1 Waves 7 Lecture 7 Longitudinal waves and Fourier Analysis. D Aims: ëSound waves: > Wave equation derived for a sound wave in a gas. ëAcoustic impedance. ëDoppler effect for sound waves. ëFourier Theory: > Description of waveforms in terms of a superposition of harmonic waves. > Fourier series (periodic functions); > Fourier transforms (aperiodic functions).

2 Waves 7 Longitudinal waves D Properties similar to other waves but:  Displacement are parallel to k, not perp.; ëNot polarised (longitudinal polarisation); ëTransmission by compression/rarefaction. > Gases and liquids cannot support shear stress and have no transverse waves! D Sound waves in a gas: ëDerivation of wave equation:  consider an element: length  x, area  S. (a column aligned along the wave direction) > determine forces on the element > apply Newton’s Laws.

3 Waves 7 Forces on the element of gas D Displacement and pressures; ëWithout the wave (i.e. in equilibrium).  With a longitudinal wave, displacement a(x) :  x becomes x+a ;  ( x+  x ) becomes ( x+  x+a+  a ). ëPressure imbalance is We need to calculate the pressure gradient

4 Waves 7 Dynamics of element D Adiabatic changes ëpressure changes in a sound wave are (normally) rapid. No heat transfer with surroundings - Adiabatic. ëDifferentiation gives: ëFor our element,  Identify pressure change, dp with  p. ëPressure gradient Usually negligible

5 Waves 7 ëForce on the element is  Apply Newton’s 2nd law to the element (mass  x  s ) ë ëNote:  dependence on molar mass, M. > speed similar to molecular velocities. Recall Wave equation for sound in a gas Wave equation Wave speed Molar mass

6 Waves 7 Waves in liquids and solids D Pressure waves: ëPrevious equation is for displacement. ëPressure is more important. We have just shown the pressure is ëfor a harmonic wave  Pressure wave leads the displacement by  /2 and has amplitude  pka o. D Solids and liquids ëPressure and displacement related by, so the velocity is ëFor solids, D Typical values: ëGases:air at STP340 ms -1. ëLiquids:1000 ms -1. ëSolids:granite5000 ms -1. Bulk modulus Young’s modulus

7 Waves 7 Acoustic impedance D Impedance for sound waves  Z =pressure due to wave / displacement velocity ë If and displacement velocity ëImpedance is ëReflection 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 Waves 7 Doppler effect D For sound ëStationary observer: > compare stationary and moving sources:  O will register f 1, 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 Waves 7 Fourier Theory ëIt is possible to represent (almost) any function as a superposition of harmonic functions. D Periodic functions: ëFourier series D Aperiodic functions: ëFourier transforms D Mathematical formalism  Function f(x), which is periodic in x, can be written: where,  Expressions for A n and B n follow from the “orthogonality” of the “basis functions”, sin and cos.

10 Waves 7 Complex notation D Example: simple case of 3 terms D Exponential representation:  with k=2  n/l.

11 Waves 7 Example D Periodic top-hat: ëN.B. Fourier transform f(x)f(x) f(x)f(x)