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Sound pressure
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The movement of the molecules of the medium generate variations in the density and pressure in the medium, which can be detected.
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Sound Pressure
Description ![](/static/icons/audio20c.png)
As sound propagates, it causes displacement of molecules at the boundary of the system, leading to impacts against the wall. These impacts transfer momentum to the wall, resulting in a force. Because the force is generated by a high number of particles, its effect depends on the surface area of the system, giving rise to a pressure.
It's important to understand that sound pressure is not the same as ambient pressure. In air, the latter is on the order of $10^5,Pa$, whereas sound pressure is typically much lower than $1,Pa$.
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Formation of pressure
Concept ![](/static/icons/audio20c.png)
If we displace a cube's face, we generate an increase or decrease in concentration, which leads to a decrease or increase in collisions of molecules with the face of the volume:
Since pressure is the transfer of momentum due to collisions of molecules with the wall, the change in volume leads to an increase or decrease in pressure.
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Model
Top ![](/static/icons/audio20c.png)
Calculations
Variables
Parameters
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Calculations
Calculations
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Equation
$ \Delta V = S \lambda $
DV = S * lambda
$ F =\displaystyle\frac{ d p }{ d t }$
F = dp / dt
$ L = 20 \log_{10}\left(\displaystyle\frac{ p }{ p_{ref} }\right)$
L = 20* log10( p / p_ref )
$ p \equiv\displaystyle\frac{ F }{ S }$
p = F / S
$ p = \rho c u $
p = rho * c * u
$ Z =\displaystyle\frac{ p }{ u }$
Z = p / u
$ Z = \rho c $
Z = rho * c
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Definition of pressure
Equation ![](/static/icons/audio20c.png)
The water column pressure ($p$) is calculated from the column force ($F$) and the column Section ($S$) as follows:
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Force exerted by molecules
Equation ![](/static/icons/audio20c.png)
According to Newton, force can be expressed as the change in momento de la partícula ($p$). This momentum is generated by the rebound of particles, which transfer momentum to the wall. Since momentum is conserved and the particle's momentum changes from $p_{particle}$ to $-p_{particle}$ upon rebound, by momentum conservation we then have:
$p_{particle} = p_{wall} - p_{particle}$
which implies
$p_{wall} = 2p_{particle}$
Thus, the change in the momento de la partícula ($p$) on the wall is the time ($t$), generating force exerted by the Molecules ($F$), which is
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Volume with molecules
Equation ![](/static/icons/audio20c.png)
When a sound wave travels through ($$), it expands and contracts over a distance on the order of ($$), resulting in a volume variation depending on the section or Area ($S$) perpendicular to the direction of propagation.
Therefore, the volume variation is equal to:
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Variation of the moment by molecules
Equation ![](/static/icons/audio20c.png)
The sound pressure ($p$) can be understood as the momentum density calculated from the mean density ($\rho$) and the molecule speed ($u$), which is then multiplied by the speed of sound ($c$) via
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The variation in momentum $dp$ is associated with the mass of the molecules $m$ and the velocity of sound $u$ through:
$dp = 2mu \approx mu$
Thus, in a time interval equal to the period $dt \approx T$, we have:
$F=\displaystyle\frac{dp}{dt}=\displaystyle\frac{mu}{T}$
Therefore, the sound pressure ($p$) can be calculated using the pressure
the speed of sound ($c$) is
$ c = \displaystyle\frac{ \lambda }{ T }$ |
and the volume with molecules ($\Delta V$) which varies
$ \Delta V = S \lambda $ |
as follows:
$p=\displaystyle\frac{1}{S} \displaystyle\frac{dp}{dt}=\displaystyle\frac{1}{S}\displaystyle\frac{mu}{T}=\displaystyle\frac{muc}{ScT}=\displaystyle\frac{muc}{S\lambda}=\displaystyle\frac{muc}{\Delta V}=\rho u c$
In the last term, both numerator and denominator are multiplied by $c$. The expression in the denominator represents the volume of gas displaced by the sound in $T$, so we can replace the mass divided by this volume with density, yielding:
$ p = \rho c u $ |
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Noise level as function of the sound pressure, air
Equation ![](/static/icons/audio20c.png)
The noise level, air ($L$) encompasses a wide range of the sound pressure ($p$), making it useful to define a scale that mitigates this difficulty. To do so, we can work with the logarithm of the pressure normalized by a value corresponding to zero on this scale. If we take the minimum pressure that a person can detect, defined as the reference pressure ($p_{ref}$), we can define a scale using:
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which starts at 0 for the audible range. In the case of air, the reference pressure ($p_{ref}$) is $20 \mu Pa$.
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Acoustic impedance
Equation ![](/static/icons/audio20c.png)
The concept of impedance ($Z$) provides a measure of the system's resistance to transmit the sound wave. It considers a pressure acting and establishes a measure in which the exposed medium is displaced. In this way, the sound pressure ($p$) is compared to the molecule speed ($u$).
Therefore, impedance ($Z$) is defined as:
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Impedance in waves
Equation ![](/static/icons/audio20c.png)
To calculate impedance ($Z$) from the mean density ($\rho$) and the speed of sound ($c$), the formula used is:
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Since impedance ($Z$) is calculated from the sound pressure ($p$) and the molecule speed ($u$) using
$ Z =\displaystyle\frac{ p }{ u }$ |
along with the expression for the sound pressure ($p$) in terms of the mean density ($\rho$) and the speed of sound ($c$),
$ p = \rho c u $ |
we obtain
$ Z = \rho c $ |
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