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Sound Intensity
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Sound intensity is the energy by area and time that helps to understand how the sound wave is distributed spatially.
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Equations
$ e =\displaystyle\frac{1}{2} \rho u ^2$
e = rho * u ^2/2
$ I = c e $
I = c * e
$ I =\displaystyle\frac{ P }{ S }$
I = P / S
$ I =\displaystyle\frac{ p ^2}{2 \rho c }$
I = p ^2/(2* rho * c )
$ I =\displaystyle\frac{1}{2} \rho c u ^2$
I = rho * c * u ^2/2
$ I_{ref} =\displaystyle\frac{ p_{ref} ^2}{2 \rho c }$
I_ref = p_ref ^2/(2* rho * c )
$ L = 10 log_{10}\left(\displaystyle\frac{ I }{ I_{ref} }\right)$
L = 10* log10( I / I_ref )
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Acoustic intensity
Equation
Intensity is the power (energy per unit time, in joules per second or watts) per area emanating from a source.
Therefore, it is defined as the sound Intensity ($I$), the ratio between the sound Power ($P$) and the section of Volume DV ($S$), so it is:
| $ I =\displaystyle\frac{ P }{ S }$ |
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Intensity based on the power density
Equation
Si se toma la energía
$W=\displaystyle\frac{E}{T}$
Si por otro lado con la variación del volumen es
| $ \Delta V = S \lambda $ |
y con section of Volume DV $m^2$, sound Intensity $W/m^2$ and sound Power $W$ la intensidad sonora es
| $ I =\displaystyle\frac{ P }{ S }$ |
por lo que
$I=\displaystyle\frac{W}{S}=\displaystyle\frac{E}{ST}=\displaystyle\frac{cE}{ScT}=\displaystyle\frac{cE}{V}$
osea con section of Volume DV $m^2$, sound Intensity $W/m^2$ and sound Power $W$ es
| $ I = c e $ |
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Sound energy density
Equation
The the energy density ($e$) is obtained from the mean density ($\rho$) and the molecule speed ($u$) as follows:
| $ e =\displaystyle\frac{1}{2} \rho u ^2$ |
The energy that a sound wave contributes to the medium in which sound propagates corresponds to the kinetic energy of the particles. With the molecule speed ($u$) and the mass of a volume of the medium ($m$) The wave energy ($E$), it equals the kinetic energy:
$E=\displaystyle\frac{1}{2}mu^2$
the energy density ($e$) is obtained by dividing the wave energy ($E$) by the volume with molecules ($\Delta V$), giving:
$e=\displaystyle\frac{E}{\Delta V}$
Introducing the mean density ($\rho$) as:
$\rho=\displaystyle\frac{m}{\Delta V}$
yields the energy density:
| $ e =\displaystyle\frac{1}{2} \rho u ^2$ |
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Intensity versus the molecule speed
Equation
Como la densidad de la energía cinética es con energy density $J/m^3$, mean density $kg/m^3$ and molecule speed $m/s$
| $ e =\displaystyle\frac{1}{2} \rho u ^2$ |
se tiene que con energy density $J/m^3$, sound Intensity $W/m^2$ and speed of sound $m/s$
| $ I = c e $ |
que la intensidad es con energy density $J/m^3$, sound Intensity $W/m^2$ and speed of sound $m/s$
| $ I =\displaystyle\frac{1}{2} \rho c u ^2$ |
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Intensity depending on the sound pressure
Equation
The sound Intensity ($I$) can be calculated from the mean density ($\rho$), the sound pressure ($p$) The molar concentration ($c$) with
| $ I =\displaystyle\frac{ p ^2}{2 \rho c }$ |
The sound Intensity ($I$) can be calculated from the mean density ($\rho$), the molecule speed ($u$), and the molar concentration ($c$) using
| $ I =\displaystyle\frac{1}{2} \rho c u ^2$ |
and since the sound pressure ($p$) is defined as
| $ p = \rho c u $ |
it follows that the sound Intensity ($I$) can be expressed in terms of the sound pressure ($p$) by
| $ I =\displaystyle\frac{ p ^2}{2 \rho c }$ |
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Noise level as function of the sound intensity
Equation
Just like in other human sensory systems, our hearing is capable of detecting pressure variations over a wide range $(10^{-5}-10^2 Pa)$. However, when we perceive a signal doubling, it doesn't correspond to double the pressure or sound intensity, but rather the square of these magnitudes. In other words, our signal detection capacity operates on a logarithmic and nonlinear scale.
Hence, the noise level, air ($L$) is indicated not in the sound Intensity ($I$) or the reference intensity, air ($I_{ref}$), but in the base ten logarithm of these magnitudes. Particularly, we take the lowest sound intensity we can perceive, the reference intensity, air ($I_{ref}$)
, and use it as a reference. The new scale is defined with as follows:
| $ L = 10 log_{10}\left(\displaystyle\frac{ I }{ I_{ref} }\right)$ |
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Intensity reference values
Equation
The sound pressure level that we can detect with our ear, denoted as the reference pressure, water ($p_{ref}$), is $2 \times 10^{-5} , Pa$.
Since the sound Intensity ($I$) is associated with the sound pressure ($p$), the mean density ($\rho$), and the speed of sound ($c$), and is equal to
| $ I =\displaystyle\frac{ p ^2}{2 \rho c }$ |
we can calculate a value for the reference intensity, air ($I_{ref}$) based on the value of the reference pressure, water ($p_{ref}$):
| $ I_{ref} =\displaystyle\frac{ p_{ref} ^2}{2 \rho c }$ |
This is achieved with a density of $1.27 , kg/m^3$ and a sound speed of $331 , m/s$, equivalent to $9.5 \times 10^{-13} , W/m^2$.
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