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If I be the wave-length of the note, and n the number of

vibrations per second producing it, then v=n) : also the length 1 of the stretched wire must be an exact multiple of 1/2. When the string is sounding its fundamental note, 1 is equal to 1/2, and


λ 21 21 M The stretching force in the present example is the weight of 25

kgm. = 25,000 x 981 dynes; and M = 4.79/200 =0.02395
Thus n =

25,000 X 981
100 N

and if we take the

= 320; vibration number of c as 256, the note emitted will be e,

for 320/256 = 5/4, and this intervall is a major third. 13. Two similar wires of the same length are stretched -the one by a weight of 4 lbs., and the other by a weight of 9 lbs. What is the interval between the notes which they produce ?

14. A stretched string 3 feet long gives the notec when vibrating transversely : what note will be given by a string i foot long stretched by the same weight and made of the same material, but of one-quarter the thickness ?

15. A vibrating string is found to give the note f when stretched by a weight of 16 lbs. What weight must be used to give the note a? and what additional weight will give c'?

16. A wire 50 cm, in length and of mass 80 gm. is stretched so that it vibrates eighty times per sec. : find the stretching force in dynes. į

1 The intervals of the diatonic scale (from c to d), and the vibration numbers of the notes, taking c= 256, are as follows :

s 8

b d



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1 t d' 융 을 3

1. 256 : 288 : 320 : 341:3 : 384 : 426-6: 480 : 512


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17. A copper wire (density 8.8) 1 metre in length and 1.8 mm. in diameter is stretched by a weight of 20 kilogrammes. Calculate the number of vibrations which it makes per second when sounding its fundamental note.

18. The e string of a violin is tuned so as to vibrate 640 times per second ; the vibrating portion has a mass off of a gramme and a length of 33 cm. What is the stretching force ?

19. An observer listens to a whistle sounded on a railway train as it comes toward him, and the pitch of the whistle appears to be f#, but just as the train passes him the pitch falls to f. Show the speed of the train may be deduced from these observations. The note actually emitted by the whistle is f, but while the

train is approaching the observer the apparent pitch is heightened, because a larger number of sound-waves enter

his ear per second (Doppler's principle). Suppose the train to be at a distance d, and moving with a

velocity of v ft. per sec. toward the observer ; also let n be
the vibration number of the note f. Consider what
happens while the engine moves through v ft. (i.e. during
I sec.) Assuming that the velocity of sound in air is 1100
ft. per sec., the first vibration, produced at a distance d,
will reach the observer in a time d/1100. When the nth
vibration is produced the train is v ft. nearer, and this last
vibration will reach him in a time d/1.100 v/1100. Thus
his ear receives n vibrations in (1 v/1100) sec., or
n/(1 v/1100) per sec., which is therefore thé vibration
number of the note f# The interval between this
and f is a minor semitöne, and is equal to 25/24, so that



1 -v/1100

24 and

v=1100/25 = 44. Thus the speed of the train is 44 ft. per sec., or 30 miles per


20. An observer listening to the whistle of an engine which is approaching him at the rate of 45 ft. per sec.,


other 170.

notices that the pitch of the note which he hears is the same as that of a tuning-fork which makes 458 vibrations per sec.

What is the actual pitch of the whistle ? (Velocity of sound in air = 1100 ft. per sec.)

21. Give a graphical illustration of the manner in which “beats" are produced, and show that the number of beats per second can be calculated from the vibration numbers of the two notes producing them.

Two open pipes are sounded together, each note consisting of its first two harmonics, together with the fundamental. One note has 256 vibrations per second, the

Show that two of the harmonics will produce beats at the rate of two per second.

22. A smoked-glass plate is held vertically in front of a vibrating fork provided with a style, and the plate is allowed to fall freely under the action of gravity, so that the style traces a wavy line upon it. Prove that if the number of waves marked in a distance d (starting from rest) be n, then the vibration-number of the fork is n 12 dg.

23. In an experiment made according to this method, it was found that in a distance of 10.9 cm. (measured from the position of rest) 681 waves were included. Find the vibration-number of the fork, taking g=980.


24. State clearly what is meant by the elasticity of water. If the elasticity of water be 2 x 1010 C.G.S. units, calculate the velocity of sound in it.

Int. Sc. Honours 1883. 25. Describe the mode of transmission of a soundwave in air. The velocity of sound in any gas is numerically equal to the square root of the ratio of the numerical value of the elasticity of the gas for constant heat,


and the density of the gas; the velocity may also be calculated from the formula v= Vem, where p is the pressure of the gas, d its density, and y the ratio of its specific heat at constant pressure and constant volume. Reconcile these statements.

N. S. Tripos. 1885. 26. Explain the reflection of sound at the end of an open and a closed organ-pipe, and deduce the possible notes for a pipe of given length.

Calculate the vibration frequency of a note sounded upon a closed organ-pipe 120 centimetres long, blown with air at a temperature of 15°C., knowing that the specific gravity of air at o°C. and 760 mm. pressure is 0.001292, the specific gravity of mercury 13.59, the acceleration of gravity 9817 the coefficient of ex

(sec.)2 pansion of air .00366, and the ratio of the two specific heats of air 1.406.

N. S. Tripos. 1885. 27. What should be the length of a glass tube, open at both ends, that it may produce the maximum resonance when a tuning-fork, making 480 complete vibrations per second, is sounding near one end ? State clearly the nature and relative extent of the motion of the air-particles in different parts of the tube, and show how the sound-waves are propagated in it. (Velocity of sound in air is 1120 feet per second.) Prel. Sc. 1887.

28. State the laws for the frequency of vibration of stretched strings.

A string of india-rubber is stretched by a force of 2 lbs., and it executes n transverse vibrations per second. The length is doubled when the stretching force is 31 lbs. What is now the frequency of the vibrations ?

Vict. Int. 1886. 29. One end of a string is attached to a prong of a tuning-fork; the string passes over a small pulley, and carries a weight at the other end. With a weight of 40 grammes the string divides into four segments when

the fork vibrates : what weight must be suspended to cause the string to vibrate in five and in six segments ?

If the string be originally in the plane of vibration of the fork, what effect will be produced by turning the fork, so that the plane of vibration of the prongs is at right angles to the string ?

B. Sc. 1885.

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