Chapter # 08 Waves Q2) Write short answers of the following questions. i) What is the difference between progressive and stationary waves? Answer: Progressive Waves 1 Progressive waves are the result of disturbance produced in a medium. The disturbance is transferred from one particle to another particle and the wave moves on. 2 Progressive waves transmit energy from one point to another as they move on. 3 The amplitude of vibration of each point is same but the phase changes continuously. 4 Unlike stationary waves, there is no node or antinode. Particles of the medium have similar motion. 5 Frequency of the progressive wave is equal to the frequency of the source producing them. Stationary Waves Stationary or standing waves are the result of superposition of two identical waves moving in opposite directions in a medium. Unlike progressive waves, there is no onward motion of the particles of the medium. Unlike progressive waves, stationary waves have no onward motion. Therefore, they do not transmit energy from one point to another. Unlike progressive wave, each particle has its own amplitude of vibration in stationary waves. At the node point, the amplitude is zero. Then it gradually increases as we go toward the antinode, where it is maximum. It again starts decreasing as we go further toward the node. In stationary waves, they have nodes and antinodes. At nodes, the particles are static while at the antinodes, the amplitude of vibration is maximum. Frequency of the stationary wave is equal to the frequency of the superposing waves. ii) What is the difference between longitudinal and transverse waves? Answer: 1 2 3 4 Transverse Waves In case of transverse waves, the particles of the medium vibrate about their mean positions perpendicular to the direction of propagation of the wave. In transverse waves there are crests and troughs coming one after the other. The distance between the two consecutive crests or troughs is called wavelength. In case of crest in a transverse wave, the particles move up and in case of trough they move downward. Longitudinal Waves In case of longitudinal waves, the particles of the medium vibrate parallel to the direction of the motion of the wave. In longitudinal waves there are compressions and rarefactions coming one after the other. The distance between two consecutive compressions or rarefactions is called a wavelength. In case of compression in a longitudinal wave, the particles of the medium come closer to one another and in case of rarefaction they move away from one another. 5 Transverse waves can move in solids and liquids only. Longitudinal waves can propagate in solids, liquids as well as in gases. Notes By: Ms. Maimoona Altaf Page 1
6 Transverse waves can be polarized. Longitudinal waves cannot be polarized. 7 8 Pressure in the transverse wave s remains constant everywhere. Transverse waves may be electromagnetic in nature as well. Pressure is maximum in the compression region and minimum in the rarefaction in longitudinal waves. Longitudinal waves are only mechanical. iii) A careful student says that he can predict the frequency of spring-mass system though he knows that how far the spring stretches when the mass is hung from it. How he justifies himself? Answer: The student is justified in his claim of prediction the frequency of spring-mass system if he knows the stretching (elongation) produced in the spring when the mass is attached to it. Suppose the elongation produced in the string when the mass is attached to it = x Now, we know that F = kx. As the frequency is Now suspend the spring and block of mass m attached to it vertically. The force applied by the mass is its weight. Therefore, mg = kx. Put value of k in equation (1) As he knows the stretching 'x' in the spring and 'π' and 'g' are already known constants. Therefore, he can find the frequency of the system. iv) Is there a transfer of energy through a medium when a stationary wave is produced in it? Explain. Answer: No, this is not possible to transfer energy through a medium when a stationary wave is produced in it. Transfer of energy in a stationary wave would take place when the wave pattern (nodes and antinodes) were moving particle to particle. As we know that in case of stationary waves, the node points are at rest so that whole of the wave is stationary. Therefore, energy cannot be transferred through a medium by a standing wave. The waves don't move in any direction and the energy is not transferred. v) Two wave pulses are travelling in opposite directions; completely cancel each other as they pass. What happens to the energy possessed by the waves? Answer: When two similar waves are traveling in opposite direction and they superpose each other, then standing waves are produced. The standing waves consist of nodes and anti-nodes. At node the amplitude of vibration is zero i.e., the particles of the medium remains at rest at the nodes. So there is no transfer of energy through the medium due to nodes. However only the inter-conversion of kinetic energy and potential energy takes place. At extreme position the potential energy is maximum while at mean position kinetic energy is maximum. vi) What are the conditions for constructive and destructive interference? Answer: Interference of two (or more) waves can be observed by seeing the waves reinforce one another at Notes By: Ms. Maimoona Altaf Page 2
some points of the region or cancel one another at the other. So it can take place when Two waves have same frequency. They have the same wavelength. The principle of linear superposition is obeyed. The direction of propagation is the same. So when they reinforce one another, the crest of one wave superpose with the crest of the other wave. Similarly, the trough of one wave overlaps with the trough of the other wave. On the other hand, when they cancel one another, the crest of one wave superpose with the trough of the other wave and vice versa. Conditions for constructive interference 1. The two waves are in phase. The initial angle of the waves at the origin must be same. And if the wavelengths and frequencies are same, the waves would have no crossings of crests and troughs. 2. Path difference is either zero or an integral multiple of the wavelength of either wave. In both cases a crest of one wave will meet a crest of the other and trough of one will meet the trough of the other as they propagate. Mathematically, path difference = d = 0, λ 2λ 3λ, Conditions for destructive interference 1. The two waves are anti-phase or out of phase. 2. The path difference of the two waves is half the wavelength or is odd multiple of half the wavelength. In both cases, trough one wave will meet the crest of the other wave and vice versa. Mathematically, path difference = d = λ/2, 3λ/2, 5λ/2, ( ) vii) How might one can locate the position of node and antinode on a vibrating string? Answer: Nodes in a standing wave produced in a string happen on equally spaced intervals. The amplitude of the waves is zero at these locations. At all these points, two anti-phase waves add together and cancel each other out. At a vibrating string the fixed ends force the wave to stop. Therefore, the fixed ends are always nodes. Along the vibrating string, other nodes occur at intervals equal to half the wavelength of the wave, i-e, λ/2. Antinodes are those points of the string which have maximum amplitudes. All antinodes are midway between the nodes. Therefore, they are also at a distance of λ/2 from each other. Thus if the ends of the vibrating string are fixed, two nodes will be formed at the fixed ends of the string and the other nodes along the string will be at distances λ/2 from one another. Similarly, the antinodes will be midway between the two consecutive nodes. The interval between the two consecutive antinodes also will be λ/2. If one end of the string is free, the free end will make an antinode. viii) Is it possible for an object which is vibrating transversely to produce sound waves? Answer: Yes, it is possible. If the surrounding medium of a vibrating object is an elastic one. It will produce sound irrespective of its transverse or longitudinal motion/vibration. We know that when an object is vibrating it does work on the surrounding. The work appears as forcing (pushing) the molecules of the surrounding medium closer to and, then away from, one another alternatively with the to and fro motion of the vibrating object. This produce compressions and rarefaction in the surrounding medium. Hence, longitudinal waves are set up in it. The sound can be heard (provided the waves are in the audible frequency range) despite the transverse vibration of the object. As an example, in all musical instruments using strings for the sound such as Sitar, Rabab, etc. The strings are vibrating in the transverse mode. But the surrounding air is experiencing compressions and rarefaction of Notes By: Ms. Maimoona Altaf Page 3
molecules and we hear the sound, a longitudinal wave phenomenon. ix) Why sound waves move faster in solids than in gases? Answer: Sound waves are mechanical waves and they need a medium for their propagation. So the effect of the nature of the medium is obvious. The speed of sound is given by the equation Where 'E' is the elasticity (Young's modulus in case of solids and Bulk modulus in case of gases.) and ρ is the density of the medium. If we consider the density aspects of solids and gases, the densities of solids is, no doubt, greater than gases. This means that unit volume of solids have more masses than gases. The density is in the denominator of the expression for the speed of sound. Thus it reduces the speed of sound in solids. The elasticity of solids is far greater than the elasticity of the gases. This means solids easily restore its size and shape within the limit of proportionality or elasticity. E is in the numerator in the expression for the speed of sound. Therefore, it increases the speed of sound in solids. So, the speed of sound depends on the ratio of E and ρ of the medium. This ratio is always high in solid substances than gaseous. Therefore, speed of sound is also greater in solids than gases. x) Why does the speed of sound wave in gas changes with temperature? Answer: Speed of sound in a gaseous medium is given by. Since γ, R and m are constants therefore the term is constant. This implies that. This means increasing the temperature of the gas increases the speed of sound. Theoretically, when we increase the temperature of the gas the molecules move with high speed. Transportation of energy from one point to another increases and the wave travel with greater speed. It is experimentally found that speed of sound increases 0.61 m/s for each degree rise in the temperature. xi) Is it possible for two astronauts to talk directly to one another even if they remove their helmets? Answer: No, this is not possible for two astronauts to talk directly in space even if they remove their helmets. Sound waves are longitudinal mechanical waves. They need a material medium for their propagation. If there is no material medium for such waves to travel in, they will not come into existence. Since there is no material medium in the space, the sound waves will not propagate in the space. In order to be in voice contact, the astronauts should have an electronic communication system. xii) Estimate the frequencies at which a test tube 15 cm long resonates when you blow across its lip. Answer: Such a test tube will behave like a closed organ pipe when air is blown into it. We know that the lowest or fundamental frequency of a closed organ pipe is given by: Here, length of the tube = 15 cm = 15 x 10-2 m Notes By: Ms. Maimoona Altaf Page 4
Let speed of the sound = 332 m/s, Then, Similarly, for the second harmonic, Putting the values, And the third harmonic is And so on. Notes By: Ms. Maimoona Altaf Page 5