Superposition of waves
• What happens about the displacement of a particle in a medium
if two or more waves propagate simultaneously in the medium ?
• The resultant displacement of a particle will be according to
principle of superposition of waves.
Principle of Super position of wavesPrinciple of super position of wave states that when two or more waves of same kind meet (superimpose) each
other, the displacement of the resultant wave is equal to the vector sum of displacement of individual waves.
Though waves are superimposed the
identity of each wave is maintained and
become prominent when they get
separated.
Y = Y1 ± Y2 ± Y3 ±……….
Analogy: crowd in Ason
https://www.youtube.com/watch?v=0JcCppja3VA
https://www.youtube.com/watch?v=LgJStYrk2fc
Application of superposition in wave phenomenon
• when two waves of same frequency and comparable amplitude
moving in the same direction superpose, interference of the wave is
formed.
• when two identical waves moving in the opposite direction are
superposed , stationary wave is formed.
• when two waves of slightly different frequency are travelling in the
same direction superpose, beats are formed.
Stationary wave
• A stationary wave ( standing wave ) is formed when two waves having
equal frequency, equal amplitudes, equal velocity but moving in the
opposite directions are superimposed.
• As two identical waves travelling in opposite direction in a confined
medium, there is no propagation of waves hence energy is localized in
the medium so wave is called standing or stationary wave.
Formation of stationary waves
Formation of stationary waves
(Graphical representation)
Nodes and anti nodes
https://www.youtube.com/watch?v=_TXl4CctTfs
https://www.youtube.com/watch?v=-HW8JcL8wms
String
Characteristics of stationary waves
1.
2.
Nodes and antinodes are alternately formed in the wave.
Node is the position of particle whose amplitude of oscillation is
minimum( zero) means particles are at rest.
3.
Antinode is the position of particle whose amplitude of
oscillation is maximum.
4.
All particles except nodes vibrate with shm whose period is
equal to the period of wave.
The amplitude of vibration gradually increased from zero at nodes to
maximum at antinodes.
5.
6.
7.
Energy of the wave does not transfer from one point to
another. It remains localized in the medium .
λ
The distance between two adjacent nodes is and the
2
λ
distance between two adjacent antinodes is also .
2
8.
The velocity and acceleration of any two particles
separated by distance λ is same at any instant of time.
Situations where stationary waves are formed
• 1. Guitar strings.
• 2. Resonance tube
• 3. Both ends open tube
• 4. stretched rubber strings
Differences between progressive wave and stationary waves
Progressive wave
Stationary wave
1. Energy is transferred from one point to another
point.
1. Energy is localized in a particular place, it does
not transfer.
2. All particles oscillates with same amplitudes.
2. Particles have different amplitudes. Maximum at
antinodes and zero at nodes.
3. At every point there is change in pressure.
3. Pressure variation is maximum at nodes and
zero at antinodes.
4. Regular phase difference exists between
successive particles.
4. All the particles between any two successive
nodes are in the same phase.
5. The value of maximum velocity for all particles of
the medium is same.
5. The maximum velocity for different particles are
different. The velocity of particle at node is
always zero.
Stationary waves in a stretched string ( Melde’s Experiment)
Stationary waves in stretched strings fixed at two ends
Transverse wave can be produced in a stretched string by plucking it. If
both ends are fixed, the stationary wave is produced in the string. The
nodes are always formed at the fixed ends. L is the length of the string.
Fig. 1
(i) The minimum frequency that can produced the stationary wave as
shown in the figure-1 is called fundamental frequency 𝑓1 .
Now, the relation between wavelength and length of wire is,
λ1
2
Fig. 2
𝑉
=𝐿
V = λ f,
𝑉
So, 𝑓1 = λ = 2𝐿
1
(Lowest possible frequency or frequency of first harmonic)
(ii) Similarly, if frequency of vibration is gradually increased, two loops are
formed as shown in fig. 2. at new frequency 𝑓2 .
λ2 = L,
𝑉
𝑉
𝑉
𝑓2 = λ = 𝐿 = 2 (2𝐿)= 2 𝑓1 ,
(Second harmonic)
2
(iii) Similarly, if frequency of vibration is further increased, three loops are
formed as shown in fig. 3. at new frequency 𝑓3 .
3λ3
2
Fig. 3
=𝐿
𝑉
3𝑉
𝑉
𝑓3 = λ = 2𝐿 = 3 (2𝐿)= 3 𝑓1 ,
3
(Third harmonic)
https://www.youtube.com/watch?v=-k2TuJfNQ9s
String holding by
two people
Stationary waves formed in a organ pipe
(i) One end closed pipe
Organ pipes are hollow metallic or wooden tubes which
are used to produce the musical sound.
Only odd harmonics are present so quality of sound is
poor than in both ends opened pipe.
Fundamental tone
Or
First harmonic
https://www.youtube.com/watch?v=g1rw6gYRB6M
third harmonic
Or
First overtone
Graph
Resonance
Fifth harmonic
Or
Second overtone
seventh harmonic
Or
Third overtone
Diagrams show how amplitude changes from
antinode to node.
(i) Both ends opened pipe
First mode of vibration
Has minimum frequency(f1) called fundamental frequency. It is also known as first harmonic
Second mode of vibration
Has double frequency (2 f1) of fundamental frequency. It is also known as second harmonic or
first overtone.
Third mode of vibration
Has Tripled frequency (3 f1) of fundamental frequency. It is also known as third harmonic or
second overtone.
All harmonics are present so quality of sound is better
than in one end closed pipe.
Pressure variation in pipes
At the sites of antinodes all molecules are moving in the same direction so the spacing among the molecules
remain constant that causes no change in pressure bur at nodes particles on both sides sometimes comes
closers ( compact) and some times go apart ( rarefied) which causes the large change in pressure.
https://www.aphysicslens.com/pressure-variation-in-stationary-sound-waves/
https://www.walterfendt.de/html5/phen/standingwavereflection_en.htm