in a longitudinal wave the particle displacement is parallel to the direction of wave propagation....
TRANSCRIPT
In a longitudinal wave the particle displacement is parallel to the direction of wave propagation.
The animation above shows a one-dimensional longitudinal plane wave propagating down a tube.
The particles do not move down the tube with the wave; they simply oscillate back and forth about their individual equilibrium positions. Pick a single particle and watch its motion!
The wave is seen as a motion of compressed regions (i.e. it is a pressure wave), which move from left to right.
Longitudinal Waves
fT
v
Transverse vs. longitudinal waveBoth propagate from left to right, but cause disturbances in different directions, y and x.
)sin()( tAty y
)sin()( tAtx x
wavelength,
wavelength,
amplitudeyA
xA amplitude
Normally the amplitudes of (harmonic) motion of the particles are much smaller than the wavelength.
Longitudinal spring waves
Waves on a Spring
A spring can support both longitudinal and transverse waves.
A wave does not have to be either purely longitudinal or purely transverse. It could be a linear combination of the two.
Harmonic waves are not the only possible type of waves!
A wave can also have a shape of a propagating pulse. True for both transverse and longitudinal waves.
A harmonic wave and a pulse are extreme cases.The intermediate case is a wave train – a finite duration sinusoidal.
How do we describe a harmonic wave mathematically?
• Features to incorporate:
in any point in space the wave produces harmonic oscillations of a type:
x
y
)cos()( tAty y
if we “freeze” the wave, we will see a harmonic function in space
)cos()( kxAxy y
- angular frequency - phase
what is this ? kif we freeze the wave and move 1 wavelength along it, we are supposed to see the same level of disturbance
y
Therefore, it must be so that 2k)()2cos())(cos()( xykxAxkAxy yy
x
y
)cos()( tAty y
the wave number measured in m-1. What is the meaning of it?
)cos()( kxAxy y
- angular frequency - phase
If we freeze the wave and ride along it, we periodically will bump into crests.
2/k tells us how many times per meter it is going to happen
tells us how many times per second we are going to fill a crest if we do not move but rather bob on the wave
2k /2 k
k
f 2/
is pretty much the same for space as is for time!k k tells us the phase change per meter and is sometimes called the
spatial frequency
x
y
)cos()( tAty y
is period in time
)cos()( kxAxy y
T/2
How do we unite the two equations (in time and in space)?
/2k
Considering only one point in space, , means taking
T is period in space
)cos(),( kxtAtxy y
0x 0kx)cos(),( 0 tAtxy y
Freezing it in time, , means taking 0t 0t )cos(),( 0 kxAtxy y
x
y
fT 2/2
- equation of a harmonic wave
/2k
)cos(),( kxtAtxy y
A crest corresponds to a point, where
)](cos[
)cos()cos(),(
vtxkA
kvtkxAtkxAtxy
y
yy
fv / vvfk //2
0)( vtxkTherefore position of the crest is given by vtx
kv
- equation of a harmonic wave on a string. All in SI units.
)23(cos1.0),( xttxy What is the maximal velocity of an element of the string?
2k3
the wave speed 3/2/ kv Velocity of an element of the string:
)23(sin3.0),(
),( xtdt
txdytxv
Maximal velocity of an element of the string: 0.3 m/s
m/s
- equation of a harmonic wave)cos(),( tkxAtxy y
)](cos[),( vtxkAtxy y - the same equation rewritten in a form emphasizing propagation and wave speed
)](cos[),( vtxkAtxy y - what would this one stand for?
is changed to , which means that the wave is propagating in
the negative x-direction, from right to leftv v
In this case, the location of a crest is given by 1)](cos[ vtxkvtxvtx 0
How can we describe a pulse? (not a harmonic wave)
Generic equation for a wave traveling in
positive x-direction with wave speed v:)(),( vtxftxy
Therefore the disturbance is the same as long as is constant, say
00 xvtxxvtx A point of constant disturbance, y(x0) , (crest, trough, etc.) moves at the
wave speed, v
Here can be ANY function. The type of the function
specifies the shape of the wave.
)(xf )(xf
How do we know it is a propagating (traveling) wave?
(the disturbance) depends on and in a VERY SPECIAL WAY:
it only depends on
y x tvtx
y vtx 0xvtx
)0,(),( 0xytxy
Example: a bell-shaped (Gaussian) curve with a peak at 0x
)exp()()( 2xxfxy A bell-shaped (Gaussian) curve with a peak at ax
])(exp[)( 2axxy What if the peak is moving along the x-axis with a speed ?vWe can plug in and getvta
])(exp[)(),( 2vtxvtxftxy
The answer we arrived at:
])(exp[)(),( 2vtxvtxftxy How do we understand it?
y(x,t) is the value of the disturbance at
the point and time of interest, x, and t
How does the profile of the disturbance
y(x,t) look at time t0?
It is ])(exp[),( 200 vtxtxy
a Gaussian function with maximum at 0vtx
0vtx
As usual for a wave, the position of the maximum is given by
vtx max
The position of the maximum, the crest, moves at the wave speed, v
Wave on a string
Any way to calculate the wave speed? What is it likely to depend on?
Amplitude of the wave? Wave length? Mechanical properties of the string?
All of those options are plausible, but it turns out the wave speed only depends on mass of the string (rope) and its tension.
Wave on a string
Waves on a string resemble very much harmonic oscillations of a mass on a spring.
Tension provides the restoring force, which wants to make the oscillations more frequent.
Mass of the string provides the inertia, which slows down both the oscillations and wave propagation.