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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.

>Model

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Iframe

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Description

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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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Concept

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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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Parameters

$F$

F

Force of the medium

N

$Z$

Z

Impedance

kg/m^2s

$p$

p

Momento de la partícula

kg m/s

$p_{ref}$

p_ref

Reference pressure

Pa

Variables

$\rho$

rho

Mean density

kg/m^3

$u$

u

Molecule speed

m/s

$L$

L

Noise level, air

dB

$p$

p

Pressure

Pa

$S$

S

Section or Area

m^2

$p$

p

Sound pressure

Pa

$c$

c

Speed of sound

m/s

$t$

t

Time

s

$\Delta V$

DV

Volume with molecules

m^3

$\lambda$

lambda

Wavelength of Sound

m

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

ID:(15453, 0)

Equation

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The water column pressure ($p$) is calculated from the column force ($F$) and the column Section ($S$) as follows:

$p \equiv\displaystyle\frac{ F }{ S }$

Conditions

Variables

$F$

Force of the medium

$N$

5815

$p$

Pressure

$Pa$

5224

$S$

Section or Area

$m^2$

5405

Relation

ID:(4342, 0)

Equation

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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

$F =\displaystyle\frac{ d p }{ d t }$

Conditions

Variables

$F$

Force of the medium

$N$

5815

$p$

Momento de la partícula

$kg m/s$

9047

$t$

Time

$s$

5264

Relation

ID:(3390, 0)

Equation

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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:

$\Delta V = S \lambda$

Conditions

Variables

$S$

Section or Area

$m^2$

5405

$\Delta V$

Volume with molecules

$m^3$

5080

$\lambda$

Wavelength of Sound

$m$

5079

Relation

ID:(3398, 0)

Equation

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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

$p = \rho c u$

Conditions

Variables

$\rho$

Mean density

$kg/m^3$

5088

$u$

Molecule speed

$m/s$

5072

$p$

Sound pressure

$Pa$

5084

$c$

Speed of sound

$m/s$

5073

Relation

Development

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:

$p = \rho c u$

ID:(3391, 0)

Equation

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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:

$L = 20 \log_{10}\left(\displaystyle\frac{ p }{ p_{ref} }\right)$

Conditions

Variables

$L$

Noise level, air

$dB$

5119

$p_{ref}$

Reference pressure

3.65e+10

$Pa$

5121

$p$

Sound pressure

$Pa$

5084

Relation

which starts at 0 for the audible range. In the case of air, the reference pressure ($p_{ref}$) is $20 \mu Pa$.

ID:(3407, 0)

Equation

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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:

$Z =\displaystyle\frac{ p }{ u }$

Conditions

Variables

$Z$

Impedance

$kg/m^2s$

5104

$u$

Molecule speed

$m/s$

5072

$p$

Sound pressure

$Pa$

5084

Relation

ID:(3414, 0)

Equation

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To calculate impedance ($Z$) from the mean density ($\rho$) and the speed of sound ($c$), the formula used is:

$Z = \rho c$

Conditions

Variables

$Z$

Impedance

$kg/m^2s$

5104

$\rho$

Mean density

$kg/m^3$

5088

$c$

Speed of sound

$m/s$

5073

Relation

Development

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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