Waves Chapter 16

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Waves Chapter 16 17

• PHYS 2326-25

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Concepts to Know

• Mass per Unit Length
• Bulk Modulus
• Youngs Modulus
• Ratio of Heat Capacities
• Ideal Gas
• Pressure
• Gas Constant
• Molecular Mass
• Absolute Temperature

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Transverse Wave

• A wave on a string is usually transverse.
  Examples include plucking or strumming a guitar
  or banjo.
• Eqn 16.18 provides the basis for transverse waves
  on a string

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General Expression

• All mechanical wave speeds follow a general
  expression
• For a string the elastic property was the tension
  T and the inertial property of the medium was the
  mass per unit length
• See section 12.4 for review information

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Bulk ModulusSound Waves in Liquid or Gas (fluid)

• Speed of sound in a medium depends upon the
  compressibility and density of the medium
• B bulk modulus (elastic property)
• ? density (inertial property)

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Youngs ModulusSound Waves in a Solid

• For longitudinal waves in a solid material
• Y Youngs modulus (elastic property)
• ? density (inertial property)

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Ideal Gas Law

• Chapter 19.5 is the ideal gas law this is
  covered in the other semester
• PV nRT (eqn 19/8)
• P pressure in pascals (N/m2)
• R universal gas constant 8.314 J/molk or
  0.08206 Latm / molK for Volume in liters and
  pressure in earth atmospheres
• n number of moles of gas
• T absolute temperature degrees kelvin

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• PV nRT N/Na RT where
• N number of molecules
• Na Avogadros number 6.022E23 atoms/mole
• PV N Kb T
• Kb Boltzmans constant R/Na
• 1.38E-23 J/K

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Longitudinal Wavein an Ideal Gas

• It is a pressure wave
• B bulk modulus change in pressure /
  fractional change in volume
• for an ideal gas relationship (for adiabatic
  condition) is pV? constant
• ? cp/cv specific heat at constant pressure
  divided by the specific heat at constant volume
• adiabatic activity where no heat enters or
  leaves
• Bad ?p so
• M molecular mass (28.8 avg. for air)

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Speed of Sound in AirAssuming Ideal Gas

• v 348m/s or 1148 ft/s
• The eqn below 17.1, v(331) sqrt(1Tc/273) m/s
• comes from this

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Energy in the Wave(String)

• Chapter 16.5 energy transfer per unit time in a
  string by a sinusoidal wave

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Sound Pressure Energy

• Like the transverse wave in a string, sound waves
  also have energy.
• Rather than an transverse amplitude A we have a
  longitudinal displacement smax

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Sound Pressure Energy

• Like the transverse wave in a string, sound waves
  also have energy.
• Rather than an transverse amplitude A we have a
  longitudinal displacement smax
• For a solid we have

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Example 1

• A mass of 2kg is suspended on a string of length
  2m and mass 10g. Find a) speed of a transverse
  wave, b) the fundamental frequency (assuming the
  wavelength is twice the length of the string, c)
  the power transferred up and down the string
  while vibrating with an amplitude of 2cm

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Example 1

• Given M 2kg, L 2m, m 0.1kg, ? 4m, A
  0.02m
• Force F mg (gravity, g 9.78m/s2)
• µ m/L (mass/length), v sqrt(F/µ) m/s
• v f?, ? 2pf, Pav ½ sqrt(µF) ?2 A2
• F (2kg)(9.78m/s2) 19.6 N
• µ m/L 0.1/2 0.005 kg/m
• v v sqrt(F/µ) sqrt(19.6/0.05) 19.8m/s
• b) frequency f v/ ? 19.8/4 4.95 Hz

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Example 1

• ? 2 pf 31.1 r/sec
• Pav ½ sqrt(0.05 19.6) (31.1)2 (0.02)2
• 0.192 W

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Example 2

• At what temeprature C is the speed of sound in
  air equal to 345 m/s. Assume air is an ideal gas
  with molecular weight of 28.8g/mole and that air
  is diatomic so ?1.40.
• M28.8E-3 kg/mole
• R 8.315 J/mole K
• v 345
• Note Tc Tk 273.15
• v sqrt (?RT/M)

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Example 2

• rearranging the v eqn,
• T M v2 / ? R 28.8 (345)2 / (1.4 8.315)
• 294.5 degrees K
• converting to centegrade
• 294.5 273.15 21.2 degrees C