Exercise 1 (Sound and Sources of Sound): For a one-dimensional problem the perfect gas equation and the momentum equation given below can be simplified as follows:

Linearise these two equations.

Utilizando o método da pertubação para linearizar as equações do gás perfeito e da quantidade de movimento, obtemos:

Temos que , e são constantes no tempo e no espaço. Assumindo que , e

são pequenos, podemos negiglenciar os termos de segunda ordem. Substituindo na equação do gás perfeito:

(1)

Substituindo na equação da quantidade de movimento:

(2)

Exercise 2 (Sound and Sources of Sound): 11Equation Section (Next)Someone tries

to estimate the distance to a storm by measuring the time delay T=7s between

lightning and thunder. The wind is blowing at an average speed of 100km/h.

Calculate the distance when

a) the effect of the wind is neglected,

b) the thunder is propagating against the wind,

c) the thunder is propagating with the wind.

Para uma atmosfera padrão ao nivel do mar:

, e ,

a) Negligênciando a influência do vento:

b) Os trovões estão propagando contra o vento:v = 100 / 3,6 =27.8

a) Os trovões estão se propagando com o vento:v = 100 / 3,6 =27.8

Exercise 3 (Sound and Sources of Sound):22Equation Section (Next) Consider a

time-harmonic plane wave at 1kHz. Calculate the frequency perceived by an

observer moving with the flow:

a) the wave is propagating upstream with M = 0:25,M = 0:5,M = 0:75,

b) the wave is propagating downstream with M = 0:25,M = 0:5,M = 0:75

Equação da onda harmônica :

Assumindo é constante e dependendo da direção do fluxo, temos:

Assim,

Para a onda se propagando na direção contrária ao fluxo,

Para M = 0.25

f '=1000∗(1+0.25)=1250Hz

Para M = 0.5

f '=1000∗(1+0.5)=1500Hz

Para M = 0.75

f '=1000∗(1+0.75)=1750Hz

Para a onda se propagando na direção do fluxo,

Para M = 0.25

f '=1000∗(1−0.25)=750Hz

Para M = 0.5

f '=1000∗(1−0.5)=500Hz

Para M = 0.75

f '=1000∗(1−0.75)=250Hz

Exercise 4 (Sound and Sources of Sound):33Equation Section (Next)44Equation

Section (Next) Describe the Lighthill Theory of Aerodynamic Sound (physics and

equations).

The sound generated by turbulence is usually called aerodynamic sound, which

is a very small byproduct of the motion of unsteady flows of high Reynolds number.

The source of aerodynamic sound was given the exact form by Lighthill. Lighthill

exactly transformed the set of fundamental equations, Navier-Stokes and continuity

equations, to an inhomogeneous wave equation whose inhomogeneous term plays the

role of the source:

(1)

where the tensor is called Lighthill’s tensor and is defined by

(2)

Here, denotes the speed of sound in a stationary acoustic medium, the air

pressure with the average , the air density with the average , and the

viscous stress tensor. It is considered that the sound wave is generated by the

quadrupole source distribution in turbulence given by the inhomogeneous term in RHS

of Equation 1 and propagates like that in the stationary acoustic medium, even though

turbulence exists. This interpretation is called Lighthill’s acoustic analogy.

Since the dissipation by can be ignored for a high Reynolds number and

adiabaticity is well held as

(3)

then the first term of Equation 2, , becomes the major term of the source. Further,

particle velocities of the sound are usually sufficiently small compared with those of the

real flow and so the source term is well approximated by that obtained from

incompressible fluid with and . Then, the sound source is given by

(4)

where and are respectively given by

(5)

(6)

For two dimensional(2D) fluid, it is further reduced into

(7)

In calculation of Lighthill’s source for 3D and 2D models, we will use the above

formulae later. For exactly incompressible fluid, an analogue to Lighthill’s equation is

written by a Poisson equation

(8)

As the analogy to the static electric field, a static pressure field is created by the

source term in RHS corresponding to the main term of Lighthill’s quadrupole source,

but the propagation speed of pressure distortion is infinite due to incompressibility. For

compressible fluid, the pressure distortion propagates at a finite speed, then the term

should be added to LHS of Equation 8 and Lighthill’s Equation 1 with the

approximations Equation 3 and Equation 4 is obtained again. Since the compressible

portion of a dynamical variable is extremely small compared with its incompressible

portion, then Lighthill’s equation Equation 1 may be well approximated in turbulence

by Equation 8.

Exercise 5 (Sound and Sources of Sound):55Equation Section (Next) Write down

the equations for the farfield pressure radiated by:

a) Monople source

(1)

b) Dipole source

(2)

c) Quadrupole source

Figure 1 – (a) Lateral quadrupole and (b) longitudinal quadrupole

Lateral

Longitudinal

Where is the fluid density, is the speed of sound, is the wave number, is the

distance from source to observation point, and and are the distances between each

pole as shown in figure 1. is a constant, termed the complex source strength and

represents the volume of fluid displaced by the source at the rate:

where is the velocity at some point on the surface of the source. For a pulsating

sphere the source strength is real, and equals the product of surface area and surface

velocity: . The pressure amplitude does not depend on angle; the

pressure produced by a monopole is the same at all points a distance r from the source.