Longitudinal waves and Fourier Analysis.

1
Lecture 7

• Longitudinal waves andFourier Analysis.
• Aims
• Sound waves
• Wave equation derived for a sound wave in a gas.
• Acoustic impedance.
• Doppler effect for sound waves.
• Fourier Theory
• Description of waveforms in terms of a
  superposition of harmonic waves.
• Fourier series (periodic functions)
• Fourier transforms (aperiodic functions).

2
Longitudinal waves

• Properties similar to other waves but
• Displacement are parallel to k, not perp.
• Not polarised (longitudinal polarisation)
• Transmission by compression/rarefaction.
• Gases and liquids cannot support shear stress and
  have no transverse waves!
• Sound waves in a gas
• Derivation of wave equation
• consider an element length Dx, area DS. (a
  column aligned along the wave direction)
• determine forces on the element
• apply Newtons Laws.

3
Forces on the element of gas

• Displacement and pressures
• Without the wave (i.e. in equilibrium).
• With a longitudinal wave, displacement a(x)
• x becomes xa
• (xDx) becomes (xDxaDa).
• Pressure imbalance is

We need to calculate the pressure gradient
4
Dynamics of element

• Adiabatic changes
• pressure changes in a sound wave are (normally)
  rapid. No heat transfer with surroundings -
  Adiabatic.
• Differentiation gives
• For our element,
• Identify pressure change, dp with Yp.
• Pressure gradient

Usually negligible
5
Wave equation for sound in a gas

• Force on the element is
• Apply Newtons 2nd law to the element (mass
  rDxDs)
• Note
• dependence on molar mass, M.
• speed similar to molecular velocities. Recall

Wave equation
Wave speed
Molar mass
6
Waves in liquids and solids

• Pressure waves
• Previous equation is for displacement.
• Pressure is more important. We have just shown
  the pressure is
• for a harmonic wave
• Pressure wave leads the displacement by p/2 and
  has amplitude gpkao.
• Solids and liquids
• Pressure and displacement related by
  , so the velocity is
• For solids,
• Typical values
• Gases air at STP 340 ms-1.
• Liquids 1000 ms-1.
• Solids granite 5000 ms-1.

Youngs modulus
Bulk modulus
7
Acoustic impedance

• Impedance for sound waves
• Zpressure due to wave / displacement velocity
• Ifand displacement velocity
• Impedance is
• Reflection and transmission coefficients follow
  as described in lecture 6.
• Open end of pipe Z0
• Closed end of pipe Z.
• Note large difference between Z in different
  materials (e.g. between air and solid) gives
  strong reflections.
• Ultrasonic scanning.

8
Doppler effect

• For sound
• Stationary observer
• compare stationary and moving sources
• O will register f1, where
• So,
• Moving observer (speed w, away from S).
• Observer sees S moving at -w and a sound speed of
  v-w. So, using 5.1
• General case (observer moving at w, source moving
  at u, w.r.t medium)

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Fourier Theory

• It is possible to represent (almost) any function
  as a superposition of harmonic functions.
• Periodic functions
• Fourier series
• Aperiodic functions
• Fourier transforms
• Mathematical formalism
• Function f(x), which is periodic in x, can be
  writtenwhere,
• Expressions for An and Bn follow from the
  orthogonality of the basis functions, sin and
  cos.

10
Complex notation

• Example simple case of 3 terms
• Exponential representation
• with k2pn/l.

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Example

• Periodic top-hat
• N.B.

Fourier transform
f(x)