EXAMPLES IN ACOUSTICS.

PERIOD, WAVE-length, and VELOCITY.

1. If an undulation travels at the rate of 100 ft. per second, and the wavelength is 2 ft., find the period of vibration of a particle, and the number of vibrations which a particle makes per second.

2. It is observed that waves pass a given point once in every 5 seconds, and that the distance from crest to crest is 20 ft. Find the velocity of the waves in feet per second.

3. The lowest and highest notes of the normal human voice have about 80 and 800 vibrations respectively per second. Find their wave-lengths when the velocity of sound is 1100 ft. per second.

4. Find their wave-lengths in water in which the velocity of sound is 4900 feet per second.

5. Find the wave-length of a note of 500 vibrations per second in steel in which the velocity of propagation is 15,000 ft. per second.

PITCH AND MUSICAL INTERVALS.

6. Show that a "fifth" added to a "fourth" makes an octave.

7. Calling the successive notes of the gamut Do1, Rei, Mi1, Fa, Soli, Lai, Si1, Do2, show that the interval from Sol, to Re, is a true "fifth.”

8. Find the first 5 harmonics of Do1.

9. A siren of 15 holes makes 2188 revolutions in a minute when in unison with a certain tuning-fork. Find the number of vibrations per second made by the fork.

10. A siren of 15 holes makes 440 revolutions in a quarter of a minute when in unison with a certain pipe. Find the note of the pipe (in vibrations per second).

REFLECTION OF SOUND, AND TONES OF PIPES.

11. Find the distance of an obstacle which sends back the echo of a sound to the source in 13 seconds, when the velocity of sound is 1100 ft. per second.

12. A well is 210 ft. deep to the surface of the water. What time will elapse between producing a sound at its mouth and hearing the echo?

13. What is the wave-length of the fundamental note of an open organ-pipe 16 ft. long; and what are the wave-lengths of its first two overtones? Find also their vibration-numbers per second.

14. What is the wave-length of the fundamental tone of a stopped organ pipe

EXAMPLES IN ACOUSTICS.

1139 5 ft. long; and what are the wave-lengths of its first two overtones? Find also their vibration-numbers per second.

15. What should be the length of a tube stopped at one end that it may resound to the note of a tuning-fork which makes 520 vibrations per second; and what should be the length of a tube open at both ends that it may resound to the same fork. [The tubes are supposed narrow, and the smallest length that will suffice is intended.]

16. Would tubes twice as long as those found in last question resound to the fork? Would tubes three times as long?

BEATS.

17. One fork makes 256 vibrations per second, and another makes 260. How many beats will they give in a second when sounding together?

18. Two sounds, each consisting of a fundamental tone with its first two harmonics, reach the ear together. One of the fundamental tones has 300 and the other 302 vibrations per second. How many beats per second are due to the fundamental tones, how many to the first harmonics, and how many to the second harmonics?

19. A note of 225 vibrations per second, and another of 336 vibrations per second, are sounded together. Each of the two notes contains the first two harmonics of the fundamental. Show that two of the harmonics will yield beats at the rate of 3 per second.

VELOCITY OF SOUND IN GASES.

20. If the velocity of sound in air at 0° C. is 33,240 cm. per second, find its velocity in air at 10° C., and in air at 100° C.

21. If the velocity of sound in air at 0° C. is 1090 ft. per second, what is the velocity in air at 10°?

22. Show that the difference of velocity for 1° of difference of temperature in the Fahrenheit scale is about 1 ft. per second.

23. If the wave-length of a certain note be 1 metre in air at 0°, what is it in air at 10°?

24. The density of hydrogen being '06926 of that of air at the same pressure and temperature, find the velocity of sound in hydrogen at a temperature at which the velocity in air is 1100 ft. per second.

25. The quotient of pressure (in dynes per sq. cm.) by density (in gm. per cubic cm.) for nitrogen at 0° C. is 807 million. Compute (in cm. per second) the velocity of sound in nitrogen at this temperature.

26. If a pipe gives a note of 512 vibrations per second in air, what note will it give in hydrogen?

27. A pipe gives a note of 100 vibrations per second at the temperature 10° C. What must be the temperature of the air that the same pipe may yield a note higher by a major fifth?

VIBRATIONS OF STRINGS.

28. Find, in cm. per second, the velocity with which pulses travel along a string whose mass per cm. of length is 005 gm., when stretched with a force of 7 million dynes.

29. If the length of the string in last question be 33 cm., find the number of vibrations that it makes per second when vibrating in its fundamental mode; also the numbers corresponding to its first two overtones.

30. The A string of a violin is 33 cm. long, has a mass of 0065 gm. per cn., and makes 440 vibrations per second. Find the stretching force in dynes.

31. The E string of a violin is 33 cm. long, has a mass of '004 gm. per cm., and makes 660 vibrations per second. Find the stretching force in dynes.

32. Two strings of the same length and section are formed of materials whose specific gravities are respectively d and d'. Each of these strings is stretched with a weight equal to 1000 times its own weight. What is the musical interval between the notes which they will yield?

33. The specific gravity of platinum being taken as 22, and that of iron as 7.8, what must be the ratio of the lengths of two wires, one of platinum and the other of iron, both of the same section, that they may vibrate in unison when stretched with equal forces?

LONGITUDINAL VIBRATIONS OF RODS.

34. If sound travels along fir in the direction of the fibres at the rate of 15,000 ft. per second, what must be the length of a fir rod that, when vibrating longitudinally in its fundamental mode, it may emit a note of 750 vibrations per second?

35. A rod 8 ft. long, vibrating longitudinally in its fundamental mode, gives a note of 800 vibrations per second. Find the velocity with which pulses are propagated along it.

EXAMPLES IN OPTICS.

PHOTOMETRY, SHADOWS, AND PLANE MIRRORS.

36. A lamp and a taper are at a distance of 4·15 m. from each other; and it is known that their illuminating powers are as 6 to 1. At what distance from the lamp, in the straight line joining the flames, must a screen be placed that it may be equally illuminated by them both?

37. Two parallel plane mirrors face each other at a distance of 3 ft., and a small object is placed between them at a distance of 1 ft. from the first mirror, and therefore of 2 ft. from the second. Calculate the distances, from the first mirror, of the three nearest images which are seen in it; and make a similar calculation for the second mirror.

38. Show that a person standing upright in front of a vertical plane mirror will just be able to see his feet in it, if the top of the mirror is on a level with his eyes, and its height from top to bottom is half the height of his eyes above his feet.

39. A square plane mirror hangs exactly in the centre of one of the walls of a cubical room. What must be the size of the mirror that an observer with his

eye exactly in the centre of the room may just be able to see the whole of the opposite wall reflected in it except the part concealed by his body?

40. Two plane mirrors contain an angle of 160°, and form images of a small object between them. Show that if the object be within 20° of either mirror there will be three images; and that if it be more than 20° from both, there will be only two.

41. Show that when the sun is shining obliquely on a plane mirror, an object directly in front of the mirror may give two shadows, besides the direct shadow.

42. A person standing beside a river near a bridge observes that the inverted image of the concavity of the arch receives his shadow exactly as a real inverted arch would do if it were in the place where the image appears to be. Explain this.

43. If a globe be placed upon a table, show that the breadth of the elliptic shadow cast by a candle (considered as a luminous point) will be independent of the position of the globe.

44. What is the length of the cone of the umbra thrown by the earth? and what is the diameter of a cross section of it made at a distance equal to that of the moon?

The radius of the sun is 112 radii of the earth; the distance of the moon from the earth is 60 radii of the earth; and the distance of the sun from the earth is 24,000 radii of the earth. Atmospheric refraction is to be neglected.

45. The stem of a siren carries a plane mirror, thin, polished on both sides, and parallel to the axis of the stem. The siren gives a note of 345 vibrations per second. The revolving plate has 15 holes. A fixed source of light sends to the mirror a horizontal pencil of parallel rays. What space is traversed in a second by a point of the reflected pencil at a distance of 4 metres from the axis of the siren? This axis is supposed vertical.

SPHERICAL MIRRORS.

46. Find the focal length of a concave mirror whose radius of curvature is 2 ft., and find the position of the image (a) of a point 15 in. in front of the mirror; (b) of a point 10 ft. in front of the mirror; (c) of a point 9 in. in front of the mirror; (d) of a point 1 in. in front of the mirror.

47. Calling the diameter of the object unity, find the diameters of the image in the four preceding cases.

48. The flame of a candle is placed on the axis of a concave spherical mirror at the distance of 154 cm., and its image is formed at the distance of 45 cm. What is the radius of curvature of the mirror?

49. On the axis of a concave spherical mirror of 1 m. radius, an object 9 cm. high is placed at a distance of 2 m. Find the size and position of the image.

50. What is the size of the circular image of the sun which is formed at the principal focus of a mirror of 20 m. radius? The apparent diameter of the sun is 30'.

51. In front of a concave spherical mirror of 2 metres' radius is placed a concave luminous arrow, 1 decimetre long, perpendicular to the principal axis, and at the distance of 5 metres from the mirror. What are the position and size of the image? A small plane reflector is then placed at the principal focus of the spherical mirror, at an inclination of 45° to the principal axis, its polished side being next the mirror. What will be the new position of the image?

REFRACTION.

(The index of refraction of glass is to be taken as §, except where otherwise specified, and the index of refraction of water as ).

52. The sine of 45° is √, or 707 nearly. Hence, determine whether a ray incident in water at an angle of 45° with the surface will emerge or will be reflected; and determine the same question for a ray in glass.

53. If the index of refraction from air into crown-glass be 12, and from air into flint-glass 1g, find the index of refraction from crown-glass into flint-glass. 54. The index of refraction from water into oil of turpentine is 1·11; find the index of refraction from air into oil of turpentine.

55. The index of refraction for a certain glass prism is 1'6, and the angle of the prism is 10°. Find approximately the deviation of a ray refracted through it nearly symmetrically.

56. A ray of light falls perpendicularly on the surface of an equilateral prism of glass with a refracting angle of 60°. What will be the deviation produced by the prism? Index of refraction of glass 1.5.

57. A speck in the interior of a piece of plate-glass appears to an observer looking normally into the glass to be 2 mm. from the near surface. What is its real distance?

58. The rays of a vertical sun are brought to a focus by a lens at a distance of 1 ft. from the lens. If the lens is held just above a smooth and deep pool of water, at what depth in the water will the rays come to a focus?

59. A mass of glass is bounded by a convex surface, and parallel rays incident nearly normally on this surface come to a focus in the interior of the glass at a distance a. Find the focal length of a plano-convex lens of the same convexity, supposing the rays to be incident on the convex side.

60. Show that the deviation of a ray going through an air-prism in water is towards the edge of the prism.

LENSES, &c.

61. Compare the focal lengths of two lenses of the same size and shape, one of glass and the other of diamond, their indices of refraction being respectively 1.6 and 2.6.

62. If the index of refraction of glass be 3, show that the focal length of an equi-convex glass lens is the same as the radius of curvature of either face.

63. The focal length of a convex lens is 1 ft. Find the positions of the image of a small object when the distances of the object from the lens are respectively 20 ft., 2 ft., and 11 ft. Are the images real or virtual?

64. When the distances of the object from the lens in last question are respectively 11 in., 10 in., and 1 in., find the distances of the image. Are the images real or virtual?

65. Calling the diameter of the object unity, find the diameter of the image in the six cases of questions 63, 64, taken in order.

66. Show that, when the distance of an object from a convex lens is double the focal length, the image is at the same distance on the other side.

67. The object is 6 ft. on one side of a lens, and the image is 1 ft. on the other side. What is the focal length of the lens?