Stationary Waves

Stationary waves are produced by superposition of two progressive waves of equal amplitude and frequency, travelling with the same speed in opposite directions.

http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html

Production of Stationary Waves

A stationary wave would be set up by causing the string to oscillate rapidly at a particular frequency.If the signal frequency is increased further, overtone patterns appear.

Properties of a stationary wave (1)

Stationary waves have nodes where there is no displacement at any time.In between the nodes are positions called antinodes, where the displacement has maximum amplitude.

A vibrating loop

N A N A N

VibratorVibrator

Properties of a stationary wave (2)

The waveform in a stationary wave does not move through medium; energy is not carried away from the source.The amplitude of a stationary wave varies from zero at a node to maximum at an antinode, and depends on position along the wave.

Vibrations of particles in a stationary waveAt t = 0, all particles are at rest because the particles reach their maximum displacements.At t = ¼T, Particles a, e, and i are at rest because they are the nodes.Particles b, c and d are moving downward. They vibrate in phase but with different amplitude.Particles f, g and h are moving upward. They vibrate in phase but with different amplitude.

t = 0

t = ¼T

t = ¼T

t = ⅜T

t = ½T

a

bc

d

ef

g

hii

ab c d

ef g h

ii

Properties of a stationary wave (2)

All particles between two adjacent nodes (within one vibrating loop) are in phase.

Video

1. Stationary waves (string)

2. Stationary waves (sound)

Standing waves in a string fixed at both ends.

Normal modes of a string

... 3, 2, 1,n 2

nL

n

... 3, 2, 1,n 2

Lvnvf

nn

Wavelength:

Frequency:

1

... 3, 2, 1,n T2

fnfLnf

n

n

Tv :Using

frequency lfundamenta thecalled is T21

1 Lf

Standing waves in a string fixed at both ends.

f1 is called the fundamental frequency

The higher frequencies fn are integer

multiples of the fundamental frequency

These normal modes are called harmonics.

f1 is the first harmonic, f2 is the second

harmonic and so on…

Modes of vibration of strings

Picture of Standing Wave Name Structure

                                              1st Harmonic

orFundamental

1 Antinode2 Nodes

                                             2nd Harmonic

or1st Overtone

2 Antinodes3 Nodes

                                             3rd Harmonic

or2nd Overtone

3 Antinodes4 Nodes

                                             4th Harmonic

or3rd Overtone

4 Antinodes5 Nodes

                                              5th Harmonic

or4th Overtone

5 Antinodes6 Nodes

L = ½λ1

f1 = v/2LL = λ2

f2 = v/LL = 1½λ3

f3 = 3v/2LL = 2λ4

f4 = 2v/L

L = 2½λ5

f5 = 5v/2L

L

Investigating stationary waves using sound waves and

microwavesMoving the detector along the line between the wave source and the reflector enables alternating points of high and low signal intensity to be found. These are the antinodes and nodes of the stationary waves.The distance between successive nodes or antinodes can be measured, and corresponds to half the wavelength λ.If the frequency f of the source is known, the speed of the two progressive waves which produce the stationary wave can be obtained.

Reflector

DetectorWave source

Resonant Frequencies of a Vibrating String

From the experiment, we find that There is a number of resonant frequencies in a vibrating string,

The lowest resonant frequency is called the fundamental frequency (1st harmonic),

The other frequencies are called overtones (2nd harmonic, 3rd harmonic etc.),

Each of the overtones has a frequency which is a whole-number multiple of the frequency of the fundamental.

Factors that determine the fundamental frequency of a

vibrating string

The frequency of vibration depends on the mass per unit length of the string, the tension in the string and, the length of the string.The fundamental frequency is given by

T

Lfo 2

1 where T = tension

= mass per unit length L = length of string

Vibrations in Air Column

When a loudspeaker producing sound is placed near the end of a hollow tube, the tube resonates with sound at certain frequencies.Stationary waves are set up inside the tube because of the superposition of the incident wave and the reflected wave travelling in opposite directions.

http://www.walter-fendt.de/ph11e/stlwaves.htm

Factors that determine the fundamental frequency of a

vibrating air columnThe natural frequency of a wind instrument is dependent upon The type of the air column, The length of the air column of the instrument.

Open tube Closed tube

Standing Waves in a String, 3This is the first normal mode

that is consistent with the boundary conditions.There are nodes at both ends.There is one antinode in the

middle.This is the longest

wavelength mode: ½ = L so = 2L

The section of the standing wave between nodes is called a loop.In the first normal mode, the

string vibrates in one loop.

Section 18.3

Standing Waves in a String, 4

Consecutive normal modes add a loop at each step. The section of the standing wave from one node to the next is called

a loop.The second mode (c) corresponds to to = L.The third mode (d) corresponds to = 2L/3.

Section 18.3

/4/4

Name

Modes of vibration for an open tube

Picture of Standing Wave Structure

                                             1st Harmonic

orFundamental

2 Antinodes1 Node

                                             2nd Harmonic

or1st Overtone

3 Antinodes2 Nodes

                                              3rd Harmonic

or2nd Overtone

4 Antinodes3 Nodes

                                             4th Harmonic

or3rd Overtone

5 Antinodes4 Nodes

                                             5th Harmonic

or4th Overtone

6 Antinodes5 Nodes

L = ½λ1

f1 = v/2L

L = λ2

f2 = v/L

L = 1½λ3

f3 = 3v/2L

L = 2λ4

f4 =2v/LL = 2½λ5

f5 = 5v/2L

Modes of vibration for a closed tube

Picture of Standing Wave Name Structure

                                              1st Harmonic

orFundamental

1 Antinode1 Node

                                              3rd Harmonic

or1st Overtone

2 Antinodes2 Nodes

                                             5th Harmonic

or2nd Overtone

3 Antinodes3 Nodes

                                             7th Harmonic

or3rd Overtone

4 Antinodes4 Nodes

                                             9th Harmonic

or4th Overtone

5 Antinodes5 Nodes

L = ¼λ1

f1 = v/4L

L = ¾λ3

f3 =3v/4L

L = 1¼λ5

f5 =5v/4L

L = 1¾λ7

f7 = 7v/4L

L = 2¼λ9

f9 =9v/4L

The quality of sound (Timbre)

The quality of sound is determined by the following factors: The particular harmonics present in addition to the fundamental vibration,

The relative amplitude of each harmonic, The transient sounds produced when the vibration is started.

1st overtone Fundamental2nd overtone

3rd overtone

resultant

http://surendranath.tripod.com/Harmonics/Harmonics.html

Chladni’s PlateChladni’s plate is an example of resonance in a plate.There are a number of frequencies at which the plate resonate. Each gives a different pattern.

Resonance. All elastic objects have natural frequencies of vibration that are determined by the materials they are made of and their shapes.

When energy is transferred at the natural frequencies, there is a dramatic increase of amplitude called resonance.

The natural frequencies are also called resonant frequencies.

When the frequency of an applied force, including the force of a sound wave, matches the natural frequency of an object, energy is transferred very efficiently. The condition is called resonance.