2.2 Equation of Continuity |
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The governing equation of the cubic can be expressed as: |
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p þ |
∂p |
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x y z þ ðpÞ y z ¼ ρ0 x y z ∙ |
∂ |
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∙ |
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v f |
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∂x |
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∂t |
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It can be simplified as: |
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x y z ¼ ρ0 x y z ∙ ∂t v f |
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∂x |
∙ |
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∂p |
∂ |
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When the above equation is compared to Newton’s law of motion, the left-hand
side of the equation is the force ( |
∂p |
∙ |
x |
y z) due to the pressure difference, and the |
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∂x |
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right-hand side of the equation is the mass (ρ0 |
x y z) multiplied by the accelera- |
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tion ( |
∂ |
v f ). |
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∂t |
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Simplifying the equation above |
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eliminating |
x y z yields the |
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one-dimensional Euler’s force equation in Cartesian coordinates: |
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∂p |
¼ ρ0 |
∂ |
v f |
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∂x |
∂t |
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The one-dimensional Euler’s force equation above can be extended to threedimensional Euler’s force as:
pðx, y, z, tÞ ¼ ρ0 |
∂ ! |
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∂t |
v f ðx, y, z, tÞ |
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where both p(x, y, z, t) and |
∂ |
! |
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v f ðx, y, z, tÞ are vectors. |
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∂t |
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2.2Equation of Continuity
The equation of continuity shows the relationship between molecular flow velocity and molecular density. The equation of continuity in Cartesian coordinates is:
∂t |
ρ0 |
ρ0 |
¼ |
∂x þ |
ey |
∂y þ |
ez |
∂z |
∙ !v f |
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∂ |
ρ |
ex |
∂ |
∂ |
∂ |
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b |
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b |
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b |
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!
where v f is the flow velocity of air molecules; variables x, y, and z are space coordinates; t is time; ρ0 is the averaged molecular density; and ρ is the instantaneous molecular density.
The equation of continuity in vector format is independent of coordinate systems. The equation of continuity in vector format is:
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2 Derivation of Acoustic Wave Equation |
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∂t |
ρ0 |
¼ |
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∂ |
ρ |
ρ0 |
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∙ !v f |
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The gradient in Cartesian |
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coordinates is a vector with bases e , e |
, and e |
: |
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∂ |
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∂ |
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∂ |
bx by |
bz |
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¼ bex |
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þ bey |
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þ bez |
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∂x |
∂y |
∂z |
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For a one-dimensional plane wave, the equation of continuity in Cartesian
coordinate can be simplified as: |
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¼ ∂x |
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∂t |
ρ0 |
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∂ |
ρ |
ρ0 |
∂ v f |
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The equation of continuity can be derived from the law of conservation of mass. To simplify the derivation of the equation of continuity, we will use the one-dimensional plane wave in Cartesian coordinate as an example:
Mass flow rate:
= 
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Mass passing through an unit area per unit time |
̂ |
+
∆ |
: Average Density |
The above figure illustrates the mass coming in and out of a cubic unit. The mass flow rate,Qρ, is defined as the mass passing through a unit area per unit time. So, the amount of mass passing through a small area, say y z, is equal to Qρ y z t.
According to the law of conservation of mass, the increase of mass in the cubic unit is equal to the “mass flow in” minus “mass flow out”, as shown in the figure below:
2.2 Equation of Continuity |
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∆ |
∆ ∆ ∆ |
+ |
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∆ ∆ ∆ ∆ |
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Increase
Mass in 
Mass out
of mass
Mass flow rate |
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Mass passing through |
: Average Density |
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an unit area per unit time |
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The equation of continuity can be concluded as:
∂ρ |
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∂Q |
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∙ t ∙ x y z ¼ Qρ ∙ |
y z ∙ |
t Qρ þ |
ρ |
x ∙ y z ∙ t |
∂t |
∂x |
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We will compare the above equation to the law of conservation of mass.
The left-hand side of the equation is the increase of mass in this cubic unit
( |
∂ρ |
∙ t ∙ x y z) after a finite time ( |
t). |
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∂t |
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y z ∙ t) minus the |
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The right-hand side of the equation is the mass flow in (Qρ ∙ |
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mass flow out ( Qρ þ |
∂Qρ |
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x ∙ y z ∙ |
t). The above equation can be simplified to: |
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∂x |
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∂ρ |
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∂Q |
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∙ t ∙ |
x y z ¼ |
ρ |
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x ∙ y z ∙ |
t |
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∂t |
∂x |
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and: |
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∂ρ |
¼ |
∂Qρ |
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∂t |
∂x |
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The mass flow Qρ is replaced with ρovf to get: |
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∂t ¼ ∂x ρov f |
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∂ρ |
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∂ |
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This is the one-dimensional equation of continuity in Cartesian coordinates.