Standing Waves

10.2Superposition of Waves Node Antinode
(a) State the principle of superposition of waves for •a point at which •a point at which
the displacement the displacement
the constructive & destructive interferences is zero where the is maximum where
destructive the constructive
Principle of superposition interference interference
whenever two or more waves are travelling occurred occurred

in the same region, the resultant (b) Use the stationary wave equation:
displacement at any point is the vector sum = 

of their individual displacement at that Characteristics of stationary waves
point

Interference
interaction (superposition) of two

or more wave motions

Constructive Destructive – The distance between adjacent 
nodes or antinodes is 2
The resultant The resultant 
displacement displacement < – The distance between a node and 4
displacement an adjacent antinode is
is >
displacement of the –  = 2  (the distance between
individual adjacent nodes orantinodes)
of the wave or equal
individual
to zero
wave    ,3 ,5 ,7 ,...
  0,2 ,4 ,6 ,...

 the formation of stationary wave A cos kx

 Stationary wave is defined as a form of  Determine the amplitude for any
wave in which the profile of the wave point along the
does not move through the medium stationary wave

 It is formed when two waves which are  It is called the amplitude formula and
travelling in opposite directions, and depends on the distance, x
which have the same speed, frequency
and amplitude are superimposed =  , =

sin t

y(x,t)= 2a cos kx sin t

Determine the time for antinodes and
nodes will occur in the stationary wave

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(c) Discuss progressive waves and stationary wave Example 10.5: A stationary wave is represented by

Progressive wave VS Stationary wave the following expression:

Progressive wave Stationary wave = 5 cos  sin 

Wave profile move Wave profile does where y and x in centimetres and t in seconds.
not move
All particles vibrate Determine
with the same Particles between two
amplitude adjacent nodes vibrate (a) the three smallest value of x (x > 0) that
Neighbouring with different amplitudes
corresponds to
particles vibrate with Particles between two
different phases adjacent nodes (i) nodes
vibrate in phase
All particles (ii) antinodes
vibrate Particles at nodes
do not vibrate at (b) the amplitude of a particle at
Produced by a
disturbance in a all (i) x = 0.4 cm

medium Produced by the (ii) x = 1.2 cm
superposition of two waves
Transmits the moving in opposite direction (iii) x = 2.3 cm
energy SOLUTION
Does not transmit (a) Compare with general equations for standing
the energy
wave:
Example 10.4: Two harmonic waves are
= 5 cos  sin 
represented by the equations below
( , ) = 3 sin( +  ) = cos sin 
( , ) = 3 sin( −  ) 2

where y1, y2 and x are in centimetre and t in =  = ,  = 2
(i) Nodes  particles with minimum
seconds.
displacement, y = 0
(a) Write an expression for the new wave when 0 = 5 cos 

both waves are superimposed.  = −1(0)
 3 5
(b) Determine the amplitude of the new wave.
SOLUTION  = 2 , 2 , 2 , …
(a) Apply the principle of superposition: = 0.5 , 1.5 , 2.5 , …

= 1 + 2 = 3 sin( +  ) + 3 sin( −  )
= 6 cos  sin  , = (4)  ℎ = 1, 3, 5, …

(b) From the expression, A = 6 cm = 0.5 , 1.5 , 2.5 , …
(ii) Antinodes  particles with maximum

displacement, y = A = 5 cm

5 = 5 cos 
 = −1(1)

 = 0, , 2, 3, …

= 1 , 2 , 3 , …


, = ( 2 )  ℎ = 0, 1, 2, 3, …
= 1 , 2 , 3 , …

(b) Apply the amplitude formula of stationary wave:

= cos = 5 cos 
(i) x = 0.4 cm

= 5 cos (0.4) , = 1.55
(ii) x = 1.2 cm, A = – 4.05 cm
(iii) x = 2.3 cm, A = 2.94 cm

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Example 10.6: An equation of a stationary wave is EXERCISE 10.2
1. Figure 10.18 shows a graph of displacement,
given by the expression below
y against distance, x for a stationary wave at
= 8 cos 2 sin  time, t where N represents the node and A
represents the antinode.
where y and x are in centimetres and t in seconds.
On the same axes, sketch the graph of
Sketch a graph of displacement, y against displacement, y against distance, x for a
0.25 cycle and 0.5 cycle later.
distance, x at t = 0.25T for a range of 0 ≤ x ≤ . 2. The expression of a stationary wave is given
by
SOLUTION
= 0.3 cos 0.5  sin 60 
Compare with general equations for standing wave: where y and x in metres and t in seconds.
(a) Write the expression for two progressive
= 8 cos 2 sin  waves resulting the stationary wave above.
(b) Determine the wavelength, frequency,
= cos sin  amplitude and velocity for both progressive
waves.
2
=  = 2,  = 1 ANS: 4 m, 30 Hz, 0.15 m, 120 m s1
2 3. A harmonic wave on a string has an
= = , = 2
amplitude of 2.0 m, wavelength of 1.2 m and
speed of 6.0 m s1 in the direction of positive
: = ( 2 ) , = 0, 1, 2, 3, … x-axis. At t = 0, the wave has a crest (peak) at
= 0, 0.5 , 1 , … x = 0.
(a) Calculate the period, frequency, angular
: = (4) , = 1, 3, 5, … frequency and wave number.
= 0.25 , 0.75 , 11.25 , … (b) Explain the motion of the wave in
mathematical equation.
Displacement at x = 0, t = 0.25(2) = 0.5 s is ANS: 0.2 s, 5 Hz, 10 rad s1 ,5.23 m1; Hint :

= 8 cos 2(0) sin (0.5) = 8 wave function

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