Fundamentals of acoustics 11

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Fundamentals of acoustics (11)

• Reflection , refraction and transmission of plane
  waves

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• When sound waves are traveling through a medium,
  they may be reflected or refracted ,diffracted or
  scattered ,interfered or absorbed .The
  transmission of sound involves the transfer of
  acoustic energy through the medium in which sound
  waves travel.

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We can define plane acoustic waves as
one-dimensional free progressive waves traveling
in the x direction. They are described by the
one-dimensional wave equation
The general solution for the one-dimensional wave
equation can be written in progressive waves form
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Which consists of two parts the first part
represents a wave of arbitrary shape traveling in
the x direction with velocity co ,and the second
part represents a wave also of arbitrary shape
traveling in the negative x direction with
velocity c0 . In complex exponential form, the
general solution can be written as
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Change in media

• When an acoustic wave traveling in one medium
  encounters the boundary of a second medium,
  reflected and transmitted (or refracted ) waves
  are generated.
• Discussion of this phenomenon is greatly
  simplified if it is assumed that both the
  incident wave and the boundary between the media
  are planar and that all media are fluid

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Transmission from one fluid to another normal
incidence

• As suggested in Fig. , let the plane x0 be the
  boundary between fluid I of characteristic
  acoustic impedance Z1 and fluid II of
  characteristic acoustic impedance Z2.
• Let there be an incident wave traveling in the
  positive x direction

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Which ,when striking the boundary , generates a
reflected wave
And a transmitted wave
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Note The transmitted wave has the same
frequency as the incident wave but , because of
the different phase speeds c1 and c2, the wave
numbers k1 in fluid I and k2 in fluid II are
different

• In the first medium, sound pressure is

In the second medium, sound pressure is
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They are described by the wave equation

• There are two boundary conditions that must be
  satisfied for all times at all point on the
  boundary

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(2)The particle velocities normal to the
boundary are equal.The second condition,
continuity of normal velocity, requires that the
fluid remain in contact.

• (1) the acoustic pressures on both sides of the
  boundary are equal
• The first condition , continuity of pressure ,
  means that there can be no net force on the
  (massless) plane separating the fluid.

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• We have defined the ratio of acoustic pressure in
  a medium to the associated particle velocity as
  the acoustic impedance.
• Zp / u, then

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From the equation of motion

• The pressure and normal particle velocity in
  fluid I are pi pr and (ui ur) so that two
  boundary conditions are

pipr pt at x0 uiurut at x0
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• We define the pressure transmission (refraction)
  and reflection coefficients, so obtain

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• From the reflection coefficient, we know that R
  is always real
• Positive when Z1ltZ2 and negative when Z1gtZ2.
  Consequently, at the boundary the acoustic
  pressure of the reflected wave is either in phase
  or 1800 out of phase with that of the incident
  wave.
• When the characteristic acoustic impedance of
  fluid II is greater than that of fluid I ,a
  positive pressure in the incident wave is
  reflected as a positive pressure.

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• when Z1Z2 then R0,and there is complete
  transmission.
• For ,the wave is
  reflected with only a slight reduction in
  amplitude and no change in phase. The resulting
  wave in fluid II has a maximum pressure amplitude
  almost twice Pi and a particle speed at the
  boundary that is almost zero. Because of this
  latter fact, such a boundary is termed rigid

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Since the intensity of a harmonic plane
progressive wave is

• The intensity transmission is real and is defined
  by

The transmission loss will be
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A wave in water incident on the water-air
interface
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Text book 3-14

• A beam of sound waves is incident normally on a
  plane interface of air and an infinite body of
  fluid of unknown impedance. If half of the sound
  energy is reflected, find the unknown impedance.

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Sound energy reflected is described by the sound
power reflection coefficient, and sound energy is
proportional to its intensity. The intensity
reflection is
Z272 MKS rayls
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Transmission form one fluid to another oblique
incidence

• Assume that the boundary separating two fluids is
  the plane x0 and that the incident, reflected ,
  and transmitted waves make the respective angles
  , shown in Fig.

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• In the first medium, p1pipr
• In the second medium, p2pt
• They can be described by wave equations

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On the boundary, they are two boundary conditions

• (1)continuity of pressure

(3-7-10)
(2) Continuity of normal velocity
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(3-7-11)
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Law of reflection
the angle of incidence is equal to the angle of
reflection, and
Snells law
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Homework 3-18