WAVES

1. WAVES B.SESHA SAI [P.G.T,PHYSICS] K.V, EOI KATHMANDU

2. WAVE MOTION • Wave motion is a form of disturbance that travels through a medium due to the repeated periodic motion of the particles of the medium about their mean positions. Here the motion of each particle is being handed over from one particle to the other.

3. TYPES OF WAVES • Mechanical and Electromagnetic waves: • The waves, which can be produced and propagated only in a material medium, are called elastic or mechanical waves. • Eg: Sound waves, waves on water surface etc. • The waves, which do not require a material medium for their propagation, are called electromagnetic or non-mechanical waves. • Eg: Light waves, X – rays, radio waves etc.

4. Transverse & longitudinal waves: (based onthe nature of propagation) Transverse wave: • It is a wave in which particles of the medium execute simple harmonic motion about their mean positionin a direction perpendicular to the direction of propagation of the wave. They travel through the medium in the form of crests and troughs. • Eg. Stretched string of sitar violin etc, Ripples in water. & all electromagnetic waves • Note: For such waves to propagate, the material medium must possess properties of elasticity (to restore the original position after being disturbed), inertia (to store energy and thus to overshoot the mean position) and very small friction or resistance (to ensure minimum loss of energy so that the wave can travel long distance).

7. CREST AND TROUGH • Crest is a portion of the medium, which is raised temporarily above the normal positions of the rest of the particles of the medium, when a transverse wave passes through it. • Trough is a portion of the medium, which is depressed temporarily below the normal positions of the rest of the particles of the medium, when a transverse wave passes through it. • Note: The center of crest or trough is the maximum displacement of the particle and so is called amplitude

8. Longitudinal wave • It is a wave in which particles of the medium vibrate about their mean position in a direction parallel to the direction of propagation of the wave. They travel through the medium in the form of compressions and rarefaction. • Eg : sound wave in air, vibration of air column in organ pipe etc

10. COMPRESSION AND RAREFACTION • A compression is a region of the medium in which particles come to a distance less than the normal distance between them causing a temporary decrease in volume and a consequent increase in density of the medium • A rarefaction is a region of the medium in which particles get apart to a distance greater than the normal distance between them causing a temporary increase in volume and a consequent decrease in density of the medium

11. WAVELENGTH () • For a transverse wave, it is the distance between two consecutive crests or troughs and for longitudinal wave it is the distance between two consecutive compressions or rarefactions. • OR • It can also be the distance traveled by the disturbance (or wave) in the time , the particle of the medium completes one vibration. • OR • It is also the minimum distance between two particles in the medium, which are in the same phase of vibration. • Note: It is measured in meters.

12. FREQUENCY () AND VELOCITY OF WAVE (v) • Frequency (): • It is the number of waves passing a particular point of the medium in unit time. • Velocity of wave (v): • It is the distance travelled by the wave in unit time. • Relation between wave velocity, frequency and wave length • As wavelength () is the distance traveled by the wave when the particle of the medium completes one vibration (in time T), • We have velocity v=

15. CHARACTERISTICS OF WAVE MOTION • i) Wave motion is disturbance that travel through a medium. • ii) Waves transfer only energy and momentum not particles. • iii) A material medium is essential for the propagation of mechanical waves and the medium must have elasticity, inertia and less friction • iv) There is a continuous phase difference between the particles of the medium, as each particle start vibrating a little later than the previous particle • v) Velocity of vibrating particle will be different at different positions but the velocity of the wave will be the • same through out a particular medium.(If the medium varies , wave velocity will also vary)

16. Sound waves: • Sound is a mechanical, longitudinal wave. Its velocity in air at STP is roughly 335m/s . Speeds greater than that of sound are called supersonic speed. As air possess volume elasticity (bulk modulus), crust and trough cannot be sustained in air and thus sound cannot travel in the form of transverse waves. • Note: Two persons n the surface of moon cannot talk each other, as the moon has no atmosphere through which sound would travel.

17. SPEED OF WAVE MOTION • i) Velocity of transverse wave: • a) Speed of transverse wave in a slid is , where is the rigidity modulus and is the density of the • material of the solid. • b) Speed of transverse wave in a stretched string is , where T is the tension and m is the linear • density (mass per unit length of the string. • ii) Velocity of longitudinal wave : • i) Velocity of longitudinal wave through a medium of modulus of elasticity E and density  is given by • V = • ii) For solids, as E = Y (Young’s modulus) V= • iii) For liquids and gases, as E = K, ( Bulk modulus ) V =

18. Newton’s formula for velocity of sound • According to Newton, the velocity of sound in gas is , where K is the bulk modulus of the medium and is the density. • As sound travels through a medium in the form of compressions and rarefactions, Newton assumed that the changes in volume and pressure of the gas during the propagation of sound are isothermal. According to him the amount of heat produced during compression is conducted to the surroundings and the amount of heat lost during rarefaction is conducted in from the surroundings so as to keep the temperature constant.

19. Thus Newton’s formula for velocity of sound becomes • From gas equation for isothermal change (PV=constant) , it can be found that Kiso= • The equation for velocity becomes • Substituting the S.T.P values of P =1.013x105 Pa & =1.293 kg/m3, it is seen that v = 280 m/s • Note: This value shows a large difference from the experimental values of velocity of sound (i.e. 332 m/s), • which could not be attributed as an experimental error rather it is a conceptual error.

20. Laplace’s correction • Laplace succeeded in explaining the discrepancy between the theoretical and experimental values of velocity of sound in a gas. He pointed out that Newton’s assumption of isothermal propagation of sound through gas was wrong and corrected it as an adiabatic change • This is because • i)As the velocity of sound in gas is quite large, compression and rarefaction follow one another so rapidly that there is no time for the exchange of heat among them to make it isothermal. • As air is a bad conductor of heat, it does not allow the free exchange of heat between compression and rarefaction.

21. Thus heat produced during compression raises the temperature of the gas and heat lost during rarefaction reduces the temperature of the gas to make it adiabatic not isothermal • Thus the velocity of sound in gas is • Using the gas equation for adiabatic change (PV =constant) , it can be shown that Kadiabatic = • The equation for velocity becomes • Substituting the S.T.P values of P =1.013x105 Pa, =1.293 kg/m3 &  = 1.41 (for air, being diatomic) it is found that speed of sound in air is v = 332.5 m/s. • Note: This value agrees well with the experimental value of velocity of sound and hence establishes Laplace’s equation as a correct relation for finding the velocity of sound in any gas.

22. Factors affecting velocity of sound in a gas • Density: • We know velocity of sound in gas • , . (i.e. as density increase, v decrease) • Note: This is why speed of sound in Hydrogen gas is more than that in Oxygen gas • Humidity: • Presence of moisture decreases the density of air and hence increases the speed. Hence sound travels faster in moist air than in dry air • Note: This is why sound travels faster on rainy days than on dry days • Temperature: • We know that for one mole a gas PV=RT and density  = Also

23. Note: This is why sound ravels faster on hot day (summer) than on cold day (winter) • If v0 is the velocity of sound at 0C and vt the velocity at tC then.(For any gas) • On applying Binomial expansion and re arranging, we get , where v0 = 332 m/s • i .e. for every 10 rise in temperature, the velocity of sound increases by 0.61 m/s • Pressure: • We know for sound • For a given gas, since , R and M are constants, when temperature is kept constant , velocity will also be a constant. • Hence velocity of sound is independent of the pressure of the gas provided temperature remains constant • Effect of nature of gas (molecular weight ) : • As , • sound has different speed through different gases and is maximum through hydrogen. • Note: Speed of sound through Oxygen is ¼ times that of hydrogen.

24. Equation for plane progressive wave: • A wave, which travels continuously in a medium in the same direction without any change in its amplitude, is called a progressive wave or a traveling wave. It may be a longitudinal or transverse wave. • Suppose a plane simple harmonic wave travels from the origin O along the positive direction of X- axis • The displacement of the particle at the origin at any instant t is given by • y = r sint , where r is the amplitude of the SHM executed by the particle and  is its angular frequency . • Let us find the displacement of another particle at point p situated at a distance x from the origin at any time t • As the particle on the positive X-axis receives disturbance a time later than that preceding it , the phase lag of the particle with respect to particle at the origin O goes on increasing as we move more and more away from the point O.

25. If is the phase lag of the particle at p with respect to particle at O , then displacement of the particle at point P at any instant t is given by • y = r sin (t-kx ) • We know that for a distance , the phase angle is 2 and hence phase difference for distance x will be = • Note: i) If the wave is traveling along the negative X-axis, replace x with - x in the above equation to get • ii) If there is a phase factor , then add  with the term inside the brackets in the above equations.

27. Principle of superposition of waves: • It states that when two waves super impose, the displacement of the resultant wave at any instant is the vector sum of the displacements due to individual waves at that instant. • Let two coherent waves be represented by y1=a1sin t and y2 = a2sin (t+), with phase difference  , • Resultant displacement, y = y1 + y2 • On substituting and proceeding, we get the equation for the resultant wave as y = A sin (t+) • and the amplitude

28. Reflection of Sound • Like light, sound waves also undergo reflection from different surfaces. • Reflection of sound from denser and rarer medium • When sound is reflected from a denser medium (such as a closed pipe), as the end is rigid and energy can not be transmitted forward , the layer of air in contact with the end must remain permanently at rest. Hence when a compression reaches he closed end the only way by which this layer can free itself from the compressional strain is by sending a wave of compression backwards. i..e . Compression is reflected as compression and hence the phase of the wave gets reversed, keeping intensity & amplitude the same as those of incident wave • In this case ,Equation of incident wave is y =A sin (t-kx) (left to right) • And the equation of reflected wave is y1 =A sin (t+ kx+) ( right to left & phase reversed) • When sound reflects from a rarer medium (such as an open pipe), as it can freely spread side ways the pulse of compression suffers lesser resistance to expansion than it has encountered inside the pipe. This causes pulse of compression to reflect as a pulse of rarefaction indicating these is no phase change. Here as reflection is partial (becaue a part of energy of the incident wave will pass out in to the open air) , the intensity and amplitude of reflected wave will be lesser than hat of incident wave. • In this case, Equation of incident wave is y =A sin (t - kx) (left to right) • And the equation of reflected wave is y1 =A sin (t+ kx) (right to left) • Note: The phenomenon of echo is also based on the reflection of sound. • To here an echo time interval between original sound and reflected sound should be greater than or equal to of a second and so minimum distance between the obstacle and source should be 16.5 m .

30. Standing Or stationary waves • When two sets of coherent progressive waves (same wavelength and amplitude) travel with the same speed through a medium along the same straight line in opposite direction superimpose, a resultant wave is formed which does not propagate in any direction and does not transfer any energy in the medium. These waves are called stationary waves or standing waves. • Note: In these waves there are certain points, which are permanently at rest, called nodes and there are points which vibrate about their mean position with maximum amplitude, called antinodes.

31. Characteristics of standing waves • Disturbance is confined to a particular region between the starting and reflecting points • There is no propagation of disturbance from one particle to the adjoining particle and also beyond the above region. • The total energy of a stationary wave is twice the energy of the incident and reflected waves. • There are nodes and antinodes in the medium and distance between any two consecutive nodes or antinodes is /2 . • The wavelength and time period of a stationary wave is same as that of the component waves.

32. Equation for a stationary wave: • When a string fixed at both ends under a tension is set to vibration, transverse harmonic wave will be propagated along its length and its reflected wave will, also exist. The incident and reflected waves when superimpose produce standing wave with the two fixed ends having nodes. • The equation for incident wave from left to right along x-axis is represented as • For the reflected wave from the right rigid boundary the equation can be written as • On superposition of these two waves, the displacement of the resultant wave pulse will be • Applying the expansion of the form • , we get • , • Where is the amplitude of the resulting wave. • Note: As the argument of trigonometric function doesn’t contain the form (vt  x) , it doesn’t represent a progressive harmonic wave . Rather it represents a new kind of waves called stationary or standing wave.

33. Where • is the amplitude of the resulting wave. • Note: As the argument of trigonometric function doesn’t contain the form (vt  x) , it doesn’t represent a progressive harmonic wave . Rather it represents a new kind of waves called stationary or standing wave.

34. Standing waves on a string fixed at both ends: • Consider a string of length L fixed at both ends is set in to vibrations so that the incident and reflected waves superimpose to form standing waves, whose equation is given by • As the string is fixed at both the ends, the ends have nodes with zero displacement. • i.e when x=0 , y=0 and when x=L , y=0 • It can be seen that eqn (1) satisfies the first condition always and second condition only when

35. i.e when , • where n=1,2,3 etc For first mode of vibration, put n=1 and thus . Frequency of vibration is given by (This frequency is called fundamental frequency or first harmonic)

36. Note: As • For second mode of vibration, put n=2 and thus Frequency of vibration is given by • Here, the frequency of vibration is twice the fundamental frequency • For third mode of vibration • , put n=3 and thus Frequency of vibration is given by • Here, the frequency of vibration is thrice the fundamental frequency • Generally for nth mode of vibration, the frequency is • Note: Here for the vibration of a string, all the harmonics are present

38. Harmonics and overtones: • Notes or sounds of frequency corresponding to integral multiple of fundamental frequency are called harmonics. • E.g: ,2,3,4…… etc • Notes or sounds of frequency greater than fundamental frequency are called overtones. • Eg: 2,3,4, ……. etc

39. Standing waves in a closed pipe • Consider a pipe of length L closed at one end in which longitudinal stationary waves are formed on • account of superposition of incident and reflected longitudinal waves. The equation for the standing wave can be expressed as • As the pipe is closed at one end this end will have a node and the open end has antinodes • i.e when x=0 , y=0 & when x=L, y= maximum • It can be seen that eqn (2) satisfies the first condition always and second condition only when • i.e when • ,

42. , where n=1,2,3 etc • For first mode of vibration, put n=1 and thus Frequency of vibration is given by .