Waves. Introduction

http://www.physicsclassroom.com/Class/waves/U10L1a.html

http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html

 

Waves are everywhere in nature. Our understanding of the physical world is not complete until we understand the nature, properties and behaviours of waves.

Waves are small amplitude perturbations which propagate through continuous media: e.g., gases, liquids, solids, or--in the special case of electromagnetic waves--a vacuum. Wave motion is a combination of oscillatory and translational motion. Waves are important because they are the means through which virtually all information regarding the outside world is transmitted to us. For instance, we hear things via sound waves propagating through the air, and we see things via light waves. Now, the physical mechanisms which underlie sound and light wave propagation are completely different. Nevertheless, sound and light waves possesses a number of common properties which are intrinsic to wave motion itself. In this section, we shall concentrate on the common properties of waves, rather than those properties which are peculiar to particular wave types.

Waves transport energy and momentum through space without transporting matter.

Wave motion involves the transfer of energy. The behaviour of this energy transfer varies with the particular medium of transport and energy form. In general, vibrations propagate in the form of waves. Mechanical waves travel in a material medium, such as a string or a membrane. Acoustic waves travel in fluids, such as air or water.

General Wave Properties

(SIMPLE HARMONIC MOTION: Click here to find out more information about this type of motion)

Travelling Waves

Elasticity and a source of energy are the preconditions for periodic motion, and when the elastic object is an extended body, then the periodic motion takes the form of traveling waves.

A pond has an equilibrium level, and gravity serves as a restoring force. When work is done on the surface to disturb its level, a transverse wave is produced.

A disturbance of the air pressure produces a spherical traveling pressure wave (sound). A sound wave in air is a longitudinal wave.

 

 

\begin{figure}\begin{center} \epsfig{file=figures/mass-spring-etc.eps, width=5in} \end{center} \vspace{-0.25in} \end{figure}

Figure 1: An infinite mass-spring system.

Longitudinal Waves

In longitudinal waves the displacement of the medium is parallel to the propagation of the wave. A wave in a "slinky" is a good visualization.

 

Sound waves in air are longitudinal waves. A single-frequency sound wave traveling through air will cause a sinusoidal pressure variation in the air. The air motion which accompanies the passage of the sound wave will be back and forth in the direction of the propagation of the sound, a characteristic of longitudinal waves.

 

 

Transverse Waves

For transverse waves the displacement of the medium is perpendicular to the direction of propagation of the wave. A ripple on a pond and a wave on a string are easily visualized transverse waves.

Transverse waves cannot propagate in a gas or a liquid because there is no mechanism for driving motion perpendicular to the propagation of the wave.

 

 

The Speed of Waves

The speed of a wave is determined by the medium. In general the speed of the wave can be thought of as being dependent on two properties of the medium

For a string the speed is given by

where T is the tension in the string and m is the mass/ lengyh of the string.

The speed of sound waves in a medium depends on the compressibility and mass density of the medium. The speed of sound can be calculated from the bulk modulus, B, and the density r. (The bulk modulus is a measure of resistance to change in volume of a solid or liquid.

The speed of sound in air is about 343 at 20 degrees C and 1 atm pressure. As the temperature increases the speed of sound in air tends to increase according to

Periodic waves

bohr.physics.hku.hk/academic/ courses/phys2222/Chap2c.ppt

If one end of a long taut string is shaken back and forth in periodic motion, then a periodic wave is generated. If a periodic wave is travelling along a taut string or any other medium, each point along the medium oscillates with the same period.

One-dimensional waves

The essential aspect of a propagating wave is that it is a self-sustaining disturbance of the medium through which it travels. For example, a pulse travels along a stretched string, as shown in figure.

Wave is a function of both position and time, and thus it can be written,

 


 

Represents the shape or profile of the wave at t=0. 

Figure at right  is a “double exposure” of a disturbance taken at the beginning and at time t.

Introduce a coordinate system S’, which travels along with the pulse at the speed v. In this system, Y    is no longer a function of time.  As we move along with S’ we see a stationary profile with the same functional form as previous equation. Here the coordinate is x’, so that

Y= f (x´); with x´= x- vt

Y = f(x-vt)

 

It represents the most general form of the one-dimensional wave function, traveling in the positive x direction.

Similarly, if the wave were traveling in the negative x-direction, the equation would become  Y = f(x+vt)

We may conclude therefore that, regardless of the shape of the disturbance, the variables x and t must appear in the function as a unit, i.e., as a single variable in the form    Y = f(x±vt)

 

 

Harmonic waves

The simplest wave form is a sine or cosine wave, that is called harmonic wave.  The profile is

Y(x,t)t=o=Y(x)= A sin kx = f(x)

Where k is a positive constant known as the propagation number or wave numbrr.   A is known as the amplitude of the wave  Replace x with x-vt, we have a progressive wave traveling to the positive x-direction with a speed of v: Y(x,t) =A sin k(x- vt)

Holding either x or t fixed results in a sinusoidal disturbance, so the wave is periodic in both space and time. The spatial period is the wave length and is denoted by l . The temporal period is the amount of time it takes for one complete wave to pass a stationary observer and is denoted as  T .

From the definition of T and l we have:

         

 

 

 

 

The inverse of the period is called the frequency f, which is the number of cycles per unit of time.  Thus

 

Angular frequency, w,  is defined as the number of radians per unit of time. Thus

 


The harmonic wave can also be written as

The most general harmonic wave equation is 

 


The whole quantity in the bracket is called the phase, f, with e the initial phase. 

The wavelength and the period describe aspects of the repetitive nature of a wave in space and time.  These concepts are equally well applied to waves that are not harmonic, as long as each wave profile is made up of a regularly repeating pattern.

 

 

 

 

The harmonic wave function is doubly periodic

Equation 

describes a pattern of motion which is periodic in both space and time

TIME PERIODICITY

The wave pattern is periodic in time,   y(x,t +nT)= y(x,t)  with period T= 2p/w

The wave period is the oscillation period of the wave disturbance at a given point in space. The wave frequency (i.e., the number of cycles per second the wave pattern executes at a given point in space) is written f= 1/T= w/2p

The quantity   w is termed the angular frequency of the wave. Finally, at any given point in space, the displacement y oscillates between + y0 and - y0 (since the maximal values of sin q are ±). Hence, y0 corresponds to the wave amplitude.

The particle vibrates with the same equation at the instants t, t+T, t+2T……

SPACE PERIODICITY

\begin{figure}\begin{center} \psfig{figure=def2.eps,width=3in}\end{center}\end{figure}The wave pattern is periodic in space, y(x+ nl,t)= y(x,t) with periodicity length  l= 2p/k

Here, l is known as the wavelength, whereas k is known as the wavenumber.

The vibration´s state for the particles at x, x+ l, x+2 l,... is the same.

Consequences:

-          Two points on a wave separated by nl are in phase.Two points on a wave with the same phase have the same ...

o        quantity of disturbance (displacement, etc.)

o        rate of change of disturbance (velocity, etc.).

-          The wave fronts are constant phase surfaces separated by one wavelength. The wave vector is normal to the wave fronts and its length is the wavenumber.

 

 

 

 

Energy and intensity of a wave

http://www.lightandmatter.com/

 

When you throw a pebble in a pond, it makes waves on the surface that move out from the place where the pebble entered the water. The waves are largest where they are formed and gradually get smaller as they move away. This decrease in size, or amplitude, of the waves is called attenuation. Seismic waves also become attenuated as they move away from the earthquake source.

The energy of a wave is related to its amplitude : Em= ½ kA2= ½ mw2A2= ½ m4p2f2A2=2mp2f2A2

So that,  E µ A2

E is the energy of the wave.
A is the amplitude of the wave.

Intensity is defined as the rate of flow of energy through unit area perpendicular to the direction of travel of the wave at the place in question. It can be thought of as an energy flux. For the intensity at a point A due to a source of spherical waves,

I =

E

s.t

 

=

P

4 pr2

µ

A02

r2

I is the intensity at point A.; and P is the power emanated by the source.
r is the distance between the source and point A.
A0 is the amplitude of waves that have just been emitted from the source. (A.r= cte)

A consequence of the above relations is that as waves spread out from a source, their amplitude naturally decreases.

 

Absorption

 

So far we have tacitly assumed that wave energy remains as wave energy, and is not converted to any other form. If this was true, then the world would become more and more full of sound waves, which could never escape into the vacuum of outer space. In reality, any mechanical wave consists of a travelling pattern of vibrations of some physical medium, and vibrations of matter always produce heat, as when you bend a coat hangar back and forth and it becomes hot. We can thus expect that in mechanical waves such as water waves, sound waves, or waves on a string, the wave energy will gradually be converted into heat. This is referred to as absorption.

The wave suffers a decrease in amplitude, as shown in figure. The decrease in amplitude amounts to the same fractional change for each unit of distance covered. That is, the reduction in amplitude is exponential.

Intensity decreases, exponentially, with the distance :  I = I0e-br

Waves propagation: Huygens´Principle

\begin{figure}\epsfysize =3in \centerline{\epsffile{huygens.eps}}\end{figure}Christian Huygen (1629-1695) is said to have gained most of his insights into wave motion by observing waves in a canal.  Huygens was the first major proponent of the wave theory of light.  Huygens had a very important insight into the nature of wave propagation which is nowadays called Huygens' principle. When applied to the propagation of light waves, this principle states that:

Every point on a wave front can be thought of as a new point source for waves generated in the direction the wave is travelling or being propagated.

Every point on a wave-front may be considered a source of secondary spherical wavelets which spread out in the forward direction at the speed of light. The new wave-front is the tangential surface to all of these secondary wavelets.

 

According to Huygens' principle, a plane light wave propagates though free space at the speed of light, $c$. The light rays associated with this wave-front propagate in straight lines, as shown in the diagram below. It is also fairly straightforward to account for the laws of reflection and refraction using Huygens' principle.


REFLECTION AND REFRACTION

Waves travel at a constant speed in a uniform medium. A change in medium, or a change in the condition of a medium, will usually result in a change in the speed of the wave.

A change in the speed of a wave results in a corresponding change in wavelength. The frequency of the wave remains constant once the wave has been generated.  The wavelength of a wave varies directly with the speed for any given frequency in the same medium. l= v/f

Reflection of Sound

The process of wave reflection may be defined as the return of all or part of a sound beam when it encounters the boundary between two media. The most important rule of reflection is that the angle of incidence is equal to the angle of reflection.

The reflection of sound follows the law "angle of incidence equals angle of reflection", sometimes called the law of reflection. The same behavior is observed with light and other waves, and by the bounce of a billiard ball off the bank of a table. The reflected waves can interfere with incident waves, producing patterns of constructive and destructive interference. This can lead to resonances called standing waves in rooms. It also means that the sound intensity near a hard surface is enhanced because the reflected wave adds to the incident wave, giving a pressure amplitude that is twice as great in a thin "pressure zone" near the surface. This is used in pressure zone microphones to increase sensitivity. The doubling of pressure gives a 6 decibel increase in the signal picked up by the microphone. Reflection of waves in strings and air columns are essential to the production of resonant standing waves in those systems.

Plane Wave Reflection

"The angle of incidence is equal to the angle of reflection" is one way of stating the law of reflection for light in a plane mirror. Sound obeys the same law of reflection .

 

 

Point source of sound reflecting from a plane surface.

When sound waves from a point source strike a plane wall, they produce reflected circular wavefronts as if there were an "image" of the sound source at the same distance on the other side of the wall. If something obstructs the direct sound from the source from reaching your ear, then it may sound as if the entire sound is coming from the position of the "image" behind the wall. This kind of sound imaging follows the same law of reflection as your image in a plane mirror.

 

Reflection from Concave Surface

Any concave surface will tend to focus the sound waves which reflect from it. This is generally undesirable in auditorium acoustics because it produces a "hot spot" and takes sound energy away from surrounding areas. Even dispersion of sound is desirable in auditorium design, and a surface which spreads sound is preferable to one which focuses it.

Refraction of Sound

Refraction is the change in direction of a wave due to a change in velocity. It happens when waves travel from a medium with a given refractive index to a medium with another. At the boundary between the media the wave changes direction, its wavelength increases or decreases but frequency remains constant. For example, a light ray will refract as it passes through glass; understanding of this concept led to the invention of the refracting telescope.

The law of  refraction states: v2senai=v1senar

Refraction is the bending of waves when they enter a medium where their speed is different. Refraction is not so important a phenomena with sound as it is with light where it is responsible for image formation by lenses, the eye, cameras, etc. But bending of sound waves does occur and is an interesting phenomena in sound

These visualizations may help in understanding the nature of refraction. A column of troops approaching a medium where their speed is slower as shown will turn toward the right because the right side of the column hits the slow medium first and is therefore slowed down. The marchers on the left, perhaps oblivious to the plight of their companions, continue to march ahead full speed until they hit the slow medium.

Not only does the direction of march change, the separation of the marchers is decreased. When applied to waves, this implies that the direction of propagation of the wave is deflected toward the right and that the wavelength of the wave is decreased. From the basic wave relationship, v=fl, it is clear that a slower speed must shorten the wavelength since the frequency of the wave is determined by its source and does not change.

Another visualization of refraction can come from the steering of various types of tractors, construction equipment, tanks and other tracked vehicle. If you apply the right brake, the vehicle turns right because you have slowed down one side of the vehicle without slowing down the other.

Diffraction of Sound

image:doubleslitdiffraction.pngDiffraction: the bending of waves around small* obstacles and the spreading out of waves beyond small* openings.

(* small compared to the wavelength)

 

Diffraction is a wave phenomenon: the apparent bending and spreading of waves when they meet an obstruction. Diffraction occurs with electromagnetic waves, such as light and radio waves, and also in sound waves and water waves. Diffraction also occurs when any group of waves of a finite size is propagating; for example, a narrow beam of light waves from a laser must, because of diffraction of the beam, eventually diverge into a wider beam at a sufficient distance from the laser. It is the diffraction of "particles", such as electrons, which stood as one of the powerful arguments in favor of quantum mechanics.

It is mathematically easier to consider the case of far-field or Fraunhofer diffraction, where the diffracting obstruction is many wavelengths distant from the point at which the wave is measured. The more general case is known as near-field or Fresnel diffraction, and involves more complex mathematics. As the observation distance in increased the results predicted by the Fresnel theory converge towards those predicted by the simpler Fraunhofer theory.

 

 

Important parts of our experience with sound involve diffraction. The fact that you can hear sounds around corners and around barriers involves both diffraction and reflection of sound. Diffraction in such cases helps the sound to "bend around" the obstacles. The fact that diffraction is more pronounced with longer wavelengths implies that you can hear low frequencies around obstacles better than high frequencies, as illustrated by the example of a marching band on the street. Another common example of diffraction is the contrast in sound from a close lightning strike and a distant one. The thunder from a close bolt of lightning will be experienced as a sharp crack, indicating the presence of a lot of high frequency sound. The thunder from a distant strike will be experienced as a low rumble since it is the long wavelengths which can bend around obstacles to get to you. There are other factors such as the higher air absorption of high frequencies involved, but diffraction plays a part in the experience.

You may perceive diffraction to have a dual nature, since the same phenomenon which causes waves to bend around obstacles causes them to spread out past small openings. This aspect of diffraction also has many implications. Besides being able to hear the sound when you are outside the door as in the illustration above, this spreading out of sound waves has consequences when you are trying to soundproof a room. Good soundproofing requires that a room be well sealed, because any openings will allow sound from the outside to spread out in the room - it is surprising how much sound can get in through a small opening. Good sealing of loudspeaker cabinets is required for similar reasons.

Another implication of diffraction is the fact that a wave which is much longer than the size of an obstacle, like the post in the auditorium above, cannot give you information about that obstacle. A fundamental principle of imaging is that you cannot see an object which is smaller than the wavelength of the wave with which you view it. You cannot see a virus with a light microscope because the virus is smaller than the wavelength of visible light. The reason for that limitation can be visualized with the auditorium example: the sound waves bend in and reconstruct the wavefront past the post. When you are several sound wavelengths past the post, nothing about the wave gives you information about the post. So your experience with sound can give you insights into the limitations of all kinds of imaging processes.

 

 

 

 

Interference of waves

In contrast to refraction, interference can only be naturally explained by thinking of light as waves. Consider two waves meeting as shown in the Figure below:

 

Interference of waves

\begin{figure} \begin{center} \leavevmode \epsfysize=7 cm \epsfbox{figs/light-9.eps} \end{center} \end{figure}

\begin{figure} \begin{center} \leavevmode \epsfysize=6 cm \epsfbox{figs/light-7b.eps} \end{center}\end{figure}
For constructive interference, the waves meet in phase, i.e. so that the crests of each wave coincide.

In destructive interference, the waves meet out of phase, so that the crest of one wave coincides with a trough of the other wave, and they cancel each other out.

This readily explains the double slit interference pattern of Figure. The two light rays start off in phase since they come from the same source. By the time they reach the screen after having been diffracted through the slits, they have travelled different distances, so that the crests of the ray that travels further (the bottom ray in Figure lag a little behind those of the top ray. If they lag behind by a half a wave length, the crest of one ray will meet the trough of the other at the screen, and destructive interference will naturally result.

 On the other hand if one ray lags behind by a full wavelength (or two or three full wavelengths, etc.) a crest will still meet a crest at the screen, and a bright spot will appear due to constructive interference.

Principle of superposition

When two or more waves overlap, the resultant wave is the algebraic sum of the individual waves

Mathematically, when there are two pulses on the string, the total wave function is the algebraic sum of the individual functions.

Superposition is a characteristic and unique property of wave motion. There is no analogous situation in Newtonian particle motion; that is, two Newtonian particles never overlap or add together in this way.

 

 

Interference of two pulses

  

 

;

Consequences:

-          Constructive interference: same phase

o       

-          Destructive interference: opposite phase

o       

Interference of two harmonic waves (same A, same frequency and same wavelength)

The phenomenon of two or more waves of the same, or almost the same, frequency superposing to produce an observable pattern in the intensity is called interference.

The result of the superposition of two harmonic waves of the same frequency depends on the phase difference between the waves.

If the two waves are in phase, then the amplitude of the resultant wave is 2A. The interference of two waves in phase is called constructive interference.

If the two waves are 180º out of phase, then the amplitude of the resultant wave is 0. This interference is called destructive interference.

A common cause of a phase difference between two waves is different path lengths between the sources of the waves and the point of interference.

 

 

Consequences:

-          Constructive interference: MAXIMUM VALUES

o       

-          Destructive interference: NODES

o       

Standing waves

If waves are confined in space, like the waves on a piano string, sound waves in an organ pipe, or light waves in a laser, reflections at both ends cause the waves to travel in both directions. These superposing waves (same A, same frequency and opposite direction) interfere in accordance with the principle of superposition. For a given string or pipe, there are certain frequencies for which superposition results in a stationary vibration pattern called a standing wave.

The necessary conditions for standing waves on a string are:

-          Each point on the string either remains at rest or oscillates with simple harmonic motion. (Those points remaining at rest are nodes).

-          The motions of any two points on the string that are not nodes oscillate either in phase or 1780º out of phase.

Nodes:

Distance between two nodes: l/2

Antinodes:

Distance between two antinodes: l/2

The distance between a node and an adjacent antinode is l/4.

 

 The Doppler Effect

If a wave source and a receiver are moving relative to each other, the received frequency is not the same as the frequency of the source. If they are moving closer together, the received frequency is greater than the source frequency; and if they are moving farther apart, the received frequency is less than the source frequency. This is called the Doppler Effect.

In the following discussion, all motions are relative to the medium. Consider the source moving with speed us, shown in the figure and a stationary receiver. The source has frequency fs (and period Ts). The received frequency fr is: fr= v/l (stationary receiver).

But  to find  fr we need to find l. l= (v±us)Ts= (v±us)/fs

When the receiver moves relative to the medium, the received frequency is different simply because the receiver moves past more or fewer wave crest in a given time. For a receiver moving with speed ur, let Tr denote the time between arrivals of successive crests. Then, during the time between the arrivals or two successive crest, each crest will have travelled a distance v.Tr, and a during the same time, the receive have travelled a distance ur.Tr.

We have fr= 1/Tr= (v±ur)/l

The correct choices for the plus or minus signs are most easily determined by remembering that the frequency tends to increase both when the source moves toward the receiver and when the receiver moves toward the source. For example, if the receiver is moving toward the source the plus sign is selected in the numerator, which tends to increase the received frequency, and if the source is moving away from the receiver the plus sign is selected in the denominator, which tends to decreased the received frequency.

 Resonance

The amplitude, and therefore, the energy, of a system in the steady state depends not only on the amplitude of the driving force, but also on its frequency. The natural frequency of an oscillator  w0, is itrs frequency when non driving or damping forces are present. (In the case of a spring, w0= (k/m)1/2). If the driving frequency is approximately equal to the natural frequency of the system, the system will oscillate with a relatively large amplitude. This phenomenon is called resonance. When the driving frequency equals the natural frequency of  the oscillator, the energy per cycle transferred to the oscillator is maxium. The natural frequency of the system is thus called the resonance frequency.

 

Summary

http://hypertextbook.com/physics/waves/periodic/

 

 
Where to find more information

 

 

http://farside.ph.utexas.edu/teaching/301/lectures/

http://hypertextbook.com/physics/waves/periodic/

http://www.physics.nmt.edu/~raymond/classes/ph13xbook/

http://www.wordiq.com/definition/Wave

http://www3.interscience.wiley.com:8100/legacy/college/cutnell/0471151831/concepts/cs/sim25.htm

http://www3.interscience.wiley.com:8100/legacy/college/cutnell/0471151831/concepts/