Topic 5 longitudinal wave

Topic 2-3 Longitudinal Waves

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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


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• 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


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• 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


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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


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• 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


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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 = angular frequency of the wave

* Note the displacement of the element is along x, in the direction of the sound wave

)cos(),( max tkxstxs


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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

maxP

maxmax BksP

P


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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


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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


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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


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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

vB

V

VBP x

itv

vABI x ˆ


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Impulse and Momentum• The change in momentum of the element of gas of mass m is

itAvvvmp xˆ

itAvvitv

vAB

pI

xx ˆˆ

• The force from the piston has provided an impulse to the element, which produces a change in momentum

B = bulk modulus of the material = density of the material

/Bv


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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

propertyelasticv


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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

15.2731)m/s3.331( cT

v


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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 /Bv

vk /


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Speed of Sound in Gases, Example Values


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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


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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

2max

2avg 2

1Power sAv


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)(sin

)]sin()][sin([

)]cos([)]sin([

ˆ)],([ˆ]),([Power

22max

2

maxmax

maxmax

tkxAsv

tkxstkxAsv

tkxst

tkxAsv

itxst

iAtxP

• Find the time average power is over one period of the oscillation

2

1

2

2sin

2

1sin

1)0(sin

1

00

2

0

2

T

TT tt

Tdtt

Tdtt

T

• For any given value of x, which we choose to be x = 0, the average value of over one period T is: )(sin 2 tkx


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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

2max

2

2

1svI


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Intensity

• In terms of the pressure amplitude,

v

PI

2

2max

• Therefore, the intensity of a periodic sound wave is proportional to the

• square of the displacement amplitude

• square of the angular frequency

2maxs

2


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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


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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

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avgavg

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PowerPower

rAI

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Topic 5 longitudinal wave