Oscillation and waves lecture notes

1. 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

2. Topic 2-3 Longitudinal Waves
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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UEEP1033 Oscillations and Waves
Speed of Sound in Gases, Example Values

17. Topic 2-3 Longitudinal Waves
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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
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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
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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
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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
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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
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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
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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
==