There are many curious and interesting facts connected with the transmission of a sound wave through air, affecting the distance at which sounds can be heard. The speed of sound in air is much influenced by the temperature of the air and by wind.
The speed of sound increases with the temperature. For every degree Fahrenheit above the melting-point of ice (32° Fahr.) the speed is increased by one foot per second. A more accurate rule is as follows: Take the temperature of the air in degrees Centigrade, and add to this number 273. In other words, obtain the value of 273 + t° where t° is the temperature of the air. Then the velocity of sound in feet per second at this temperature is equal to the value of the expression—
1090 √273 + t°/273
There is one point in connection with the velocity of propagation of a sound wave which should not be left without elucidation. It has been explained that the velocity of a wave in any medium is numerically given by the number obtained by dividing the square root of the elasticity of the medium by the square root of its density. The number representing the elasticity of a gas is numerically the same as that representing its absolute pressure per square unit of surface. The volume elasticity of the air may therefore be measured by the absolute pressure it exerts on a unit of area such as 1 square foot. At the earth’s surface the pressure of the air at 0° C. is equal to about 2116·4 lbs. per square foot. The absolute unit of force in mechanics is that force which communicates a velocity of 1 foot per second to a mass of 1 lb. after acting upon it for 1 second. If we allow a mass of 1 lb. to fall from rest under the action of gravity at the earth’s surface, it acquires after 1 second a velocity of 32·2 feet per second. Hence the force usually called “a pressure of 1 lb.” is equal to 32·2 absolute units of force. Accordingly, the atmospheric pressure at the earth’s surface is 2116·4 × 32·2 = 68,148 absolute units of force in that system of measurement in which the foot, pound, and second are the fundamental units.
The absolute density of the air is the mass of 1 cubic foot: 13 cubic feet of air at the freezing-point, and when the barometer stands at 30 inches, weigh nearly 1 lb. More exactly, 1 cubic foot of air under these conditions weighs 0·080728 lb. avoirdupois. If, then, we divide the number representing the absolute pressure of the air by the number representing the absolute density of air, we obtain the quotient 844,168; and if we take the square root of this, we obtain the number 912·6.
The above calculation was made first by Newton; and he was unable to explain how it was that the velocity of the air wave, calculated in the above manner from the general formula for wave-speed, gave a value for the velocity, viz. 912·6, which was so much less than the observed velocity of sound, viz. 1090 feet per second at 0° C. The true explanation of this difference was first given by the celebrated French mathematician Laplace. He pointed out that in air, as in all other gases, the elasticity, when it is compressed slowly, is less than that when it is compressed quickly. A gas, when compressed, is heated, and if we give this heat time to escape, the gas resists the compression less than if the heat stays in it. Hence air is a little more resilient to a very sudden compression than to a slow one. Laplace showed that the ratio of the elasticity under sudden compression was to that under slow compression in the same ratio as the quantities of heat required to raise a unit mass of air 1° C. under constant pressure and under constant volume. This ratio is called “the ratio of the two specific heats,” and is a number close to 1·41. Hence the velocity, as calculated above, must be corrected by multiplying the number 844,168 by the number 1·41, and then taking the square root of the product. When this calculation is made, we obtain, as a result, the number 1091, which is exactly the observed value of the velocity of sound in feet per second at 0° C. and under atmospheric pressure. The velocity of sound is much affected by wind or movement of the air. Sound travels faster with the wind than against it. Hence the presence of wind distorts the shape of the sound wave by making portions of it travel faster or slower than the rest.
These two facts explain how it happens that loud sounds are sometimes heard at great distances from the source, but not heard at places close by.
Consider the case of a loud sound made near the surface of the earth. If the air were all at rest, and everywhere at the same temperature, the sound waves should spread out in hemispherical form. But if, as is generally the case, the temperature near the ground is higher than it is up above, then the part of the wave near the earth travels more quickly than that in the higher regions of the air. It follows that the sound wave will have its direction altered, and instead of proceeding near the earth in a direction parallel to the ground, it will be elevated, so as to strike in an upward direction. Again, it may be brought down by meeting with a current of air which blows against the lower portion and so retards that to a greater extent than it does the upper part. So it comes to pass that a sound wave may, as it were, “play leap-frog” over a certain district, being lifted up and then let down again; and persons in that region will not hear the sound, although others further off will do so.
Fig. 47 (reproduced by permission of proprietors of Knowledge).—Map of South of England, showing places (black dots) at which sound of funeral guns was heard, February 1, 1901.