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
Pressure Variation in Sound 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 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
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
Producing a Periodic Sound Wave 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
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: s ( x, t ) s cos( kx t ) max s max = maximum position of the element relative to equilibrium (or displacement amplitude of the wave) k = wave number = angular frequency of the wave * Note the displacement of the element is along x, in the direction of the sound wave
Periodic Sound Waves, Pressure The variation in gas pressure, P, is also periodic P P max sin( kx t ) P max = pressure amplitude (i.e. the maximum change in pressure from the equilibrium value) The pressure can be related to the displacement: P max Bks max B is the bulk modulus of the material
Periodic Sound Waves A sound wave may be considered either a displacement wave or a pressure wave The pressure wave is 90 o out of phase with the displacement wave The pressure is a maximum when the displacement is zero, etc
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
Speed of Sound in a Gas After a time period, Δt, the piston has moved to the right at a constant speed v x. 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
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: I F t A P t iˆ The change in pressure can be related to the volume change and the bulk modulus: P B V V B v v x I AB v v x t iˆ
Impulse and Momentum The change in momentum of the element of gas of mass m is p m v I vv p x A t iˆ The force from the piston has provided an impulse to the element, which produces a change in momentum AB v v x t iˆ vv x A t iˆ v B / B = bulk modulus of the material = density of the material
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: v elastic inertial property property For a solid rod, the speed of sound depends on Young s modulus and the density of the material
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 v (331.3 m/s ) 1 73 T c.15 331.3 m/s = the speed at 0 o C T C = air temperature in Celsius
Relationship Between Pressure and Displacement The pressure amplitude and the displacement amplitude are related by: ΔP max = B k s max 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: ΔP max = ρ v ω s max v B / k / v
Speed of Sound in Gases, Example Values
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
Power of a Periodic Sound Wave The rate of energy transfer is the power of the wave Power F v x The average power is over one period of the oscillation Power avg 1 Av s max
Power [ P ( x, t ) A ] iˆ t [ s ( x, t )] iˆ [ v As max sin( kx t )] t [ s max cos( kx t )] [ v As max sin( kx t )][ s max sin( kx t )] v As max sin ( kx t ) Find the time average power is over one period of the oscillation For any given value of x, which we choose to be x = 0, the average value of sin ( kx t ) over one period T is: 1 T 0 T sin (0 t ) dt 1 T 0 T sin tdt 1 T t sin t T 0 1
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 I Power A avg Example: wave in air I 1 v s max
Intensity Therefore, the intensity of a periodic sound wave is proportional to the square of the displacement amplitude square of the angular frequency s m ax In terms of the pressure amplitude, I P max v
Reflection and Transmission of Sound Waves at Boundaries