Waves

Waves

Waves- By Aditya Abeysinghe 1

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A wave allows energy to be transferred from one point to another some distance away without any particles of the medium travelling between the two points.

E.g.:

2Waves- By Aditya Abeysinghe

Characteristics of a waveLet’s take a simple example.

You may have seen that there is a repetition of the shape and the position of particles over a certain distance.


Displacement

Distance

Crest

Trough

Wave length (λ)

amplitude

The distance between two such particles are to be said be in the same phase and the distance between these two particles is called the wave length. The maximum height achieved from the median position is called the amplitude of the wave.

Thus, the distance between corresponding points in successive waveforms, such as two successive crests or twosuccessive troughs , is called the wavelength, λ.

Within a single vibration of this wave, the waveform moves a distance λ.


So, in one second, when f vibrations occur, the waveform moves a distance fλ. So, the distance travelled in unit time is fλ

However, by the definition of speed,

Speed = distance travelled in unit time.

So, the speed of the wave = fλ

Therefore, V = fλ

This relationship between V, f and λ is true for any type of wave , i.e. mechanical, sound or electromagnetic.


Types of wavesThere are various types of waves. Some types are visible while others are invisible. Some waves are tangible while others are intangible.

However, in this presentation I have focussed only on the mechanical waves. Mechanical waves, like sound waves, need a medium of propagation.

Depending on the characteristics of waves, mechanical waves can mainly be divided to two types as :

1. Transverse waves

2. Longitudinal waves


Transverse wavesA wave which is propagated by vibrations perpendicular to the direction of travel of the wave is called a transverse wave.

Some of the examples of transverse waves are:


The propagation of a transverse wave can be illustrated as

follows:


Displacement

Distance

Trough

Longitudinal WavesA longitudinal wave is a wave in which the vibrations occur in the same direction as the direction of travel of the wave.

Some of the examples of longitudinal waves are:


The propagation of a longitudinal wave can be illustrated as

follows:


Rarefaction Rarefaction RarefactionCompression Compression

distance

11Progressive waves

Progressive waves are the waves in which particles travel along with the speed of the wave. As opposed to progressive waves, stationary waves (which will be described later in this presentation) also move along the speed of the wave.

However, in a stationary wave, the waveform is reflected back along the direction of initial propagation, after travelling some distance.

In a progressive wave, the waveform is never reflected back.

Furthermore, if you are interested in applying the general equation of speed for a wave (V = fλ) , you can apply it only for a progressive wave. This is due to the fact that we are considering the whole motion of the wave within a given time. (In a stationary wave, this might not be so as within the time interval specified, the wave might have reflected back!!)

Waves- By Aditya Abeysinghe

Principle of superpositionPrinciple:

The resultant displacement at any point is the sum of the separate displacements due to the two waves.

Used for:

When two waves travel through a medium, their combined effect at any point can be found by the principle of superposition.

Consider two waves in two occasions, where the amplitudes are similar and dissimilar.

(i) When the amplitudes are similar-

When the amplitudes are similar but opposite in direction, the total displacement of the wave at any point of similar phase is zero.

See the diagram:


(ii) When the amplitudes are dissimilar-

When the amplitudes are dissimilar but opposite in direction, the total displacement of the wave at any point of similar phase is not zero.

See the diagram:


(iii) When the amplitudes are either similar or dissimilar, but the two waves are travelling in the same direction:


The two waves travel in the same direction

The two waves meet at some point

This results in an increase of amplitude. The amplitudes of the two individual waves are added up. This results in an unstable equilibrium

Finally, stability occurs when the two waves start travelling in opposite ways.

Stationary or Standing wavesConsider the following apparatus:

When the mass is kept constant, the tension of the string is constant. Furthermore, the pulley acts as a barrier for the further propagation of the wave.

However, the vibrator continously produces vibrations on the string surface. Therefore, at the pulley end, the wave returns or is reflected along the initial direction of propagation.


Original wave

Reflected wave

Pulley

Mass

Light string

Vibrator vibrating at constant frequency

This time the wave-like profile on the string does not move along the medium, and the wave is therefore called a stationary (or standing) wave.

The stationary wave is due to the superposition of two waves of equal frequency and amplitude travelling in opposite directions along the string.

The figure shows how the motion or the behavior or the appearance of a stationary wave changes with time.


Waves- By Aditya Abeysinghe17

t = 0

t = T/8

t = T/4

t = 3T/8

t = T/2

Properties of stationary or standing waves

Consider the stationary wave below:

The following points are important in understanding the behavior of a standing wave:

1. There are points where the displacement is permanently zero. These points are called the nodes of the stationary waves.

2. At points between successive nodes the vibrations are in phase.


Original wave

Reflected wave

3. Each point along the wave has a different amplitude of vibration from neighboring points. Points with the greatest amplitude are called antinodes.

4. The wavelength, λ, of any type of stationary wave is twice the distance between successive nodes or successive antinodes. Thus, the distance between a node or an antinode and the next node or the antinode is λ/2 and the distance between a node and a neighboring antinode is λ/4.

Note:

The second and third points are in sharp contrast to the behavior of a progressive wave, where the phase of points near each other are all different and every point vibrates with the same amplitude.


Stationary longitudinal wavesStationary longitudinal waves can be studied when considering the wave patterns inside a closed pipe.

In a closed pipe, the displacement of the particles near the closed end should be zero and the displacement near the open end should be maximum. So, the node of the wave formed inside a closed pipe is on the closed end and the antinode is near the open end.


Node Antinode

Stationary transverse wavesStationary transverse waves behave in a similar vein to that of stationary longitudinal waves.

Stationary transverse waves can be observed when a string is tied at both ends and a vibration is made on one of its ends.


Node Node

Antinode

Pressure in stationary wave

Consider the diagram below.

At the node, the particles on either side produce a compression (increase of pressure), from the direction of their displacement. At the same time, the particles near an antinode are zero. Thus, the pressure is normal (decrease of pressure)


displacement

N N N N NA A A A

pressure

N N N N NA A A A

Normal pressure

Date post:	16-May-2015
Category:	Education
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Waves

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