Velocity of Sound

Refraction of Sound-Waves.

The optical analogy which applies to reflection of sound may, as we have already seen, apply to refraction also. When plane-waves of sound cross the bounding surface between two different media having different wave-velocities and the direction of propagation is changed in accordance with the sine law, viz., sin 02 = c2 where and are the angles of incidence and refraction respectively. Hence the sound "rays" in passing from one medium to another are bent towards or away from the normal according as the velocity of the wave in the first medium is greater or less than in the second medium. Thus, for example, sound is refracted towards the normal when passing from air into The "critical angle", when the sound is totally reflected, and there is no refracted beam, is given by sin ci/c2. The critical angle from air to water, 'Jo() ft./sec. and c2= 476o ft./sec., is therefore 131° approximately; above this angle no sound what ever can enter the water (we have already shown, p. 23, that even at normal incidence the reflected amplitude is o•99943 of the incident amplitude). From air to solid materials (steel, glass, wood, etc.) = o.o65 approx. whence the critical angle is 31°. Sondhauss (1852) demonstrated the refraction of sound through prisms containing various gases, and determined the refractive indexµ relative to air, ,u= sin 01/sin ci/c2.

He was also successful in demonstrating the focusing action of a convex lens of carbon dioxide enclosed in a thin envelope of collodion.

Refraction by Wind and Temperature Gradients.—It is well known that sound travels better with the wind than against it. This effect is due to the increase of velocity of the wind from the earth's surface upwards. The effective velocity of sound is equal to its normal velocity 1/(7p/p) plus or minus the velocity of the medium. In still air the wave-front of a sound-beam will travel over the earth's surface parallel to its initial direction, say at right angles to the ground. If, however, a wind is blowing in the same direction, the upper part of the wave where the wind velocity is greater, travels faster than that part near the ground, with the result that the wave-front tends to bend down wards towards the ground. An observer therefore hears by a direct ray which starts with a slightly upward inclination. Similarly a horizontal ray travelling against the wind is bent upwards and, at a moderate distance, passes over the head of an observer. A similar effect may be noticed when there is a gradual change in the temperature of the air from the ground upwards. The warmer the air, the greater the velocity of sound. If the temperature increases upwards, the wave-front will be bent downwards towards the ground; conversely, when the tempera ture diminishes upwards the sound-beam will be deflected upwards into the higher atmosphere and lost. The curvature of the ray is given by 1/R = i/c.ac/ay where ac/ay is the velocity gradient with respect to height—this gradient may be positive or negative.

Audibility of Fog Signals. Tyndall (Phil. Trans., 1874) made extensive researches on the audibility of fog signals across the Channel from South Foreland, and came to the conclusion that "temperature refraction" and a "flocculent" condition of the atmosphere, arising from unequal heating or moisture, were responsible for large fluctuations in the observed ranges of audibility. Contrary to general opinion at the time Tyndall found that the presence of fog favoured the transmission of sound signals—the atmosphere being then in a more homo geneous condition, particularly in regard to temperature grad ients. The problem of long-distance transmission of signals has recently been studied by King, Tucker, Paris and others. Using a doubly-resonated hot-wire microphone, Tucker and Paris have made intensity measurements at varying distances and orientations with respect to diaphones and sirens mounted in light-houses at sea. In a particular experiment it was found that the ratio of sound intensities at two miles from the source, with the wind and against it was 25 to I. The effects of tem

perature refraction were also partly responsible for this high value of the ratio at such a short range. Certain fluctuations of intensity were ascribed to moving eddies in the atmosphere. Player, however, as a result of recent observations of a similar nature, reached the conclusion that humidity is the only factor in which the variations are at all comparable with the large and sudden variations in the range of audible transmission.

Zones of Silence. The sound of a large explosion is sometimes observed at very great distances, whilst at intermediate distances nothing at all is heard. Thus the explosion of a large ammunition dump in Holland (Jan. 1923) was recorded at a distance of 85o km. whereas at too to 18o km. no sound could be detected. These intermediate zones of silence are not uncommon in such cases. They doubtless arise as a result of peculiar meteorological conditions at the time of the explosion. As we have seen, the action of temperature and wind gradients may cause upward or downward refraction of the sound. Esclangon (Comptes Rendus, 1924) has shown that these two factors are sufficient to account for one or a succession of zones of silence, with reinforcement of sound in particular directions. Zones of silence are in certain cases due to the interference between sound-waves reaching the observer by different paths. (See Interference above.) Effect of Temperature Gradients on Sound Propagation Under Water.—The sea is a much more homogeneous medium for sound transmission than the atmosphere. Sounds of moderate power may be heard at long ranges, 40 or 5o miles, without the corres ponding fluctuations of intensity which are so troublesome in air. The effects of tidal currents (analogous to "wind"), say io ft./sec., are in most cases negligible, the velocity of sound in water being about 5,000 ft./sec. Seasonal variations in range of signals have been observed (see H. Lichte, Phys. Zeits., Sept. 1919) these being ascribed to refraction produced by temperature gradients in horizontal layers of the sea. Such a temperature gradient may cause a sound "ray" to curve upwards to the surface where it is reflected down again, only to rise once more. In this way it travels forward along a cycloidal or "festoon" type of track. A temperature gradient of opposite sign causes a bending down wards with analogous effects at the sea bed. Surface and bottom reflections play an important part in the long-range transmission of sound in the sea. With a particular sound-transmitter Lichte and Barkhausen (Ann. d. Physik, 62, July 192o) noted a change from io km. in summer to 20 km. in winter in the Baltic sea.

Interference.

The passage of sound-waves through a medium is in no way affected by the passage of other sound-waves through the same part of the medium. This important principle involv ing the independence of separate trains of waves is known as the Principle of Superposition and was first propounded by Huyghens in dealing with light-waves. The resultant displace ment of a particle of the medium is obtained by adding the separate displacements vectorically (see p. 20). Thus two periodic motions of the same frequency, of amplitudes and and phase difference E combine to form a periodic motion having an amplitude whose square is cose). The principle of superposition applied to the squares of ampli tudes (i.e., energies) would account for the first two terms only and the results would be erroneous. It is found under certain conditions that two trains of sound-waves may, at certain points in the medium, neutralize each other's effects and pro duce silence where previously there was a definite sound due to either of the trains of waves. Thus in the above case, if E= i8o°, i.e., the two independent sets of vibrations of the particle are in opposite phase cos€ = , and the resultant amplitude is which becomes zero when if €=360° or o°,