Sensitivity of the Ear to Intensity and Frequency Variation.— The relation between sensation (loudness) and stimulus (intensity) applicable to all sensations, is generally expressed by Weber's kw—"The increase of stimulus necessary to produce the mini mum perceptible increase of sensation is proportional to the pre-existing stimulus." From this law Fechner derived the rela tion S=k log E or as/aE=k/E, where S is the magnitude of the sensation, E the intensity of the stimulus and k a constant. The "law" obviously approximates to the truth, for the sensitiveness of the ear as/aE diminishes rapidly with increase of the total intensity E of the sound. Feeble sounds which are easily heard at night when E is small, cannot be distinguished from the general noise in the daytime when E is much greater. Wood and Young (Proc. Roy. Soc., A. ioo, p. 264 and p. 266, 1921), in judging equality of two sounds of the same pitch, remark that under favourable conditions it was possible to distinguish a dif ference of intensity of io%. A similar conclusion was reached by V. 0. Knudsen (Phys. Rev., 21 Jan. 1923). Measuring the "intensity sensitivity " of the ear 6E/ E where bE is the least per ceptible difference of energy and E the total energy of the tone, Knudsen found 6EIE to be about o• io for moderate and large intensities E, but increased to the limiting value of unity as the intensity decreased to the threshold value (the minimum energy of audibility). To include this region of very feeble intensities Knudsen proposed a modification of the Weber Fechner law, namely, 6E/ E= k+ (1— k) E)n where k=o.io app. and n varies somewhat with frequency, being 1.65 for 200 cycles/sec. and 1.05 for i,000 cycles/sec., nevertheless at the same loudness level e.g., 6E1 E is nearly independent of frequency. More recentl., R. R. Reisz (Phys. Rev., 31, May, 1928) has found that 6EIE lies between 0.05 and o.15 according to frequency. He found also that 6E/ E is a minimum at 2,500 p.p.s.—this minimum being more sharply defined at the smaller sound-intensities. This frequency corresponds to the region of greatest absolute sensitivity of the ear. Fig.12 indicates the range of the average human ear with regard to both intensity and frequency. The upper curve gives the sound-pressures (root mean square) which produce a sensation of feeling, and serves as a practical upper limit to the range of auditory sensation (Wegel, Bell System. Tech. Journ., 1, Nov. 1922) whilst the lower curve indicates the pressures at the threshold of audibility. It will be seen that the ear is most sensitive in the region of frequency 500 to 5,000 p.p.sec. The range of intensity (proportional to square of pressure) appreciated by the ear in this range is of the order The smallest detectable pressure-amplitude is of the order io -3 i.e., of atmospheric pressure, corres ponding to a displacement-amplitude of the order of cm. Knudsen, Fletcher and Reisz independently conclude that near a frequency of 2,00o the ear can distinguish, under favourable conditions, from 30o to 400 gradations of loudness between the threshold of audibility and the threshold of feeling (a painful intensity times as great)—each step being recognisable by the ear as just perceptibly louder than the one before it.
Audible Limits of Frequency.— The lower and upper limits of frequency for tones audible to the human ear vary according to different observers. A very good average range of fre quency may be taken as 20 to 20,000 cycles/sec. Very high and very low pitched sounds of great intensity are felt rather than heard. The ear loses its power to discriminate variations of pitch at very high fre quency. The frequency range employed in music extends from about 4o to 4,000 cycles/sec. When two notes within the range 50o to 4,00o cycles/sec. are sounded alternately, the ear can detect a difference of frequency of about 1 in 300. When the two notes are sounded together the discrimination is greatly improved—a frequency difference of one in 20,000 being readily discernible by "beats" (see p. 6). Kohlrausch demonstrated experimentally that the sensation of pitch may be excited even with so few as two vibrations.
a sound proceeding from the right or the left is readily determined with fair accuracy, but there is little difference observable tween a sound approaching from behind or ahead. For high pitched sounds of short wave-length these directional effects might be explained by the difference of intensity of the sound reaching the two ears, since the head acts to some extent as a screen to the ear which is more remote from the source of sound. When the wave-length of the sound exceeds the perimeter of the head, however, the intensity difference must be very slight and we must look for another explanation of the directional effect. Rayleigh (Sound, Vol. II., p. 44o) who has examined this question carefully, arrived at the conclusion that the perception of direction is dependent on the relative phase of the sounds as they reach the two ears, a small difference of phase being sufficient to indicate the required direction. He found that if the same tone be led by different paths to the two ears, the sound could be made to appear to come from the right or the left at will, by adjusting the path-lengths and consequently the relative phase. The origin of the sound was always attributed to that side on which the phase is in advance (by less than half a period). No explanation of this effect has yet been given.
Pressure of Sound-Waves.—Radiometers.—Sound-waves, like light-waves, exert a slight pressure on any surface upon which they fall; and radiometers for measuring sound-intensity have been constructed on this principle. Rayleigh (Phil. Mag., io, 1905) has determined this pressure on theoretical grounds, but a more simple treatment due to Larmor will suffice here. Plane waves are incident on a perfectly reflecting wall free to move, the wall being pushed with velocity v to meet the advancing sound-waves of velocity c and mean energy density E (see p. 28). The total energy density in front of the wall, if stationary, would be 2E due to the incident and reflected trains of waves. The length of the wave-train incident per second on the ad vancing wall is (c+v) this being compressed during transmission into a length (c—v) due to the approach of the wall. The energy density in the reflected-wave is therefore increased in the ratio The increase in the total energy in the region of length c in front of the moving wall is consequently c.6E, and this must inevitably be due to the work done by the wall in compressing the radia tion. If P is the radiation pressure, the work done by the wall per second is Pv, whence Pv= c.6E= E. 271, that is, P = 2E, the mean radiation pressure being equal to the energy density in the medium in front of the wall. If the wall is a perfect absorber there will be no reflected-wave and P = E. Now the intensity of the sound is equal to the energy density X the wave-velocity c (see page 22). Provided therefore that the radiation pressure can be measured, and the reflection characteristics of the wall can be determined, we have here an absolute method of measuring sound-intensity. Radiometers for measuring the intensity of high frequency sound waves under water have been constructed on the above principle by Langevin, Wills, Boyle and others. Boyle (Proc. Roy. Soc., Canada, 1925) constructed torsion pendulums for use with high frequency quartz oscillators under water. The pressure of the "ultra sonic" radiation on the pendulum vane causes a deflection which is reduced to zero by a torsion head. This gives a measure of the twisting moment and consequently the radiation pressure. Comparative measurements with such radiometers confirm the theoretical deduction that the radiation pressure is proportional to the energy density. All absolute measurements of radiation pressure must, of course, make allowance for diffraction at the edge of the reflecting vane. Using the radiometer method, Boyle measured the energy output of high frequency quartz oscillators and the reflection and transmission coefficients of various mate rials under water. Torsion radiometers have similarly been used to measure sound-intensity in air (Altberg, Ann. d. Phys., II, 1903). They are only of value, however, in cases where the sound-in tensity is very great.