Добавил:
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:
акустика / lin_h_bengisu_t_mourelatos_zp_lecture_notes_on_acoustics_and.pdf
Скачиваний:
154
Добавлен:
04.05.2023
Размер:
12.68 Mб
Скачать

2.2 Equation of Continuity

 

 

31

The governing equation of the cubic can be expressed as:

 

 

p þ

p

 

x y z þ ðpÞ y z ¼ ρ0 x y z

 

 

 

 

v f

x

 

t

It can be simplied as:

 

x y z ¼ ρ0 x y z t v f

 

 

 

x

 

 

 

 

p

 

 

When the above equation is compared to Newtons law of motion, the left-hand

side of the equation is the force (

p

x

y z) due to the pressure difference, and the

 

 

 

 

x

 

 

 

 

 

right-hand side of the equation is the mass (ρ0

x y z) multiplied by the accelera-

tion (

v f ).

 

 

 

 

 

 

 

 

 

 

 

 

t

 

 

 

 

 

Simplifying the equation above

by

eliminating

x y z yields the

one-dimensional Eulers force equation in Cartesian coordinates:

 

 

 

p

¼ ρ0

v f

 

 

 

 

 

 

 

 

 

 

x

t

 

The one-dimensional Eulers force equation above can be extended to threedimensional Eulers force as:

pðx, y, z, tÞ ¼ ρ0

!

t

v f ðx, y, z, tÞ

where both p(x, y, z, t) and

!

 

 

 

v f ðx, y, z, tÞ are vectors.

t

2.2Equation of Continuity

The equation of continuity shows the relationship between molecular ow 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

ρ

ex

 

 

 

 

b

 

 

b

 

b

 

 

!

where v f is the ow 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:

32

 

 

 

 

 

 

2 Derivation of Acoustic Wave Equation

t

ρ0

¼

 

 

 

 

 

 

ρ

ρ0

 

 

 

!v f

 

 

 

The gradient in Cartesian

 

 

 

 

 

 

 

coordinates is a vector with bases e , e

, and e

:

 

 

 

 

bx by

bz

 

 

¼ bex

 

þ bey

 

 

þ bez

 

 

 

 

x

y

z

 

 

 

For a one-dimensional plane wave, the equation of continuity in Cartesian

coordinate can be simplied as:

 

 

¼ x

 

t

ρ0

 

 

ρ

ρ0

v f

 

 

 

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:= :

Mass passing through an unit area per unit time

̂

+

: Average Density

The above gure illustrates the mass coming in and out of a cubic unit. The mass ow rate,Qρ, is dened 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 ow inminus mass ow out, as shown in the gure below:

2.2 Equation of Continuity

33

∆ ∆ ∆

+

 

∆ ∆ ∆ ∆

 

 

 

 

Increase

Mass in Mass out

of mass

Mass flow rate

 

 

 

 

Mass passing through

: Average Density

an unit area per unit time

 

The equation of continuity can be concluded as:

∂ρ

 

 

 

Q

 

t x y z ¼ Qρ

y z

t Qρ þ

ρ

x y z t

t

x

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 nite time (

t).

 

 

 

 

 

 

 

 

 

 

t

 

 

 

 

 

 

y z t) minus the

 

 

The right-hand side of the equation is the mass ow in (Qρ

mass ow out ( Qρ þ

Qρ

 

x y z

t). The above equation can be simplied to:

x

 

 

 

∂ρ

 

 

 

Q

 

 

 

 

 

t

x y z ¼

ρ

 

x y z

t

 

 

 

t

x

and:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂ρ

¼

Qρ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

t

x

 

 

 

The mass ow Qρ is replaced with ρovf to get:

 

 

 

 

 

 

 

 

t ¼ x ρov f

 

 

 

 

 

 

 

 

∂ρ

 

 

 

 

 

This is the one-dimensional equation of continuity in Cartesian coordinates.