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Oscillation and waves lecture notes

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- 1. Topic 2-3 Longitudinal Waves 1 UEEP1033 Oscillations and Waves Topic 5 Longitudinal Waves waves in which the particle or oscillator motion is in the same direction as the wave propagation Longitudinal waves propagate as sound waves in all phases of matter, plasmas, gases, liquids and solids
- 2. Topic 2-3 Longitudinal Waves 2 UEEP1033 Oscillations and Waves • Motion of one-dimensional longitudinal pulse moving through a long tube containing a compressible gas • When the piston is suddenly moved to the right, the gas just in front of it is compressed – Darker region in b – The pressure and density in this region are higher than before the piston was pushed Pressure Variation in Sound Waves
- 3. Topic 2-3 Longitudinal Waves 3 UEEP1033 Oscillations and Waves • When the piston comes to rest, the compression region of the gas continues to move – This corresponds to a longitudinal pulse traveling through the tube with speed v Pressure Variation in Sound Waves
- 4. Topic 2-3 Longitudinal Waves 4 UEEP1033 Oscillations and Waves Producing a Periodic Sound Wave • A one-dimensional periodic sound wave can be produced by causing the piston to move in simple harmonic motion • The darker parts of the areas in the figures represent areas where the gas is compressed and the density and pressure are above their equilibrium values • The compressed region is called a compression
- 5. Topic 2-3 Longitudinal Waves 5 UEEP1033 Oscillations and Waves • When the piston is pulled back, the gas in front of it expands and the pressure and density in this region ball below their equilibrium values • The low-pressure regions are called rarefactions • They also propagate along the tube, following the compressions • Both regions move at the speed of sound in the medium • The distance between two successive compressions (or rarefactions) is the wavelength Producing a Periodic Sound Wave
- 6. Topic 2-3 Longitudinal Waves 6 UEEP1033 Oscillations and Waves Periodic Sound Waves, Displacement • As the regions travel through the tube, any small element of the medium moves with simple harmonic motion parallel to the direction of the wave • The harmonic position function: smax = maximum position of the element relative to equilibrium (or displacement amplitude of the wave) k = wave number ω = angular frequency of the wave * Note the displacement of the element is along x, in the direction of the sound wave )cos(),( max tkxstxs ω−=
- 7. Topic 2-3 Longitudinal Waves 7 UEEP1033 Oscillations and Waves Periodic Sound Waves, Pressure • The variation in gas pressure, , is also periodic = pressure amplitude (i.e. the maximum change in pressure from the equilibrium value) • The pressure can be related to the displacement: B is the bulk modulus of the material )sin(max tkxPP ω−∆=∆ maxP∆ maxmax BksP =∆ P∆
- 8. Topic 2-3 Longitudinal Waves 8 UEEP1033 Oscillations and Waves Periodic Sound Waves • A sound wave may be considered either a displacement wave or a pressure wave • The pressure wave is 90o out of phase with the displacement wave • The pressure is a maximum when the displacement is zero, etc
- 9. Topic 2-3 Longitudinal Waves 9 UEEP1033 Oscillations and Waves Speed of Sound in a Gas • Consider an element of the gas between the piston and the dashed line • Initially, this element is in equilibrium under the influence of forces of equal magnitude – force from the piston on left – another force from the rest of the gas – These forces have equal magnitudes of PA • P is the pressure of the gas • A is the cross-sectional area of the tube element of the gas
- 10. Topic 2-3 Longitudinal Waves 10 UEEP1033 Oscillations and Waves Speed of Sound in a Gas • After a time period, Δt, the piston has moved to the right at a constant speed vx. • The force has increased from PA to (P+ΔP)A • The gas to the right of the element is undisturbed since the sound wave has not reached it yet
- 11. Topic 2-3 Longitudinal Waves 11 UEEP1033 Oscillations and Waves Impulse and Momentum • The element of gas is modeled as a non-isolated system in terms of momentum • The force from the piston has provided an impulse to the element, which produces a change in momentum • The impulse is provided by the constant force due to the increased pressure: • The change in pressure can be related to the volume change and the bulk modulus: ( )itPAtFI ˆ∆∆=∆= ∑ v v B V V BP x = ∆ −=∆ it v v ABI x ˆ ∆=⇒
- 12. Topic 2-3 Longitudinal Waves 12 UEEP1033 Oscillations and Waves Impulse and Momentum • The change in momentum of the element of gas of mass m is ( )itAvvvmp x ˆ∆ρ=∆=∆ ( )itAvvit v v AB pI x x ˆˆ ∆ρ= ∆ ∆= • The force from the piston has provided an impulse to the element, which produces a change in momentum B = bulk modulus of the material ρ = density of the material ρ=⇒ /Bv
- 13. Topic 2-3 Longitudinal Waves 13 UEEP1033 Oscillations and Waves Speed of Sound Waves, General • The speed of sound waves in a medium depends on the compressibility and the density of the medium • The compressibility can sometimes be expressed in terms of the elastic modulus of the material • The speed of all mechanical waves follows a general form: • For a solid rod, the speed of sound depends on Young’s modulus and the density of the material propertyinertial propertyelastic =v
- 14. Topic 2-3 Longitudinal Waves 14 UEEP1033 Oscillations and Waves Speed of Sound in Air • The speed of sound also depends on the temperature of the medium – This is particularly important with gases • For air, the relationship between the speed and temperature is 331.3 m/s = the speed at 0o C TC = air temperature in Celsius 15.273 1)m/s3.331( cT v +=
- 15. Topic 2-3 Longitudinal Waves 15 UEEP1033 Oscillations and Waves Relationship Between Pressure and Displacement • The pressure amplitude and the displacement amplitude are related by: ΔPmax = B k smax • The bulk modulus is generally not as readily available as the density of the gas • By using the equation for the speed of sound, the relationship between the pressure amplitude and the displacement amplitude for a sound wave can be found: ΔPmax = ρ v ω smax ρ= /Bv vk /ω=
- 16. Topic 2-3 Longitudinal Waves 16 UEEP1033 Oscillations and Waves Speed of Sound in Gases, Example Values
- 17. Topic 2-3 Longitudinal Waves 17 UEEP1033 Oscillations and Waves Energy of Periodic Sound Waves • Consider an element of air with mass Δm and length Δx • Model the element as a particle on which the piston is doing work • The piston transmits energy to the element of air in the tube • This energy is propagated away from the piston by the sound wave
- 18. Topic 2-3 Longitudinal Waves 18 UEEP1033 Oscillations and Waves Power of a Periodic Sound Wave • The rate of energy transfer is the power of the wave • The average power is over one period of the oscillation xvF ⋅=Power ( ) 2 max 2 avg 2 1 Power sAvωρ=
- 19. Topic 2-3 Longitudinal Waves 19 UEEP1033 Oscillations and Waves )(sin )]sin()][sin([ )]cos([)]sin([ ˆ)],([ˆ]),([Power 22 max 2 maxmax maxmax tkxAsv tkxstkxAsv tkxs t tkxAsv itxs t iAtxP ω−ωρ= ω−ωω−ωρ= ω− ∂ ∂ ω−ωρ= ∂ ∂ ⋅∆= • Find the time average power is over one period of the oscillation 2 1 2 2sin 2 1 sin 1 )0(sin 1 0 0 2 0 2 = ω ω +=ω=ω− ∫∫ T TT tt T dtt T dtt T • For any given value of x, which we choose to be x = 0, the average value of over one period T is:)(sin2 tkx ω−
- 20. Topic 2-3 Longitudinal Waves 20 UEEP1033 Oscillations and Waves Intensity of a Periodic Sound Wave • Intensity of a wave I = power per unit area = the rate at which the energy being transported by the wave transfers through a unit area, A, perpendicular to the direction of the wave • Example: wave in air ( ) A I avgPower = 2 max 2 2 1 svI ωρ=
- 21. Topic 2-3 Longitudinal Waves 21 UEEP1033 Oscillations and Waves Intensity • In terms of the pressure amplitude, ( ) v P I ρ ∆ = 2 2 max • Therefore, the intensity of a periodic sound wave is proportional to the • square of the displacement amplitude • square of the angular frequency 2 maxs 2 ω
- 22. Topic 2-3 Longitudinal Waves 22 UEEP1033 Oscillations and Waves A Point Source • A point source will emit sound waves equally in all directions - this can result in a spherical wave • This can be represented as a series of circular arcs concentric with the source • Each surface of constant phase is a wave front • The radial distance between adjacent wave fronts that have the same phase is the wavelength λ of the wave • Radial lines pointing outward from the source, representing the direction of propagation, are called rays
- 23. Topic 2-3 Longitudinal Waves 23 UEEP1033 Oscillations and Waves Intensity of a Point Source • The power will be distributed equally through the area of the sphere • The wave intensity at a distance r from the source is: • This is an inverse-square law The intensity decreases in proportion to the square of the distance from the source ( ) ( ) 2 avgavg 4 PowerPower rA I π ==

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