Basics of Fluid Mechanics, 2014a

4.3. PRESSURE AND DENSITY IN A GRAVITATIONAL FIELD 85
density is a function of the depth. For constant bulk modulus, it was shown in “Fundamentals
of Compressible Flow” by this author that the speed of sound is given by
√
B T
c =
(4.VII.a)
ρ
Calculate the time it take for a sound wave to propagate perpendicularly to the surface
to a depth D (perpendicular to the straight surface). Assume that no variation of
the temperature exist. For the purpose of this exercise, the salinity can be completely
ignored.
Solution
The equation for the sound speed is taken here as correct for very local point. However,
the density is different for every point since the density varies and the density is a
function of the depth. The speed of sound at any depth point, x, is to be continue
??????????????
B T
c =
=
√ ρ 0 B T
B T − gρ 0 z
√
B T − gρ 0 z
ρ 0
The time the sound travel a small interval distance, dz is
dτ =
dz
√
BT − gρ 0 z
ρ 0
(4.VII.b)
(4.VII.c)
The time takes for the sound the travel the whole distance is the integration of infinitesimal
time
⎧ - D
t =
⎪⎭
- 0
dz
√
BT − gρ 0 z
The solution of equation (4.VII.d) is
t = √ ρ 0
(2 √ B T − 2 √ )
B T − D
ρ 0
(4.VII.d)
(4.VII.e)
The time to travel according to the standard procedure is
t =
√ D = D √ ρ 0
√
BT BT
ρ 0
The ratio between the corrected estimated to the standard calculation is
√ ( √
ρ0 2 BT − 2 √ B T − D )
correction ratio =
D √ ρ 0
√
BT
(4.VII.f)
(4.VII.g)