FUNDAMENTALS OF ACOUSTICS 14

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FUNDAMENTALS OF ACOUSTICS (14)

• Spherical Waves and Cylindrical Waves
• April 25, 2004

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• Thus far we have developed a set of differential
  equations that describe the local behavior of
  sound waves.
• We become acquainted with some other important
  types of wave behavior ----- spherical waves .
• Our emphasis here is on the waves themselves and
  in the following chapter we will pursue the
  interaction of these waves with their sources.

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• Spherical spreading describes the decrease in
  level when a sound wave propagates away from a
  source uniformly in all directions.

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• Sound levels are therefore constant on spherical
  surfaces surrounding the sound source.
• Sound levels decrease rapidly as sound spreads
  out from a sphere with a radius of r0 to a larger
  sphere with a radius r.

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Spherical waves are represented in complex form by

• The wave diminish in amplitude as the distance
  from the source increase.

the particle velocity is not in phase with the
pressure
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The Acoustic impedance
Separating za into real and imaginary parts, we
have
The first term is the acoustic resistance, and
the second term is the acoustic reactance.
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• The acoustic impedance of spherical wave
• Real , imaginary parts and phase

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• We emphasize again that the acoustic impedance
  depends on the frequency , as well as on the
  medium, the wave type, and the geometric
  position.
• Neither the frequency nor the distance alone is
  so important as is the dimension combination kr
• For kr gtgt1, Za is predominantly resistive,
  meaning that energy fed into such waves
  propagates away from the source, never to return.

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• The larger the value of kr, the less curved are
  the wave fronts and the more nearly do these
  waves act just like plane waves.
• But for kr ltlt 1 the reactive component is large,
  and it is positive ( masslike )
• This means that pushing on such a wave is mainly
  a matter of storing kinetic energy in motion of
  the liquid, where it can be recovered later by
  the source. (like energy in an electrical
  inductor )

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The intensity and power of spherical waves

• There is an alternative view that suggests a
  simpler insight into the spherical wave.
• If we had considered the ratio u/p instead of p/u
  , the resulting admittance would be a much
  simpler function.

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• For a harmonic spherical wave, the acoustic
  intensity is

Notice that the formula has been found to be
exactly true for both plane and spherical waves.
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• The acoustic intensity can be expressed in
  another form

The average rate at which energy flows through a
closed spherical surface of radius r surrounding
a source of symmetrical spherical waves is
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The average rate of energy flow through any
spherical surface surrounding the origin is
independent of the radius of the surface, a
conclusion that is consistent with conservation
of energy in a lossless medium
Since
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example

• A diverging spherical wave has a peak acoustic
  pressure of 2Pa at a distance of 1m from the
  source at standard atmospheric pressure and
  temperature. What is its intensity at a distance
  of 10m from the source?

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• Assume the source is emitting a constant amount
  of energy to the sound wave. For diverging
  spherical waves, the area of the wavefront
  increases as the waves are traveling farther and
  farther from the source. Hence intensity of such
  waves diminishes with distance of propagation.
• At a distance of 1m from the source,

Where p0 1.21kg/m3 is the density of air, and
c343m/s is the speed of sound in air at standard
atmospheric pressure and temperature
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• At a distance of 10m from the source, the
  effective sound pressure will change but the
  power radiated will remain the same.

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3-10 Cylindrical Waves

• Sound waves radiate from something acting more or
  less as a line source rather than a point source.
  Then it is natural to describe such waves in
  cylindrical coordinates

The simplest solution using these coordinates
will be when p does not depend on
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• A simple approximation for spreading loss in a
  medium with upper and lower boundaries can be
  obtained by assuming that the sound is
  distributed uniformly over the surface of a
  cylinder having a radius equal to the range r and
  a height H equal to the depth of the ocean.

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• In this case , the pressure is a function of
  radial distance and time , Then the wave equation
  simplifies to

If the waves are harmonic, such waves are
represented in complex form by
The equation (3-10-3) is called the Bessel
equation with the general solution
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• Here J0(kr) is the zero order Bessel function
• N0(kr) is the zero order Neumann function

Introduce Hankel function
For the diverging waves
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The diverging waves are

• When kr gtgt1

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The acoustic intensity is
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The relative intensity level decrease less
rapidly for cylindrical than for spherical
spreading.

• The total power crossing a cylinder surrounding
  the source equals the intensity times the area of
  the cylinder

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The acoustic impedance
When kr ltlt1
When kr gtgt1
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When kr gtgt1, the acoustic impedance

• In this limit the acoustic impedance again
  approaches p0c0, since very large cylindrical
  wave fronts will be practically indistinguishable
  from plane waves.
• We may remark that , like the spherical waves,
  cylindrical waves take on a much simpler form
  when kr becomes large.
• The pressure amplitude of cylindrical waves
  approach

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Homework

• Textbook P276 3-8, 3-10
• Spherical acoustic waves of frequency 125Hz are
  emitted from a small source. At a radial distance
  of 1.5m from the source, what is the phase angle
  between acoustic pressure and particle velocity?
  Find the magnitude of the acoustic impedance at
  this point.