Longitudinal Waves
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.
v

T
 f
Transverse vs. longitudinal wave
Both propagate from left to right, but cause disturbances in
different directions, Dy and Dx.
Dy (t )  Ay sin( t )
wavelength, 
Ay
amplitude
Dx(t )  Ax sin( t )
wavelength, 
Ax
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?
y
x

• Features to incorporate:
in any point in space the wave produces harmonic oscillations of a type:
y (t )  Ay cos(t   )  - angular frequency

- phase
if we “freeze” the wave, we will see a harmonic function in space
y ( x)  Ay cos( kx   )
what is this
k
if we freeze the wave and move 1 wavelength
to see the same level of disturbance y
?

along it, we are supposed
k  2 so that
y ( x   )  Ay cos( k ( x   )   )  Ay cos( kx  2   )  y ( x)
Therefore, it must be
y
x

y (t )  Ay cos(t   )
 - angular frequency
k
y ( x)  Ay cos( kx   )

- phase
k  2  k  2 / 
the wave number measured in m-1. What is the meaning of it?
If we freeze the wave and ride along it, we periodically will bump into crests.
k / 2
tells us how many times per meter it is going to happen
 / 2  f
k
k
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

is pretty much the same for space as
is for time!
tells us the phase change per meter and is sometimes called the
spatial frequency
y
x
y (t )  Ay cos(t   )
  2 / T
T
is period in time

y ( x)  Ay cos( kx   )
k  2 / 
 is period in space
How do we unite the two equations (in time and in space)?
y ( x, t )  Ay cos(t  kx)
Considering only one point in space, x0 , means taking    kx0
y ( x0 , t )  Ay cos(t   )
Freezing it in time, t 0 , means taking   t 0
y ( x, t0 )  Ay cos( kx   )
y
x

y ( x, t )  Ay cos(t  kx)
- equation of a harmonic wave
  2 / T  2 f
k  2 / 
  v/ f
k  2 f / v   / v
  kv
y( x, t )  Ay cos( kx  t )  Ay cos( kx  kvt) 
 Ay cos[ k ( x  vt)]
k ( x  vt)  0
Therefore position of the crest is given by x  vt
A crest corresponds to a point, where
y ( x, t )  0.1cos (3t  2 x)
- equation of a harmonic wave
on a string. All in SI units.
What is the maximal velocity of an element of the string?
k 2
3
the wave speed
v   / k  2/ 3
m/s
Velocity of an element of the string:
dy ( x, t )
v ( x, t ) 
 0.3sin (3t  2 x)
dt
Maximal velocity of an element of the string: 0.3 m/s
y ( x, t )  Ay cos( kx  t )
- equation of a harmonic wave
y ( x, t )  Ay cos[ k ( x  vt)]
- the same equation rewritten
in a form emphasizing
propagation and wave speed
y ( x, t )  Ay cos[ k ( x  vt)]
- what would this one stand for?
v is changed to  v , which means that the wave is propagating in
the negative x-direction, from right to left
In this case, the location of a crest is given by
x  vt  0  x  vt
cos[ k ( x  vt)]  1
How can we describe a pulse? (not a harmonic wave)
Generic equation for a wave traveling in
positive x-direction with wave speed v:
Here
f (x)
y( x, t )  f ( x  vt)
can be ANY function. The type of the function
f (x)
specifies the shape of the wave.
How do we know it is a propagating (traveling) wave?
y (the disturbance) depends on x and t in a VERY SPECIAL WAY:
it only depends on x  vt
Therefore the disturbance y is the same as long as
is constant, say x  vt  x0
x  vt  x0
 x  vt  x0
x  vt
y( x, t )  y( x0 ,0)
A point of constant disturbance, y(x0) , (crest, trough, etc.) moves at the
wave speed, v
Example: a bell-shaped (Gaussian)
curve with a peak at x  0
y( x)  f ( x)  exp(  x )
2
A bell-shaped (Gaussian) curve with
a peak at x  a
y( x)  exp[ ( x  a) ]
2
What if the peak is moving along the xaxis with a speed v ?
a  vt and get
2
y( x, t )  f ( x  vt)  exp[ ( x  vt) ]
We can plug in
The answer we arrived at:
y( x, t )  f ( x  vt)  exp[ ( x  vt) ]
2
How do we understand it?
x  vt0
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?
y ( x, t0 )  exp[ ( x  vt0 ) 2 ]
a Gaussian function with maximum at x  vt0
It is
As usual for a wave, the position of the maximum is given by
xmax  vt
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.