4.1 Internal Sounds in Brass Instruments
107
If A is a real number, the first term on the right-hand side of Eq. 4.8 is purely real,
and the second term is purely imaginary. Only the real part represents the physically
measurable value of the variable. The pressure in the forward-going wave, which is
of course a real quantity, is
p + = A cos(ωt − kx).
(4.9)
At time t = 0 the dependence of the acoustic pressure on distance along the tube
is, from Eq. 4.9,
p + = A cos(−kx) = A cos(kx).
(4.10)
Comparison of Eq. 4.10 and Fig. 4.4a shows that Eq. 4.3 does indeed represent the
illustrated travelling wave, with A = 1.
An expression for the acoustic particle velocity v + in the wave can be obtained
from Eq. 4.3 by making use of the Euler equation :
ρ
∂v
∂t
= −∇p,
(4.11)
where ρ is the density of air. This equation is essentially Newton’s second law of
motion applied to a fluid particle: the left-hand side is proportional to the mass
of the particle times its acceleration, while the right-hand side is proportional to
the net external force. In general particle velocity and pressure gradient are vector
quantities, but in the present case, both vectors are directed along the x axis, and
Eq. 4.11 can be rewritten as
ρ
∂v
∂t
= −
∂p
∂x
.
(4.12)
The spatial pressure gradient can be found by differentiating Eq. 4.3 with respect
to x. Substituting this into Eq. 4.12 and integrating both sides of the resulting
equation with respect to t yields an expression for the particle velocity in the
forward-going wave:
v + =
A
ρc
e
j (ωt−kx)
=
1
ρc
p + .
(4.13)
The ratio p/v = ρc is called the specific acoustic impedance of the wave.
The fact that the specific acoustic impedance is real means that there is no phase
difference between the plane travelling pressure wave and the associated particle
velocity wave.
The amplitude A in Eq. 4.9 is written as a constant. In reality the amplitude of
the wave decreases as it travels down the tube because some sound energy is lost
through friction between the air and the wall and transfer of thermal energy to the
wall. For the time being, we will ignore these viscothermal losses.
107
If A is a real number, the first term on the right-hand side of Eq. 4.8 is purely real,
and the second term is purely imaginary. Only the real part represents the physically
measurable value of the variable. The pressure in the forward-going wave, which is
of course a real quantity, is
p + = A cos(ωt − kx).
(4.9)
At time t = 0 the dependence of the acoustic pressure on distance along the tube
is, from Eq. 4.9,
p + = A cos(−kx) = A cos(kx).
(4.10)
Comparison of Eq. 4.10 and Fig. 4.4a shows that Eq. 4.3 does indeed represent the
illustrated travelling wave, with A = 1.
An expression for the acoustic particle velocity v + in the wave can be obtained
from Eq. 4.3 by making use of the Euler equation :
ρ
∂v
∂t
= −∇p,
(4.11)
where ρ is the density of air. This equation is essentially Newton’s second law of
motion applied to a fluid particle: the left-hand side is proportional to the mass
of the particle times its acceleration, while the right-hand side is proportional to
the net external force. In general particle velocity and pressure gradient are vector
quantities, but in the present case, both vectors are directed along the x axis, and
Eq. 4.11 can be rewritten as
ρ
∂v
∂t
= −
∂p
∂x
.
(4.12)
The spatial pressure gradient can be found by differentiating Eq. 4.3 with respect
to x. Substituting this into Eq. 4.12 and integrating both sides of the resulting
equation with respect to t yields an expression for the particle velocity in the
forward-going wave:
v + =
A
ρc
e
j (ωt−kx)
=
1
ρc
p + .
(4.13)
The ratio p/v = ρc is called the specific acoustic impedance of the wave.
The fact that the specific acoustic impedance is real means that there is no phase
difference between the plane travelling pressure wave and the associated particle
velocity wave.
The amplitude A in Eq. 4.9 is written as a constant. In reality the amplitude of
the wave decreases as it travels down the tube because some sound energy is lost
through friction between the air and the wall and transfer of thermal energy to the
wall. For the time being, we will ignore these viscothermal losses.