理想媒质中的声波方程

ρ = [ ( ρu x ) + ( ρu y ) + ( ρu z )] t x y z
t → 0
ρ ρ lim = t → 0 t t
We obtain the equation of continuity
ρ = [ ( ρu x ) + ( ρu y ) + ( ρu z )] t x y z
= i+ j+ k x y z
is gradient operator
p p p p = i + j+ k x y z
2. The equation of continuity
restatement of the law of the conservation of matter To relate the motion of the fluid to its compression or dilatation, we need a functional relationship between the particle velocity u and the instantaneous density p .
ρ ρ0
We obtain
u x ( ρu x ) ( ρ 0u x ) = ρ 0 x x x
Similar expressions gibe the net influx for the y and z directions,
u y ( ρu y ) ρ 0 y y
u z ( ρu z ) ρ 0 z z
pressure and velocity, velocity, Consider a fluid element V = xyz
y
C
D
y
G
H
F1
B A
z
F2
F x E
x
z
When the sound waves pass, the pressure is
P0 + p ( x, y , z , t )
Consider a small rectangularrectangularparallelepiped volume element dV=dxdydz which is fixed in space and through which elements of the fluid travel. The net rate with which mass flows into the volume through its surface must equal the rate with the mass within the volume increases.
dP 1 d 2P P＝P(ρ0 ) + ( )S,ρ0 ( ρ ρ0 ) + ( 2 )Ｓ,ρ０ ( ρ ρ0 ) 2 + dρ 2! d ρ
Where S is adiabatic process, the partial derivatives are constants determined for adiabatic compression and expansion of the fluid about its equilibrium density.
THREE BASIC EQUATIONS 理想媒质中的三个基本方程
1. The equation of motion
1. the equation of motion ( Euler's equation) First, we write the relation between sound
ρ = ρ 0 u t
u x u y u z ρ = ρ0 ( + + ) t x y z
Where
i
is the divergence operator
ax a y az i a = ( + + ) x y z
ρ = ρ 0iu t
3. THale Waihona Puke Baidue equation of state
We need one more relation in order to
P = P0 + p
ρ = ρ0 + ρ '
dp 2 dρ ' =c dt dt
This is the equation of state, gives the relationship between the pressure fluctuation and the change in density.
du x u x dt t du x u x u x dx u x dy u x dz = + + + dt t x dt y dt z dt
m = ρV = ρx y z
According to Newton’s second law F=ma, Newton’
the acceleration of small volume in x direction will be
Speed of sound in fluids
We get a thermodynamic expression for the speed of sound
dP c= dρ s, ρ0
Where the partial derivative is evaluated at
equilibrium conditions of pressure and density. For a sound wave propagates through a perfect gas, the speed of sound is :
If the fluctuations are small, only the lowest order term in ρ ρ0 Need be retained .This gives a linear relationship between the pressure fluctuation and the change in density
y
C
y
H
G
D
F1
F2
B A
z z E
F x
x
So the force on area ABCD will be
F1 = ( P0 + p )yz
P0 + p is the force of per unit area
The force on area EFGH will be
F2 = ( P0 + p + P )yz
Similar expressions give the net influx for the y and z directions,
( ρu y ) xyzt y
( ρu z ) xyzt z
So that the total influx must be
m = ρxyz = [ ( ρ u x ) + ( ρ u y ) + ( ρ u z )]xyz t x y z
The net force experienced by the volume dV in the x direction is
Fx = F1 F2 = Pyz
For small amplitude , we can neglect the second order variable terms,
When
u x Pyz = ρxyz t
u x P =ρ x t
x → 0
P P lim = x → 0 x x
For small amplitude u x P = ρ0 t x
u y P = ρ0 y t
ρ ρ0
Similarly ,in the direction of y and z, we can obtain
PV = P0V0
γ γ
Here r is the ratio of specific heat at constant pressure to that at constant volume. Air , for instance ,has r =1.4 at normal conditions
For idea gas,
P＝ P ( ρ )
Generally the adiabatic equation of state is complicated. In these cases it is preferable to determine experimentally the isentropic (等 熵)relationship between pressure and density fluctuations. We write a Taylor’s expansion Taylor’
[ ρu x + ( ρu x ) x ]yzt x
That the net influx of mass into this spatially fixed volume,resulting from flow in the x direction , is
x ( ρ u x ) x y z t
Note that the equation is nonlinear;
the right term involves the product of particle velocity and instantaneous density, both of which are acoustic variables. Consider a small amplitude sound wave, if we write p=p0(1+s) .Use the fact that p0 is a constant in both space and time, and assume that s is very small,
dP P＝P(ρ0 ) + ( )S,ρ0 (ρ ρ0 ) dρ
dP dP = ( ) S , ρ0 d ρ dρ
We suppose
dP c = ( ) S , ρ0 dρ
2
In the case of gases at sufficiently low
density, their behavior will be well approximated by the ideal gas law. An adiabatic process in an ideal gas is governed by
determine the three functions P ,ρ, and u. It is provided by the condition that we have an adiabatic(绝热的) process, ( there is adiabatic(绝热的) insignificant exchange of thermal energy from one particle of fluid to another). Under these conditions, it is conveniently expressed by saying that the pressure p is uniquely determined as a function of the density ρ ( rather than a depending separately on both ρ and T)
PV = P0V0
γ γ
ρ0V0 = ρV
γ
γ
ρ ρ0 + d ρ γ P0 P = P0 = d ρ + P0 ≈ P0 + ρ0 ρ0 ρ0
In the sound field of small amplitude
dρ ＜＜ 1 ρ0
γ P0 p = dP = P P0 ≈ d ρ = c2d ρ ρ0
u z P = ρ0 t z
Now let the motion be three –dimensional,
so write
u P = ρ0 t
P ≡ p
Since P0 is a constant, and obtain
du p = ρ 0 dt This is the linear inviscid equation of motion, valid for acoustic processes of small amplitude