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Lecture 7

Lecture 7. Longitudinal waves and Fourier 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.

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Lecture 7

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  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 Newton’s 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 x+a; • (x+Dx) becomes (x+Dx+a+Da). • 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 Newton’s 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. Young’s modulus Bulk modulus

  7. Acoustic impedance • Impedance for sound waves • Z=pressure due to wave / displacement velocity • Ifand displacement velocity • Impedance is • Reflection and transmission coefficients follow as described in lecture 6. • Open end of pipe Z=0; • 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)

  9. 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 written:where, • 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 k=2pn/l.

  11. Example • Periodic top-hat: • N.B. Fourier transform f(x)

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