11. Waves
Any disturbance that propagates with a well-defined velocity is called a wave.
Waves can be classified in the following three categories:
1.
2.
3.
Mechanical waves: can exist only within a material medium such as air,
water, etc. The medium’s mass and elasticity influence the propagation
properties of mechanical waves.
Electromagnetic waves: involve propagating disturbances in the electric and
magnetic field governed by Maxwell’s equations and do not require a material
medium. Common examples are radio waves, light, x-rays, gamma rays.
Matter waves: associated with all microscopic particles such as electrons,
protons, neutrons, atoms, etc (de Broglie rule).
Transverse
wave – the
disturbance is
perpendicular
to the wave
propagation
velocity.
A
Figures from HRW,2
A wave in a stretched string
A sound wave in a pipe
Longitudinal
wave – the
disturbance is
parallel to the
wave propag.
velocity.
1
The disturbance can be described by  ( x ,t ) function giving its magnitude at point
x and instant t. If the oscillations of a wave’s source are given by the harmonic
function St   A sint then one can write  0,t   St  .
The motion of a given element at a position x and instant t is the same as the
motion of a source at x = 0 and an earlier instant t   t  x / v (v - the wave speed).


x
v


 x ,t    0,t    0 sin t    0 sin   t     0 sin t 
 x ,t    0 sin(t  kx )
where
k

v

2
2

T v

x 

v 
(11.1)
k – wave number, λ – wavelength,
T – period of oscillations
A
A snapshot (fixed instant of time) of a
transverse wave in a string
(in this case ψ(x,t) = y(x,t) in eq.(11.1)).
A wavelength λ measured from an
arbitrary position x1 is indicated.
2
11.1. The phase velocity of a wave
The argument of the sine function in eq.(11.1) is called the phase:
 x ,t   kx   t
(11.2)
To keep a constant phase with increasing time t, it follows from (11.2) that x must
also increase what means that phase (11.2) describes a wave travelling in the positive
direction of coordinate x.
Calculating the differential of a phase φ(x,t) one obtains:


(11.3)
d 
dx 
dt  k dx   dt
x
t
For a constant phase dφ=0, then kdx  dt  0
or dx  v  
dt
k
The quantity vφ (or v) is the phase velocity
(the speed of propagation of a given phase).
Eq.(11.4) can be more generally written as
v k  
(11.4)
 k 
A
A wave moving
with velocity v
(two snapshots
are shown).
k
what is called a relation of dispersion. The phase velocity can be a function of k (or λ)
what means that a wave being a superposition of waves with different k udergoes
dispersion. In this case it changes a shape during propagation, as the constituent3
waves move with different velocities.
11.2. The wave equation
In one dimension the wave equation can be written as
 2
1  2

x 2 v 2 t 2
(11.5)
partial differential equation
Eq.(11.5) is fulfilled by the superposition of waves with constant velocity v.
Now we prove an assumption that eq.(11.5) is obeyed by the superposition of waves
moving in +x and –x directions
(11.6)
 x ,t    0 sink x  vt 
Calculating the derivatives one obtains

 k  v  0 cosk x  vt 
t
 2
 k 2v 2 0 sink x  vt 
2
t

 k 0 cosk x  vt 
x
 2
 k 2 0 sink x  vt 
2
x
Substituting (11.6a) and (11.6b) into (11.5) we get
 k 2v 2 0 sin k x  vt   v 2  k 2  0 sin k x  vt 
what means that eq.(11.5) is obeyed by expression (11.6).
(11.6a)
A
(11.6b)
4
11.3. The principle of superposition
The wave equation is linear. For this type of equations the following theorem obeys:
if ψ1 and ψ2 are solutions of the equation, the function
Figure from HRW,2
c1 ψ1 + c2 ψ2 is also a solution (c1 and c2 are constants).
The direct consequence of the linearity of the wave
equation is the principle of superposition:
Two different waves propagate independently of each
other through a medium; the resulting disturbance
at any point in space at any instant is the superposition
(sum) of the disturbances due to each wave.
If one wave is characterised by a displacement
y1(x,t) and the second by a displacement y2(x,t),
the net wave after overlapping is obtained by
algebraic summation
y x,t   y1 x,t   y2 x,t 
A
The superposition of two pulses
travelling in opposite directions.
Overlapping waves do not alter each other.
5
Interference of waves
The waves with a constant in time phase shift interfere (all types of waves). The
sources sending these waves are called coherent sources. Typical examples are the
sources of radio waves or microwaves. Natural sources of light are incoherent. In this
case to observe the phenomenon of diffraction one has to use a diaphragm with slits
(Young’s experiment, 1801).
The small slit S0 in the first diaphragm makes the
sources S1 and S2 coherent. The waves from sources
S1 and S2 are of equal phases, frequencies and
amplitudes. At point P the waves have different
phases and this difference depends on the position
of point P on the screen. As a result one obtains the
sequence of interference fringes.
The superposition of waves at point P gives
 1   2   0 sinkr1  t    0 sinkr2  t 
A
(11.7)
The phase difference for these waves is
  k r2  r1  
2

r2  r1 
therefore depends on the path length difference r2 – r1.
(11.8)
6
Interference of waves, cont.
For the conditions:
1.   2 n, n  0,1,2,...  we have the greatest possible amplitude – fully constructive
interference. The above condition is equivalent to
2
(11.9)
r  r   2 n or r2  r1  n
 2 1
what means that the difference in path length is an integer number of wavelength.
2.   2n  1 , n  0,1,2,... 
2

r2  r1   2n  1 or
- fully destructive interference. This is equivalent to
r2  r1  2n  1

(11.10)
2
in this case the diference in path length is an odd number of half of wavelength.
For other conditions we have
an intermediate interference.
Example
Interference of two sinusoidal waves
y1=Asin(kx-ωt) and
y2= Asin(kx-ωt+φ) what gives
y = y1+y2 = 2Acos(φ/2)sin(kx-ωt+φ/2).
The resultant waves for three
different phase shifts φ are shown.
A
7
Figure from HRW,2
Interference of waves, cont.
The positions of interference extrema in Young’s experiment are function of the
diffraction angle θ, because the path difference for two rays is equal
  r2  r1  d sin
(11.11)
According to (11.10) the main maxima (bright fringes)
are then obtained for diffraction angles
(11.12)
d sin  n
To determine the light intensity in two-slit interference
we can use phasor (wskaz) diagrams
The wave disturbance can be
represented as a projection of
the vector amplitude rotating
around an origin with the
angular frequency  of the wave.
Interferens fringes for two very
narrow (a << λ) slits. The phase
shift, taking into account (11.11)
can be determined as
 / 2A  r2  r1  /   d sin / 


   1   2   0 sin( kr  t )   0 sinkr  t     2 0 cos sin( kr  t  )   m sin( kr  t   )
2
From the condition that
I   m 


I0   0 
2
one obtains
where
2
I  4I0 cos 2   Im cos 2 


2

d
sin 

(11.13)
8
(11.13a)
Interference of waves, cont.
The intensities of interference fringies in double-slit experiment, calculated from
eqs. (11.13) and (11.13a) are shown in the figure below
I0 - intensity for one source
2I0 - intensity for two incoherent sources
4I0 - intensity for two coherent sources
(vawe intensity is a power passing through
the unit surface area)
Standing waves
We analyze the interference of two waves travelling in opposite directions. From the
superposition principle one obtains
(11.14)
 ( x ,t )   0 sin( kx  t )   0 sin( kx  t )  2 0 sin kx cos t  A' ( x ) cosAt
We obtained the equation of an oscillatory motion with the amplitude A’(x) varying
with position x. This is called a standing wave. The position dependent amplitude
is zero for kx = nπ (nodes) and has a maximum value for kx = (n + ½) π (antinodes).
9
Standing waves, cont.
Figure from HRW,2
Five snapshots at
indicated times of a
wave travelling to the
left, travelling to the
right and the
superposition of both
waves.
antinode
node
A standing wave can be set up by reflecting
a travelling wave from the obstacle.
When reflecting wave looses energy, the standing
wave has no point nodes. In this case the SWR
(standing wave ratio) is defined
SWR 
Amax
A  Ar
 i
Amin
Ai  Ar
Ar- amplitude of a reflected wave
A
Ai- amplitude of an incident wave
Ten snapshots of a standing wave.
Both adjacent nodes and adjacent
antinodes are separated by λ/2.
10
Standing waves, resonance
When we are trying to set up a standing wave in a string clamped on both sides,
only at selected frequencies the interference produces a standing wave . These
frquencies are called resonant frequencies of the system and the phenomenon itself
is known as a resonance.
Generally the resonance occurs when the standing wave
satisfies the boundary conditions of the system.
For the case shown in the figure these conditions are:
the amplitudes at point A and B must be zero (nodes) as at
these points the string is fastened.
The lowest resonant frequency possible in the system is
shown in fig.(a).
L/2
A
B
A
B
A
B
   2L or f  v / 2L
The second standing wave has three nodes and frequency
f v/ L 2
v
2L
A
fig.(b)
Generally for the system of a string clamped on both sides
one obtains
v
fn  n
2L
Figure from HRW,2
11