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A particle of the medium undergoes displacement according to the particle velocity of the sound wave traveling through the medium, while the sound wave itself moves at the speed of sound , equal to 343 m/s in air at 20 °C .
Mathematical definition
[ edit ]
Particle displacement, denoted δ , is given by[3]
δ
=
∫
t
v
d
t
{\displaystyle \mathbf {\delta } =\int _{t}\mathbf {v} \,\mathrm {d} t}
where v is the particle velocity .
Progressive sine waves
[ edit ]
The particle displacement of a progressive sine wave is given by
δ
(
r
,
t
)
=
δ
sin
(
k
⋅
r
−
ω
t
+
φ
δ
,
0
)
,
{\displaystyle \delta (\mathbf {r} ,\,t)=\delta \sin(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}),}
where
δ
{\displaystyle \delta }
is the amplitude of the particle displacement;
φ
δ
,
0
{\displaystyle \varphi _{\delta ,0}}
is the phase shift of the particle displacement;
k
{\displaystyle \mathbf {k} }
is the angular wavevector ;
ω
{\displaystyle \omega }
is the angular frequency .
It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave x are given by
v
(
r
,
t
)
=
∂
δ
(
r
,
t
)
∂
t
=
ω
δ
cos
(
k
⋅
r
−
ω
t
+
φ
δ
,
0
+
π
2
)
=
v
cos
(
k
⋅
r
−
ω
t
+
φ
v
,
0
)
,
{\displaystyle v(\mathbf {r} ,\,t)={\frac {\partial \delta (\mathbf {r} ,\,t)}{\partial t}}=\omega \delta \cos \!\left(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}+{\frac {\pi }{2}}\right)=v\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{v,0}),}
p
(
r
,
t
)
=
−
ρ
c
2
∂
δ
(
r
,
t
)
∂
x
=
ρ
c
2
k
x
δ
cos
(
k
⋅
r
−
ω
t
+
φ
δ
,
0
+
π
2
)
=
p
cos
(
k
⋅
r
−
ω
t
+
φ
p
,
0
)
,
{\displaystyle p(\mathbf {r} ,\,t)=-\rho c^{2}{\frac {\partial \delta (\mathbf {r} ,\,t)}{\partial x}}=\rho c^{2}k_{x}\delta \cos \!\left(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}+{\frac {\pi }{2}}\right)=p\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{p,0}),}
where
v
{\displaystyle v}
is the amplitude of the particle velocity;
φ
v
,
0
{\displaystyle \varphi _{v,0}}
is the phase shift of the particle velocity;
p
{\displaystyle p}
is the amplitude of the acoustic pressure;
φ
p
,
0
{\displaystyle \varphi _{p,0}}
is the phase shift of the acoustic pressure.
Taking the Laplace transforms of v and p with respect to time yields
v
^
(
r
,
s
)
=
v
s
cos
φ
v
,
0
−
ω
sin
φ
v
,
0
s
2
+
ω
2
,
{\displaystyle {\hat {v}}(\mathbf {r} ,\,s)=v{\frac {s\cos \varphi _{v,0}-\omega \sin \varphi _{v,0}}{s^{2}+\omega ^{2}}},}
p
^
(
r
,
s
)
=
p
s
cos
φ
p
,
0
−
ω
sin
φ
p
,
0
s
2
+
ω
2
.
{\displaystyle {\hat {p}}(\mathbf {r} ,\,s)=p{\frac {s\cos \varphi _{p,0}-\omega \sin \varphi _{p,0}}{s^{2}+\omega ^{2}}}.}
Since
φ
v
,
0
=
φ
p
,
0
{\displaystyle \varphi _{v,0}=\varphi _{p,0}}
, the amplitude of the specific acoustic impedance is given by
z
(
r
,
s
)
=
|
z
(
r
,
s
)
|
=
|
p
^
(
r
,
s
)
v
^
(
r
,
s
)
|
=
p
v
=
ρ
c
2
k
x
ω
.
{\displaystyle z(\mathbf {r} ,\,s)=|z(\mathbf {r} ,\,s)|=\left|{\frac {{\hat {p}}(\mathbf {r} ,\,s)}{{\hat {v}}(\mathbf {r} ,\,s)}}\right|={\frac {p}{v}}={\frac {\rho c^{2}k_{x}}{\omega }}.}
Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by
δ
=
v
ω
,
{\displaystyle \delta ={\frac {v}{\omega }},}
δ
=
p
ω
z
(
r
,
s
)
.
{\displaystyle \delta ={\frac {p}{\omega z(\mathbf {r} ,\,s)}}.}
See also
[ edit ]
References and notes
[ edit ]
^
John Eargle (January 2005). The Microphone Book: From mono to stereo to surround – a guide to microphone design and application . Burlington, Ma: Focal Press. p. 27. ISBN 978-0-240-51961-6 .
Related Reading:
External links
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Particle_displacement&oldid=1194556617 "
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