IB – PHYSICS (AHL)– Wave Phenomena
Topic 11.1 Standing Waves
A standing wave, also known as a stationary wave, is a wave that remains in a constant
position.
Two opposing waves combine to form a standing wave.
The main difference between travelling wave and standing wave is that in the latter no energy or
momentum is transferred.
One characteristic of every standing wave pattern is that there are points along the medium
which appear to be standing still. These points, sometimes described as points of no
displacement, are referred to as nodes. There are other points along the medium which undergo
vibrations between a large positive and and large negative displacement. These are the points
which undergo the maximum displacement during each vibrational cycle of the standing wave.
In a sense, these points are the opposite of nodes, and so they are called antinodes. A standing
wave pattern always consists of an alternating pattern of nodes and antinodes.
Points D,C and B indicate nodes and points A,E and F indicate antinodes.
IB – PHYSICS (AHL)– Wave Phenomena
Characteristics of standing waves:

In stationary waves, there are certain points called nodes where the particles are
permanently at rest and certain other points called antinodes where the particles vibrate
with maximum amplitude. The nodes and antinodes are formed alternately.

All the particles of the medium except those at the nodes, vibrate simple harmonically
with a time period equal to that of the component waves.

The amplitude of vibration increases gradually from zero to maximum from a node to an
antinode.

The medium is split up into segments. The particles in a segment vibrate in phase. The
particles in one segment are out of phase with the particles in the neighbouring segment
by 180o.

Velocity and acceleration of all the particles separated by a distance l, are the same at a
given instant.

In a given segment, the particles attain their maximum or minimum velocity and
acceleration at the same instant.

There is no net transport of energy in the medium.

Compressions and rarefactions do not travel forward as in progressive waves. They
appear and disappear alternately, at the same place.

During each vibration, all the particles pass simultaneously through their mean positions
twice, with maximum velocity which is different for different particles.
Harmonics
Standing wave patterns are produced within the medium when it is vibrated at certain
frequencies. Each frequency is associated with a different standing wave pattern. These
frequencies and their associated wave patterns are referred to as harmonics. A careful study of
the standing wave patterns reveal a clear mathematical relationship between the wavelength of
the wave which produces the pattern and the length of the medium in which the pattern is
displayed. Furthermore, there is predictability about this mathematical relationship that allows
one to generalize and deduce a statement concerning this relationship. To illustrate, consider the
first harmonic standing wave pattern for a vibrating rope as shown below.
The pattern for the first harmonic reveals a single antinode in the middle of the rope. This
antinode position along the rope vibrates up and down from a maximum upward displacement
IB – PHYSICS (AHL)– Wave Phenomena
from rest to a maximum downward displacement as shown. The vibration of the rope in this
manner creates the appearance of a loop within the string. A complete wave in a pattern could be
described as starting at the rest position, rising upward to a peak displacement, returning back
down to a rest position, then descending to a peak downward displacement and finally returning
back to the rest position. The animation below depicts this familiar pattern. As shown in the
animation, one complete wave in a standing wave pattern consists of two loops. Thus, one loop
is equivalent to one-half of a wavelength.
In comparing the standing wave pattern for the first harmonic with its single loop to the diagram
of a complete wave, it is evident that there is only one-half of a wave stretching across the length
of the string. That is, the length of the string is equal to one-half the length of a wave. Put in the
form of an equation:
Now consider the string being vibrated with a frequency that establishes the standing wave
pattern for the second harmonic.
The second harmonic pattern consists of two anti-nodes. Thus, there are two loops within the
length of the string. Since each loop is equivalent to one-half a wavelength, the length of the
string is equal to two-halves of a wavelength. Put in the form of an equation:
The same reasoning pattern can be applied to the case of the string being vibrated with a
frequency that establishes the standing wave pattern for the third harmonic.
The third harmonic pattern consists of three anti-nodes. Thus, there are three loops within the
length of the string. Since each loop is equivalent to one-half a wavelength, the length of the
string is equal to three-halves of a wavelength. Put in the form of an equation:
IB – PHYSICS (AHL)– Wave Phenomena
When inspecting the standing wave patterns and the length-wavelength relationships for the first
three harmonics, a clear pattern emerges. The number of antinodes in the pattern is equal to the
harmonic number of that pattern. The first harmonic has one antinode; the second harmonic has
two antinodes; and the third harmonic has three antinodes. Thus, it can be generalized that the
nth harmonic has n antinodes where n is an integer representing the harmonic number.
Furthermore, one notices that there are n halves wavelengths present within the length of the
string. Put in the form of an equation:
Observe that each consecutive harmonic is characterized by having one additional node and
antinode compared to the previous one. The table below summarizes the features of the standing
wave patterns for the first several harmonics.
Harmonic
No of Nodes
No of Antinodes
1st
2
1
2nd
3
2
3rd
4
3
4th
5
4
5th
6
5
6th
7
6
nth
n+1
n
Pattern
--
Comparison between Travelling and Standing Waves
�Travelling waves
�Standing waves
�The disturbance produced in the medium
travels onward, it being handed over from
one particle to the next. Each particle
executes the same type of vibration as the
�There is no onward motion of the disturbance as
no particle transfers its motion to the next. Each
particle has its own characteristic vibration.
IB – PHYSICS (AHL)– Wave Phenomena
preceding one, though not at the same time.
�The amplitudes of the different particles are
different, ranging from zero at the nodes to
�The amplitude of each partide is the same maximum at the antinodes. All the particles in a
but the phase changes continuously,
given segment vibrate in phase but in opposite
phase relative to the particles in the adjacent
segment.
�No particle is parmanently at rest.
Different particles attain the state of
momentary rest at different instants,
�The particles at the nodes are permanently at
rest but other particles attain their position of
momentary rest simultaneously.
All the particles attain the same maximum
velocity when they pass through their mean
positions.
All the particles attain their own maximum
velocity at the same time when they pass through
their mean positions.
�In the case of a longitudinal progressive
wave all the parts of the medium undergo
similar variation of density one after the
other. At every point there will be a density
variation.
�In the case of a longitudinal stationary wave the
variation of density is different at different points
being maximum at the nodes and zero at the
antinodes.
�There is a flow of energy across every
plane in the direction of propagation.
�Energy is not transported across any plane.
The Doppler Effect:
The Doppler effect (or Doppler shift), named after Austrian physicist Christian Doppler who proposed it
in 1842, is the change in frequency of a wave for an observer moving relative to the source of the wave.
It is commonly heard when a vehicle sounding a siren or horn approaches, passes, and recedes from an
observer. The received frequency is higher (compared to the emitted frequency) during the approach, it
is identical at the instant of passing by, and it is lower during the recession.
IB – PHYSICS (AHL)– Wave Phenomena
Doppler Effect Equation in Sound
Kinds of source:
Doppler effect is true for all kinds of waves that is sound waves and light waves.It may be noted
that
1). Doppler's effect give the change in frequency of vibrations of source and it does not give any
information about the intensity of vibrations of source.
2). For sound waves doppler effect depends on the relative motion of source and observer and
also it depends on whether the source or the observer is moving.
Doppler effect in sound is studied in four parts:
Source moving toward stationary observer
When the source of sound is moving toward you, the pitch you hear is higher than what was
emitted from the source and the wavelength is shorter than emitted.
The equation for the observed frequency of sound when the source is traveling toward you is:
fo = fv/(v − vt)
where





fo is the observed frequency
f is the emitted frequency
v is the velocity of sound
vt is the velocity of the source toward you
v > vt (vt is less than v)
IB – PHYSICS (AHL)– Wave Phenomena
Wavelength
Also, since the velocity of the wave equals the frequency times the wavelength (v = fλ or f =
v/λ), the equation for the observed wavelength when the source is traveling toward you is:
λo = λ(1 − vt/v)
where



λo is the observed wavelength
λ is the emitted wavelength (Greek symbol lambda)
v > vt (vt less than but not equal to v)
Velocity
f you know the resulting frequency, you can find the speed of the source moving toward you:
vt = v(f/fo + 1)
Source moving away from a stationary observer
When the source of sound is moving away from you, the pitch you hear is lower, the frequency is
slower and the wavelength is longer than what was emitted from the source.
Note that the equations are the same as when the source is moving toward you, except that the
"−" sign is replaced by a "+" sign to indicate the change in direction of the source. In the case
of velocity, the "+" sign is replaced by a "−" sign.
Frequency
The equation for the observed frequency of a waveform when the source is traveling away from
you is:
fo = fv/(v + va)
where va is the velocity of the source away from you.
Wavelength
The equation for the observed wavelength when the source is traveling away from you is:
λo = λ(1 + va/v)
IB – PHYSICS (AHL)– Wave Phenomena
Velocity
The equation for the velocity of the source, when it is traveling away from you is:
va = v(f/fo − 1)
2. Source at Rest and observer in motion:
Suppose S is a sound-source at rest and O is observer. The true frequency of source is
n. if observer is also at rest, he will hear n waves per second. Therefore, true wavelength .But the
source is moving with velocity of observer.
The number of waves in distance is λ'.The total number of waves received by observer in one
𝑣
second.The apparent frequency equation is fo = fs(1 + )
𝑐
3. Source and observer both in motion:
Suppose source and observer both are moving with velocities vs and v0 respectively in
the direction in which waves are being received. There will be change in frequency.
The apparent frequency equation is n' =