to this Message

Natural Science Forum / Physics / Acoustics / August 2006

Attenuation Coefficient for Sound