1 8 4 D O C . 3 9 P R O P A G A T I O N O F S O U N D
First, we look at the purely mechanical part of the problem. The (Eulerian) dif-
ferential equation of motion for a plane wave is (taking into account the simplifi-
cations customary for sound problems)
. (1)
Here, is the infinitesimal deviation of pressure from its value at equilibrium,
is the density (at equilibrium), is the elongation of an air particle in the direc-
tion of the X-axis, resp., wave normal. The excess pressure is related to the com-
pression , which is connected with the elongation according to the equation
. (2)
We now look for the propagation law of an attenuated plane sine wave for which
we put
(3)
,
where , , , , , are real constants. The phase difference corresponds
to the energy dissipation.
Instead of the real Ansatz (3), we use in customary manner the complex Ansatz
}
,
(4)
where we used the abbreviation
1Pertinent
experimental investigations were already carried out in 1910 on in the
Nernst laboratory (see F. Keutel, Berlin dissertation, 1910). It is already pointed out there
that the velocity of sound depends upon the reaction velocity.
J2 J
J.1
+
N2O4
∂π
∂x
------ –
ρ--------
∂2u
∂t2
=
π p
ρ u
π
Δ u
Δ
∂u
∂x
- –ρ-----
=
[2]
[3]
π π
°
ω t
x⎞
v⎠
-- - –
⎝
⎛
φ +
e–βx}
Δ Δ
°
ω t
x⎞
v⎠
-- - –
⎝
⎛ e–βx
cos =
cos =
{1}
π
°
Δ
°
ω v φ β φ
π π
°
e
j( ω t ax φ) + –
Δ Δ
°
e
j( ω t ax) –
=
=