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1
I n o n e d i m e n s i o n
T o g g l e I n o n e d i m e n s i o n s u b s e c t i o n
1 . 1
E q u a t i o n
1 . 2
S o l u t i o n
1 . 3
D e r i v a t i o n
2
I n t h r e e d i m e n s i o n s
T o g g l e I n t h r e e d i m e n s i o n s s u b s e c t i o n
2 . 1
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2 . 2
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2 . 2 . 1
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2 . 2 . 3
S p h e r i c a l c o o r d i n a t e s
3
S e e a l s o
4
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A c o u s t i c w a v e e q u a t i o n
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Equation for the propagation of sound waves through a medium
In physics , the acoustic wave equation is a second-order partial differential equation that governs the propagation of acoustic waves through a material medium resp. a standing wavefield . The equation describes the evolution of acoustic pressure p or particle velocity u as a function of position x and time t . A simplified (scalar) form of the equation describes acoustic waves in only one spatial dimension, while a more general form describes waves in three dimensions. Propagating waves in a pre-defined direction can also be calculated using a first order one-way wave equation .
For lossy media, more intricate models need to be applied in order to take into account frequency-dependent attenuation and phase speed. Such models include acoustic wave equations that incorporate fractional derivative terms, see also the acoustic attenuation article or the survey paper.[1]
In one dimension
[ edit ]
Equation
[ edit ]
The wave equation describing a standing wave field in one dimension (position
x
{\displaystyle x}
) is
∂
2
p
∂
x
2
−
1
c
2
∂
2
p
∂
t
2
=
0
,
{\displaystyle {\partial ^{2}p \over \partial x^{2}}-{1 \over c^{2}}{\partial ^{2}p \over \partial t^{2}}=0,}
where
p
{\displaystyle p}
is the acoustic pressure (the local deviation from the ambient pressure), and where
c
{\displaystyle c}
is the speed of sound .[2]
Solution
[ edit ]
Provided that the speed
c
{\displaystyle c}
is a constant, not dependent on frequency (the dispersionless case), then the most general solution is
p
=
f
(
c
t
−
x
)
+
g
(
c
t
+
x
)
{\displaystyle p=f(ct-x)+g(ct+x)}
where
f
{\displaystyle f}
and
g
{\displaystyle g}
are any two twice-differentiable functions. This may be pictured as the superposition of two waveforms of arbitrary profile, one (
f
{\displaystyle f}
) traveling up the x-axis and the other (
g
{\displaystyle g}
) down the x-axis at the speed
c
{\displaystyle c}
. The particular case of a sinusoidal wave traveling in one direction is obtained by choosing either
f
{\displaystyle f}
or
g
{\displaystyle g}
to be a sinusoid, and the other to be zero, giving
p
=
p
0
sin
(
ω
t
∓
k
x
)
{\displaystyle p=p_{0}\sin(\omega t\mp kx)}
.
where
ω
{\displaystyle \omega }
is the angular frequency of the wave and
k
{\displaystyle k}
is its wave number .
Derivation
[ edit ]
Derivation of the acoustic wave equation
The derivation of the wave equation involves three steps: derivation of the equation of state, the linearized one-dimensional continuity equation, and the linearized one-dimensional force equation.
The equation of state (ideal gas law )
P
V
=
n
R
T
{\displaystyle PV=nRT}
In an adiabatic process , pressure P as a function of density
ρ
{\displaystyle \rho }
can be linearized to
P
=
C
ρ
{\displaystyle P=C\rho \,}
where C is some constant. Breaking the pressure and density into their mean and total components and noting that
C
=
∂
P
∂
ρ
{\displaystyle C={\frac {\partial P}{\partial \rho }}}
:
P
−
P
0
=
(
∂
P
∂
ρ
)
(
ρ
−
ρ
0
)
{\displaystyle P-P_{0}=\left({\frac {\partial P}{\partial \rho }}\right)(\rho -\rho _{0})}
.
The adiabatic bulk modulus for a fluid is defined as
B
=
ρ
0
(
∂
P
∂
ρ
)
a
d
i
a
b
a
t
i
c
{\displaystyle B=\rho _{0}\left({\frac {\partial P}{\partial \rho }}\right)_{adiabatic}}
which gives the result
P
−
P
0
=
B
ρ
−
ρ
0
ρ
0
{\displaystyle P-P_{0}=B{\frac {\rho -\rho _{0}}{\rho _{0}}}}
.
Condensation, s , is defined as the change in density for a given ambient fluid density.
s
=
ρ
−
ρ
0
ρ
0
{\displaystyle s={\frac {\rho -\rho _{0}}{\rho _{0}}}}
The linearized equation of state becomes
p
=
B
s
{\displaystyle p=Bs\,}
where p is the acoustic pressure (
P
−
P
0
{\displaystyle P-P_{0}}
).
The continuity equation (conservation of mass) in one dimension is
∂
ρ
∂
t
+
∂
∂
x
(
ρ
u
)
=
0
{\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial }{\partial x}}(\rho u)=0}
.
Where u is the flow velocity of the fluid.
Again the equation must be linearized and the variables split into mean and variable components.
∂
∂
t
(
ρ
0
+
ρ
0
s
)
+
∂
∂
x
(
ρ
0
u
+
ρ
0
s
u
)
=
0
{\displaystyle {\frac {\partial }{\partial t}}(\rho _{0}+\rho _{0}s)+{\frac {\partial }{\partial x}}(\rho _{0}u+\rho _{0}su)=0}
Rearranging and noting that ambient density changes with neither time nor position and that the condensation multiplied by the velocity is a very small number:
∂
s
∂
t
+
∂
∂
x
u
=
0
{\displaystyle {\frac {\partial s}{\partial t}}+{\frac {\partial }{\partial x}}u=0}
Euler's Force equation (conservation of momentum) is the last needed component. In one dimension the equation is:
ρ
D
u
D
t
+
∂
P
∂
x
=
0
{\displaystyle \rho {\frac {Du}{Dt}}+{\frac {\partial P}{\partial x}}=0}
,
where
D
/
D
t
{\displaystyle D/Dt}
represents the convective, substantial or material derivative , which is the derivative at a point moving along with the medium rather than at a fixed point.
Linearizing the variables:
(
ρ
0
+
ρ
0
s
)
(
∂
∂
t
+
u
∂
∂
x
)
u
+
∂
∂
x
(
P
0
+
p
)
=
0
{\displaystyle (\rho _{0}+\rho _{0}s)\left({\frac {\partial }{\partial t}}+u{\frac {\partial }{\partial x}}\right)u+{\frac {\partial }{\partial x}}(P_{0}+p)=0}
.
Rearranging and neglecting small terms, the resultant equation becomes the linearized one-dimensional Euler Equation:
ρ
0
∂
u
∂
t
+
∂
p
∂
x
=
0
{\displaystyle \rho _{0}{\frac {\partial u}{\partial t}}+{\frac {\partial p}{\partial x}}=0}
.
Taking the time derivative of the continuity equation and the spatial derivative of the force equation results in:
∂
2
s
∂
t
2
+
∂
2
u
∂
x
∂
t
=
0
{\displaystyle {\frac {\partial ^{2}s}{\partial t^{2}}}+{\frac {\partial ^{2}u}{\partial x\partial t}}=0}
ρ
0
∂
2
u
∂
x
∂
t
+
∂
2
p
∂
x
2
=
0
{\displaystyle \rho _{0}{\frac {\partial ^{2}u}{\partial x\partial t}}+{\frac {\partial ^{2}p}{\partial x^{2}}}=0}
.
Multiplying the first by
ρ
0
{\displaystyle \rho _{0}}
, subtracting the two, and substituting the linearized equation of state,
−
ρ
0
B
∂
2
p
∂
t
2
+
∂
2
p
∂
x
2
=
0
{\displaystyle -{\frac {\rho _{0}}{B}}{\frac {\partial ^{2}p}{\partial t^{2}}}+{\frac {\partial ^{2}p}{\partial x^{2}}}=0}
.
The final result is
∂
2
p
∂
x
2
−
1
c
2
∂
2
p
∂
t
2
=
0
{\displaystyle {\partial ^{2}p \over \partial x^{2}}-{1 \over c^{2}}{\partial ^{2}p \over \partial t^{2}}=0}
where
c
=
B
ρ
0
{\displaystyle c={\sqrt {\frac {B}{\rho _{0}}}}}
is the speed of propagation.
In three dimensions
[ edit ]
Equation
[ edit ]
Feynman[3] provides a derivation of the wave equation for sound in three dimensions as
∇
2
p
−
1
c
2
∂
2
p
∂
t
2
=
0
,
{\displaystyle \nabla ^{2}p-{1 \over c^{2}}{\partial ^{2}p \over \partial t^{2}}=0,}
where
∇
2
{\displaystyle \nabla ^{2}}
is the Laplace operator ,
p
{\displaystyle p}
is the acoustic pressure (the local deviation from the ambient pressure), and
c
{\displaystyle c}
is the speed of sound .
A similar looking wave equation but for the vector field particle velocity is given by
∇
2
u
−
1
c
2
∂
2
u
∂
t
2
=
0
{\displaystyle \nabla ^{2}\mathbf {u} \;-{1 \over c^{2}}{\partial ^{2}\mathbf {u} \; \over \partial t^{2}}=0}
.
In some situations, it is more convenient to solve the wave equation for an abstract scalar field velocity potential which has the form
∇
2
Φ
−
1
c
2
∂
2
Φ
∂
t
2
=
0
{\displaystyle \nabla ^{2}\Phi -{1 \over c^{2}}{\partial ^{2}\Phi \over \partial t^{2}}=0}
and then derive the physical quantities particle velocity and acoustic pressure by the equations (or definition, in the case of particle velocity):
u
=
∇
Φ
{\displaystyle \mathbf {u} =\nabla \Phi \;}
,
p
=
−
ρ
∂
∂
t
Φ
{\displaystyle p=-\rho {\partial \over \partial t}\Phi }
.
Solution
[ edit ]
The following solutions are obtained by separation of variables in different coordinate systems. They are phasor solutions, that is they have an implicit time-dependence factor of
e
i
ω
t
{\displaystyle e^{i\omega t}}
where
ω
=
2
π
f
{\displaystyle \omega =2\pi f}
is the angular frequency . The explicit time dependence is given by
p
(
r
,
t
,
k
)
=
Real
[
p
(
r
,
k
)
e
i
ω
t
]
{\displaystyle p(r,t,k)=\operatorname {Real} \left[p(r,k)e^{i\omega t}\right]}
Here
k
=
ω
/
c
{\displaystyle k=\omega /c\ }
is the wave number .
Cartesian coordinates
[ edit ]
p
(
r
,
k
)
=
A
e
±
i
k
r
{\displaystyle p(r,k)=Ae^{\pm ikr}}
.
Cylindrical coordinates
[ edit ]
p
(
r
,
k
)
=
A
H
0
(
1
)
(
k
r
)
+
B
H
0
(
2
)
(
k
r
)
{\displaystyle p(r,k)=AH_{0}^{(1 )}(kr )+\ BH_{0}^{(2 )}(kr )}
.
where the asymptotic approximations to the Hankel functions , when
k
r
→
∞
{\displaystyle kr\rightarrow \infty }
, are
H
0
(
1
)
(
k
r
)
≃
2
π
k
r
e
i
(
k
r
−
π
/
4
)
{\displaystyle H_{0}^{(1 )}(kr )\simeq {\sqrt {\frac {2}{\pi kr}}}e^{i(kr-\pi /4)}}
H
0
(
2
)
(
k
r
)
≃
2
π
k
r
e
−
i
(
k
r
−
π
/
4
)
{\displaystyle H_{0}^{(2 )}(kr )\simeq {\sqrt {\frac {2}{\pi kr}}}e^{-i(kr-\pi /4)}}
.
Spherical coordinates
[ edit ]
p
(
r
,
k
)
=
A
r
e
±
i
k
r
{\displaystyle p(r,k)={\frac {A}{r}}e^{\pm ikr}}
.
Depending on the chosen Fourier convention, one of these represents an outward travelling wave and the other a nonphysical inward travelling wave. The inward travelling solution wave is only nonphysical because of the singularity that occurs at r=0; inward travelling waves do exist.
See also
[ edit ]
References
[ edit ]
^ S. P. Näsholm and S. Holm, "On a Fractional Zener Elastic Wave Equation," Fract. Calc. Appl. Anal. Vol. 16, No 1 (2013), pp. 26-50, DOI: 10.2478/s13540-013--0003-1 Link to e-print
^ Richard Feynman , Lectures in Physics, Volume 1, Chapter 47: Sound. The wave equation , Caltech 1963, 2006, 2013
^ Richard Feynman , Lectures in Physics, Volume 1, 1969, Addison Publishing Company, Addison
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Acoustic_wave_equation&oldid=1191462053 "
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