Page 1 Introductory Exercise 16.4 Note Students are advised to solve this exercise after studying Young's double slit experiment in class 12 th . Q 1. Two sound waves emerging from a source reach a point simultaneously along two paths. When the path difference is 12 cm or 36 cm then there is silence at that point. If the speed of sound in air be 330 m/s then calculate least possible frequency of the source. Q 2. A wave of frequency 500 cycle/s has a phase velocity of 350 m/s. (a) How far apart are two points 60° out of phase? (b) What is the phase difference between two displacements at a certain point at times 10 3 s apart? Q 3. A source S and a detector D of high frequency waves are a distance d apart on the ground. The direct wave from S is found to be in phase at D with the wave from S that is reflected from a horizontal layer at an altitude H. The incident and reflected rays make the same angle with the reflecting layer. When the layer rises a distance h, no signal is detected at D. Neglect absorption in the atmosphere and find the relation between d, h, H and the wavelength ? of the waves. Q 4. Two sound speakers are driven in phase by an audio amplifier at frequency 600 Hz. The speed of sound is 340 m/s. The speakers are on the yaxis, one at y = + 1.0m and the other at y = 1.0 m. A listener begins at y = 0 and walks along a line parallel to the yaxis at a very large distance x away. (a) At what angle ? (between the line from the origin to the listener at the xaxis) will she first hear a minimum sound intensity? (b) At what angle will she first hear a maximum (after ? = 0°) sound intensity? (c) How many maxima can she possibly hear if she keeps walking in the same direction? Q 5. Two speakers separated by some distance emit sound of the same frequency. At some point P the intensity due to each speaker separately is l 0 . The path difference from P to one of the speakers is 1 2 ? greater than that from P to the other speaker. What is the intensity at P if (a) the speakers are coherent and in phase; (b) the speakers are incoherent; and (c) the speakers are coherent but have a phase difference of 180°? Q 6. Two loudspeakers radiate in phase at 170 Hz. An observer sits at 8 m from one speaker and 11 m from the other. The intensity level from either speaker acting alone is 60 dB. The speed of sound is 340 m/s. (a) Find the observed intensity level when both speakers are on together. (b) Find the observed intensity level when both speakers are on together but one has its leads reversed so that the speakers are 180° out of phase. Page 2 Introductory Exercise 16.4 Note Students are advised to solve this exercise after studying Young's double slit experiment in class 12 th . Q 1. Two sound waves emerging from a source reach a point simultaneously along two paths. When the path difference is 12 cm or 36 cm then there is silence at that point. If the speed of sound in air be 330 m/s then calculate least possible frequency of the source. Q 2. A wave of frequency 500 cycle/s has a phase velocity of 350 m/s. (a) How far apart are two points 60° out of phase? (b) What is the phase difference between two displacements at a certain point at times 10 3 s apart? Q 3. A source S and a detector D of high frequency waves are a distance d apart on the ground. The direct wave from S is found to be in phase at D with the wave from S that is reflected from a horizontal layer at an altitude H. The incident and reflected rays make the same angle with the reflecting layer. When the layer rises a distance h, no signal is detected at D. Neglect absorption in the atmosphere and find the relation between d, h, H and the wavelength ? of the waves. Q 4. Two sound speakers are driven in phase by an audio amplifier at frequency 600 Hz. The speed of sound is 340 m/s. The speakers are on the yaxis, one at y = + 1.0m and the other at y = 1.0 m. A listener begins at y = 0 and walks along a line parallel to the yaxis at a very large distance x away. (a) At what angle ? (between the line from the origin to the listener at the xaxis) will she first hear a minimum sound intensity? (b) At what angle will she first hear a maximum (after ? = 0°) sound intensity? (c) How many maxima can she possibly hear if she keeps walking in the same direction? Q 5. Two speakers separated by some distance emit sound of the same frequency. At some point P the intensity due to each speaker separately is l 0 . The path difference from P to one of the speakers is 1 2 ? greater than that from P to the other speaker. What is the intensity at P if (a) the speakers are coherent and in phase; (b) the speakers are incoherent; and (c) the speakers are coherent but have a phase difference of 180°? Q 6. Two loudspeakers radiate in phase at 170 Hz. An observer sits at 8 m from one speaker and 11 m from the other. The intensity level from either speaker acting alone is 60 dB. The speed of sound is 340 m/s. (a) Find the observed intensity level when both speakers are on together. (b) Find the observed intensity level when both speakers are on together but one has its leads reversed so that the speakers are 180° out of phase. (c) Find the observed intensity level when both speakers are on and in phase but the frequency is 85 Hz. Q 7. Two identical speakers emit sound waves of frequency 680 Hz uniformly in all directions with a total audio output of 1 mW each. The speed of sound in air is 340 m/s. A point P is a distance 2.00 m from one speaker and 3.00 m from the other. (a) Find the intensities l 1 and l 2 from each speaker at point P separately. (b) If the speakers are driven coherently and in phase, what is the intensity at point P? (c) If they are driven coherently but out of phase by 180°, what is the intensity at point P? (d) If the speakers are incoherent, what is the intensity at point P? Solutions 1. 2. (a) = 0.1166 m =11.7 cm (b) 3. Solving these two equations we get, 4. (a) = 8.14° (b) d sin ? = ? = 16.5° Page 3 Introductory Exercise 16.4 Note Students are advised to solve this exercise after studying Young's double slit experiment in class 12 th . Q 1. Two sound waves emerging from a source reach a point simultaneously along two paths. When the path difference is 12 cm or 36 cm then there is silence at that point. If the speed of sound in air be 330 m/s then calculate least possible frequency of the source. Q 2. A wave of frequency 500 cycle/s has a phase velocity of 350 m/s. (a) How far apart are two points 60° out of phase? (b) What is the phase difference between two displacements at a certain point at times 10 3 s apart? Q 3. A source S and a detector D of high frequency waves are a distance d apart on the ground. The direct wave from S is found to be in phase at D with the wave from S that is reflected from a horizontal layer at an altitude H. The incident and reflected rays make the same angle with the reflecting layer. When the layer rises a distance h, no signal is detected at D. Neglect absorption in the atmosphere and find the relation between d, h, H and the wavelength ? of the waves. Q 4. Two sound speakers are driven in phase by an audio amplifier at frequency 600 Hz. The speed of sound is 340 m/s. The speakers are on the yaxis, one at y = + 1.0m and the other at y = 1.0 m. A listener begins at y = 0 and walks along a line parallel to the yaxis at a very large distance x away. (a) At what angle ? (between the line from the origin to the listener at the xaxis) will she first hear a minimum sound intensity? (b) At what angle will she first hear a maximum (after ? = 0°) sound intensity? (c) How many maxima can she possibly hear if she keeps walking in the same direction? Q 5. Two speakers separated by some distance emit sound of the same frequency. At some point P the intensity due to each speaker separately is l 0 . The path difference from P to one of the speakers is 1 2 ? greater than that from P to the other speaker. What is the intensity at P if (a) the speakers are coherent and in phase; (b) the speakers are incoherent; and (c) the speakers are coherent but have a phase difference of 180°? Q 6. Two loudspeakers radiate in phase at 170 Hz. An observer sits at 8 m from one speaker and 11 m from the other. The intensity level from either speaker acting alone is 60 dB. The speed of sound is 340 m/s. (a) Find the observed intensity level when both speakers are on together. (b) Find the observed intensity level when both speakers are on together but one has its leads reversed so that the speakers are 180° out of phase. (c) Find the observed intensity level when both speakers are on and in phase but the frequency is 85 Hz. Q 7. Two identical speakers emit sound waves of frequency 680 Hz uniformly in all directions with a total audio output of 1 mW each. The speed of sound in air is 340 m/s. A point P is a distance 2.00 m from one speaker and 3.00 m from the other. (a) Find the intensities l 1 and l 2 from each speaker at point P separately. (b) If the speakers are driven coherently and in phase, what is the intensity at point P? (c) If they are driven coherently but out of phase by 180°, what is the intensity at point P? (d) If the speakers are incoherent, what is the intensity at point P? Solutions 1. 2. (a) = 0.1166 m =11.7 cm (b) 3. Solving these two equations we get, 4. (a) = 8.14° (b) d sin ? = ? = 16.5° (c) d sin ? = n? = 3.5 ? Maximum 3 maximas can be heard corresponding to n = 1, 2 and 3 (beyond ? = 0°) 5. (a) = ? + 0 = ? So, they will interfere destructively and (b) No interference will take place. Hence, I net = I 1 + I 2 = 2I 0 (c) In both conditions they interference constructively. Hence, 6. (a) ...(i) Substituting value of ?, we get I = 0 (b) Now, Hence, from Eq. (i) I = 4I 0 L 1 = 66 dB (c) Sources are incoherent Hence, I = I 0 + I 0 = 2I 0 L 1  60 = 10 log 10 2 = 3 or L 1 = 63 dB 7. (a) Page 4 Introductory Exercise 16.4 Note Students are advised to solve this exercise after studying Young's double slit experiment in class 12 th . Q 1. Two sound waves emerging from a source reach a point simultaneously along two paths. When the path difference is 12 cm or 36 cm then there is silence at that point. If the speed of sound in air be 330 m/s then calculate least possible frequency of the source. Q 2. A wave of frequency 500 cycle/s has a phase velocity of 350 m/s. (a) How far apart are two points 60° out of phase? (b) What is the phase difference between two displacements at a certain point at times 10 3 s apart? Q 3. A source S and a detector D of high frequency waves are a distance d apart on the ground. The direct wave from S is found to be in phase at D with the wave from S that is reflected from a horizontal layer at an altitude H. The incident and reflected rays make the same angle with the reflecting layer. When the layer rises a distance h, no signal is detected at D. Neglect absorption in the atmosphere and find the relation between d, h, H and the wavelength ? of the waves. Q 4. Two sound speakers are driven in phase by an audio amplifier at frequency 600 Hz. The speed of sound is 340 m/s. The speakers are on the yaxis, one at y = + 1.0m and the other at y = 1.0 m. A listener begins at y = 0 and walks along a line parallel to the yaxis at a very large distance x away. (a) At what angle ? (between the line from the origin to the listener at the xaxis) will she first hear a minimum sound intensity? (b) At what angle will she first hear a maximum (after ? = 0°) sound intensity? (c) How many maxima can she possibly hear if she keeps walking in the same direction? Q 5. Two speakers separated by some distance emit sound of the same frequency. At some point P the intensity due to each speaker separately is l 0 . The path difference from P to one of the speakers is 1 2 ? greater than that from P to the other speaker. What is the intensity at P if (a) the speakers are coherent and in phase; (b) the speakers are incoherent; and (c) the speakers are coherent but have a phase difference of 180°? Q 6. Two loudspeakers radiate in phase at 170 Hz. An observer sits at 8 m from one speaker and 11 m from the other. The intensity level from either speaker acting alone is 60 dB. The speed of sound is 340 m/s. (a) Find the observed intensity level when both speakers are on together. (b) Find the observed intensity level when both speakers are on together but one has its leads reversed so that the speakers are 180° out of phase. (c) Find the observed intensity level when both speakers are on and in phase but the frequency is 85 Hz. Q 7. Two identical speakers emit sound waves of frequency 680 Hz uniformly in all directions with a total audio output of 1 mW each. The speed of sound in air is 340 m/s. A point P is a distance 2.00 m from one speaker and 3.00 m from the other. (a) Find the intensities l 1 and l 2 from each speaker at point P separately. (b) If the speakers are driven coherently and in phase, what is the intensity at point P? (c) If they are driven coherently but out of phase by 180°, what is the intensity at point P? (d) If the speakers are incoherent, what is the intensity at point P? Solutions 1. 2. (a) = 0.1166 m =11.7 cm (b) 3. Solving these two equations we get, 4. (a) = 8.14° (b) d sin ? = ? = 16.5° (c) d sin ? = n? = 3.5 ? Maximum 3 maximas can be heard corresponding to n = 1, 2 and 3 (beyond ? = 0°) 5. (a) = ? + 0 = ? So, they will interfere destructively and (b) No interference will take place. Hence, I net = I 1 + I 2 = 2I 0 (c) In both conditions they interference constructively. Hence, 6. (a) ...(i) Substituting value of ?, we get I = 0 (b) Now, Hence, from Eq. (i) I = 4I 0 L 1 = 66 dB (c) Sources are incoherent Hence, I = I 0 + I 0 = 2I 0 L 1  60 = 10 log 10 2 = 3 or L 1 = 63 dB 7. (a) = 19.9 × 10 6 W/m 2 (b) (c) (d) Introductory Exercise 16.5 Q 1. The fundamental frequency of a closed organ pipe is 220 Hz. (a) Find the length of this pipe. (b) The second overtone of this pipe has the same wavelength as the third harmonic of an open pipe. Find the length of this open pipe. Take speed of sound in air 345 m/s. Q 2. Standing sound waves are produced in a pipe that is 0.8 m long, open at one end, and closed at the other. For the fundamental and first two overtones, where along the pipe (measured from the closed end) are (a) the displacement antinodes (b) the pressure antinodes. Q 3. An organ pipe has two successive harmonics with frequencies 400 and 560 Hz. The speed of sound in air is 344 m/s. (a) Is this an open or a closed pipe? (b) What two harmonics are these? (c) What is the length of the pipe? Q 4. The atomic mass of iodine is 127 g/mol. A standing wave in iodine vapour at 400 K has nodes that are 6.77 cm apart when the frequency is 1000 Hz. At this temperature, is iodine vapour monoatomic or diatomic. Q 5. A tuning fork whose natural frequency is 440 Hz is placed just above the open end of a tube that contains water. The water is slowly drained from the tube while the tuning fork remains in place and is kept vibrating. The sound is found to be enhanced when the air column is 60 cm long and when it is 100 cm long. Find the speed of sound in air. Solutions 1. (a) = 0.392 m (b) Page 5 Introductory Exercise 16.4 Note Students are advised to solve this exercise after studying Young's double slit experiment in class 12 th . Q 1. Two sound waves emerging from a source reach a point simultaneously along two paths. When the path difference is 12 cm or 36 cm then there is silence at that point. If the speed of sound in air be 330 m/s then calculate least possible frequency of the source. Q 2. A wave of frequency 500 cycle/s has a phase velocity of 350 m/s. (a) How far apart are two points 60° out of phase? (b) What is the phase difference between two displacements at a certain point at times 10 3 s apart? Q 3. A source S and a detector D of high frequency waves are a distance d apart on the ground. The direct wave from S is found to be in phase at D with the wave from S that is reflected from a horizontal layer at an altitude H. The incident and reflected rays make the same angle with the reflecting layer. When the layer rises a distance h, no signal is detected at D. Neglect absorption in the atmosphere and find the relation between d, h, H and the wavelength ? of the waves. Q 4. Two sound speakers are driven in phase by an audio amplifier at frequency 600 Hz. The speed of sound is 340 m/s. The speakers are on the yaxis, one at y = + 1.0m and the other at y = 1.0 m. A listener begins at y = 0 and walks along a line parallel to the yaxis at a very large distance x away. (a) At what angle ? (between the line from the origin to the listener at the xaxis) will she first hear a minimum sound intensity? (b) At what angle will she first hear a maximum (after ? = 0°) sound intensity? (c) How many maxima can she possibly hear if she keeps walking in the same direction? Q 5. Two speakers separated by some distance emit sound of the same frequency. At some point P the intensity due to each speaker separately is l 0 . The path difference from P to one of the speakers is 1 2 ? greater than that from P to the other speaker. What is the intensity at P if (a) the speakers are coherent and in phase; (b) the speakers are incoherent; and (c) the speakers are coherent but have a phase difference of 180°? Q 6. Two loudspeakers radiate in phase at 170 Hz. An observer sits at 8 m from one speaker and 11 m from the other. The intensity level from either speaker acting alone is 60 dB. The speed of sound is 340 m/s. (a) Find the observed intensity level when both speakers are on together. (b) Find the observed intensity level when both speakers are on together but one has its leads reversed so that the speakers are 180° out of phase. (c) Find the observed intensity level when both speakers are on and in phase but the frequency is 85 Hz. Q 7. Two identical speakers emit sound waves of frequency 680 Hz uniformly in all directions with a total audio output of 1 mW each. The speed of sound in air is 340 m/s. A point P is a distance 2.00 m from one speaker and 3.00 m from the other. (a) Find the intensities l 1 and l 2 from each speaker at point P separately. (b) If the speakers are driven coherently and in phase, what is the intensity at point P? (c) If they are driven coherently but out of phase by 180°, what is the intensity at point P? (d) If the speakers are incoherent, what is the intensity at point P? Solutions 1. 2. (a) = 0.1166 m =11.7 cm (b) 3. Solving these two equations we get, 4. (a) = 8.14° (b) d sin ? = ? = 16.5° (c) d sin ? = n? = 3.5 ? Maximum 3 maximas can be heard corresponding to n = 1, 2 and 3 (beyond ? = 0°) 5. (a) = ? + 0 = ? So, they will interfere destructively and (b) No interference will take place. Hence, I net = I 1 + I 2 = 2I 0 (c) In both conditions they interference constructively. Hence, 6. (a) ...(i) Substituting value of ?, we get I = 0 (b) Now, Hence, from Eq. (i) I = 4I 0 L 1 = 66 dB (c) Sources are incoherent Hence, I = I 0 + I 0 = 2I 0 L 1  60 = 10 log 10 2 = 3 or L 1 = 63 dB 7. (a) = 19.9 × 10 6 W/m 2 (b) (c) (d) Introductory Exercise 16.5 Q 1. The fundamental frequency of a closed organ pipe is 220 Hz. (a) Find the length of this pipe. (b) The second overtone of this pipe has the same wavelength as the third harmonic of an open pipe. Find the length of this open pipe. Take speed of sound in air 345 m/s. Q 2. Standing sound waves are produced in a pipe that is 0.8 m long, open at one end, and closed at the other. For the fundamental and first two overtones, where along the pipe (measured from the closed end) are (a) the displacement antinodes (b) the pressure antinodes. Q 3. An organ pipe has two successive harmonics with frequencies 400 and 560 Hz. The speed of sound in air is 344 m/s. (a) Is this an open or a closed pipe? (b) What two harmonics are these? (c) What is the length of the pipe? Q 4. The atomic mass of iodine is 127 g/mol. A standing wave in iodine vapour at 400 K has nodes that are 6.77 cm apart when the frequency is 1000 Hz. At this temperature, is iodine vapour monoatomic or diatomic. Q 5. A tuning fork whose natural frequency is 440 Hz is placed just above the open end of a tube that contains water. The water is slowly drained from the tube while the tuning fork remains in place and is kept vibrating. The sound is found to be enhanced when the air column is 60 cm long and when it is 100 cm long. Find the speed of sound in air. Solutions 1. (a) = 0.392 m (b) = 0.47 m 2. (a) In second figure, l = Displacement antinode is at (b) Pressure antinode is displacement node: In third figure Pressure antinode or displacement node is at, 3. (a) (b) Since these are odd harmonics (5 and 7). Hence pipe is closed. (c) Given frequency are integer multiples of 80 Hz. Hence fundamental frequency is 80 Hz. = 1.075 m 4. v = f ? = (1000)(2 × 6.77 × 10 2 ) = 135.4 m/s = 0.7 So, we will have to substitute, M = 2 × 0.127 The ? comes out to be 1.4 which is the ? of a diatomic gas. 5. = 352 m/sRead More
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