Download Acoustic wave equation

Acoustic wave equation
In physics, the acoustic wave equation governs the propagation of acoustic waves through a material
medium. The form of the equation is a second order partial differential equation. The equation
describes the evolution of acoustic pressure p or particle velocity u as a function of position r and
time t. A simplified form of the equation describes acoustic waves in only one spatial dimension,
while a more general form describes waves in three dimensions.
In one dimension
Equation
Feynman derives the wave equation that describes the behaviour of sound in matter in one
dimension (position ) as:
𝜕2 𝑝
𝜕𝑥 2
−
1 𝜕2
𝑐 2 𝜕𝑡 2
=0
Where p is the acoustic pressure (the local deviation from the ambient pressure), and where c is the
speed of sound.
Solution
Provided that the speed c is a constant, not dependent on frequency (the dispersionless case), then
the most general solution is
𝑝 = 𝑓(𝑐𝑡 − 𝑥 ) + 𝑔(𝑐𝑡 + 𝑥)
Where f and g are any two twice-differentiable functions. This may be pictured as the
superposition of two waveforms of arbitrary profile, one (f) travelling up the x-axis and the other (g)
down the x-axis at the speed c. The particular case of a sinusoidal wave travelling in one direction is
obtained by choosing either f or g to be a sinusoid, and the other to be zero, giving
𝑝 = 𝑝0 sin(𝜔 ∓ 𝑘𝑥)
Where ω is the angular frequency of the wave and k is its wave number, related to the wavelength
by the equation
𝑘=
2𝜋
𝜆
Wavelength of a sine wave, λ, can be measured between any two consecutive points with the same
phase, such as between adjacent crests, or troughs, or adjacent zero crossings with the same
direction of transit, as shown.
Derivation
The wave equation can be developed from the linearized one-dimensional continuity equation, the
linearized one-dimensional force equation and the equation of state.
The equation of state (ideal gas law)
𝑝𝑉 = 𝑛𝑅𝑇
In an adiabatic process, pressure p as a function of density ρ can be linearized to
𝑝 = 𝐶𝑝
where C is the specific heat for constant pressure. Breaking the pressure and density into their mean
and total components and noting that :
𝐶=
𝜕𝑝
𝜕ρ
𝜕𝑃
𝑝 − 𝑝0 = ( ) (ρ − ρ0 )
𝜕ρ
The adiabatic bulk modulus (współczynnik sprężystości objętościowej) for a fluid is defined as
𝜕𝑃
𝐵 = 𝜌0 ( )
𝜕ρ 𝑎𝑑𝑖𝑎𝑏𝑎𝑡𝑖𝑐
which gives the result
𝜌 − 𝜌0
𝑝 − 𝑝𝑜 = 𝐵
𝜌0
Condensation, s, is defined as the change in density for a given ambient fluid density.
𝜌 − 𝜌0
𝑠=
𝜌0
The linearized equation of state becomes
𝑃 = 𝐵𝑠
where P is the acoustic pressure (𝑝 − 𝑝0 ).
The continuity equation (conservation of mass) in one dimension is
𝜕𝜌 𝜕
(𝜌𝑢) = 0
+
𝜕𝑡 𝜕𝑥
Again the equation must be linearized and the variables split into mean and variable components.
𝜕
𝜕
(𝜌 + 𝜌0 𝑠) +
(𝜌 𝑢 + 𝜌0 𝑠𝑢) = 0
𝜕𝑡 0
𝜕𝑥 0
Rearranging and noting that ambient density does not change with time or position and that the
condensation multiplied by the velocity is a very small number:
𝜕𝑠 𝜕
+ 𝑢=0
𝜕𝑡 𝜕𝑥
Euler's Force equation (conservation of momentum) is the last needed component. In one
dimension the equation is:
𝐷𝑢 𝜕𝑃
𝜌
+
=0
𝐷𝑡 𝜕𝑥
Where
𝑫
represents the convective, substantial or material derivative (operator Stokesa), which is
𝑫𝒕
the derivative at a point moving with medium rather than at a fixed point.
Linearizing the variables:
𝜕
∂
∂
(ρ0 + ρ0 s) ( + u ) u + (p0 + P) = 0
𝜕𝑡
∂x
∂x
Rearranging and neglecting small terms, the resultant equation becomes:
𝜕𝑢 𝜕𝑝
ρ0
+
=0
𝜕𝑡 𝜕𝑥
Taking the time derivative of the continuity equation and the spatial derivative of the force equation
results in:
𝜕2𝑠 𝜕2𝑢
+
=0
𝜕𝑡 2 𝜕𝑥𝜕𝑡
𝜕2𝑢 𝜕2𝑃
𝜌0
+
=0
𝜕𝑥𝜕𝑡 𝜕𝑥 2
Multiplying the first by 𝛒𝟎 , subtracting the two, and substituting the linearized equation of state,
ρ0 𝜕 2 𝑃 𝜕 2 𝑃
−
+ 2=0
2
𝐵 𝜕𝑡
𝜕𝑥
The final result is
𝜕2𝑃 1 𝜕2𝑃
− 2 2 =0
2
𝜕𝑥
𝑐 𝜕𝑡
Where 𝒄
But!
=√
𝑩
𝛒𝟎
is the speed of propagation.
In three dimensions
Equation
Feynman derives the wave equation that describes the behaviour of sound in matter in three
dimensions as:
1 ∂2 P
∇ P− 2 2 =0
c ∂t
2
Where 𝛁 𝟐 is the Laplace operator,
(The Laplace operator is a second order differential operator in the n-dimensional Euclidean space,
defined as the divergence (∇·) of the gradient (∇ƒ). Thus if ƒ is a twice-differentiable real-valued
function, then the Laplacian of ƒ is defined by
∆𝑓 = 𝛻 2 𝑓 = 𝛻 ∙ 𝛻𝑓
Equivalently, the Laplacian of ƒ is the sum of all the unmixed second partial derivatives in the
Cartesian coordinates 𝑥𝑖 :
∆𝑓 =
𝜕2 𝑓
𝑛
∑𝑖=1 2 )
𝜕𝑥
𝑖
P is the acoustic pressure (the local deviation from the ambient pressure), and where c is the speed
of sound.
Solution
The following solutions are obtained by separation of variables in different coordinate systems. They
are phasor solutions (wykres wektorowy), that is they have an implicit time-dependence factor of 𝒆𝒊𝝎𝒕
where 𝝎 = 𝟐𝝅𝒇 is the angular frequency. The explicit time dependence is given by
𝑝(𝑟, 𝑡, 𝑘) = 𝑅𝑒𝑎𝑙[𝑝(𝑟, 𝑘)𝑒 𝑖𝜔𝑡 ]
Here 𝒌 = 𝝎/𝒄 is the wave number.
Cartesian coordinates
𝑝(𝑟, 𝑘) = 𝐴𝑒 ±𝑖𝑘𝑟
Spherical coordinates
𝐴 ±𝑖𝑘𝑟
𝑝(𝑟, 𝑘) = 𝑒
𝑟
Depending on the chosen Fourier convention, one of these represents an outward travelling wave
and the other an unphysical inward travelling wave. The inward travelling solution wave is only
unphysical because of the singularity that occurs at r=0; inward travelling waves do exist.