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EXAMINATION PAPERS.                        291
marked 1 will be points of rest, since they are on both sets of nodal lines. The points 2 also are nodal points, for they have equal ampli­tudes in opposite directions. A little consideration will show also that the lines 2, 1, 2 connecting these points of rest will be nodal lines. Hence the sand figure will be the square 2, 2, 2, 2. It can be obtained by clamping the plate at 2 and bowing close to one of the corners. Now (b) let fig. D represent the plate B, but vibrating in opposite phase as shown. Let E represent D superposed on A. Then the rectangles a a b b will have twice the amplitude due to each set of vibrations singly. The points 1, 1, 1, 1 are of course nodal points: the four corners 3, 3, 3, 3 and the centre 0 are also nodal points, since they have equal and opposite amplitudes. Consequently the sand figure will be the cross along the diagonals 3, 0, 3. It can be obtained by clamping the plate at the centre and bowing near the middle of one of the sides.
6. (a) What is a good form for a resonator? Explain {b) why a well-made resonator is not set into active vibration, even by a loud tone, unless the tone is accurately of the pitch to which the resonator is tuned ; (<?) in what way resonators provide a means for the analysis of compound sounds.
Am.—(a) See p. 65. (b) See pp. 60, 61, and 62. (e) See pp. 66 and 69.
Afternoon. 2 to 5.
1.    Describe the structures of the inner ear, and explain how they aid in the perception of pitch, the quality, and the direction of sounds.
Am— See pp. 25, 26, 27, 28.
2.    What is the reason why the ear is more sensitive to slight varia­tions of pitch in moderately high than in low notes.
Am.—In the Cochlea (see pp. 27 and 28) the radial fibres are probably tuned with fair regularity, like the forks of a Tonometer (see p. 151). Suppose for the sake of explanation they are tuned at say about 2 vibrations difference between successive fibres. In that case the ear would be able to detect difference of vibration number with equal keenness in all parts of the scale. But two vibrations at low pitch means a much larger interval than at high pitch. For example, 2 vibrations difference at d = 30 would mean a semitone, while 2 vibrations above d = 300 would mean only a small fraction of a semitone. Thus the ear would be much more sensitive to difference of pitch at the latter pitch than at the former.
3.    In arithmetic, a fourth and a fifth added together do not make an eighth; yet in music a "fourth" and a "fifth" added together make an " octave." Also in music a major " third " and a " fourth " added together make a sixth ; yet in arithmetic neither do three and four make six, nor do a third and a fourth added together make a