Kinetic Energy in rate of transfer of energy per unit area of cross section of the wave may be regarded as a physical measure of the intensity of the sound transmitted. The kinetic energy of a layer of unit area and thickness ox is The maximum value of auat is na (from eqn. [5]). Consequently the maximum kinetic energy per unit area and unit length (i.e. unit volume) of the wave = (15). Since the sum of kinetic and potential energies is a constant, must also be the total energy of the wave motion per unit volume. This quantity may be described as the energy density in the wave. Now the wave travels a distance c per second, therefore the transmission of energy per second per unit area of (i6). This may be regarded as a measure of the intensity of the The intensity is therefore equal to the product of energy density and wave velocity c. The maximum particle velocity avai is from (5) equal to na and the maximum condensation s= that is, equal to the maximum particle velocity divided by the wave velocity. The expression for the intensity may therefore be written = Now = KI p and therefore the intensity or energy ,flow/sec. = 2pc (i7) a useful expression giving the intensity of the sound-wave in terms of maximum pressure variation in the path of the wave.
Power of Sound Source. Radiation Impedance.—.The energy thus present in the sound-wave must be derived from the vibrating source. The rate at which the source does work, that is the power of the source, in producing sound-waves, is equal to the product of pressure variation and particle velocity, i.e., to Opxavat per unit area of wave-front. Now bp= KS= K/C•avat, and ic/p whence ap=11Kp.avat. (i8) Therefore the power expended by the source per unit area of wave front = (Kp)•(avaw (iv). The relations expressed in (i8) and (to) are closely analogous to the relations between e.m.f., current, and resistance in electrical circuits. if we regard bp, -V Kp, and 'vat as corresponding to e.m.f. E, resistance (or impedance) R, and current i respectively we see that eqn. (i8) is the mechanical analogue of Ohm's law in electricity, and that (19) is analogous to power dissipation Rig in an electrical cir cuit. The quantity I Kp or pc is consequently designated the radiation impedance per unit area of the medium transmitting the sound-wave. This quantity is important in the consideration of the transmission of sound waves through a succession of different media (see p. 18).
results derived above for the energy in plane-waves hold equally well for spherical-waves at a suffi cient distance from the source. Elementary considerations at once indicate that the energy density in the wave will vary inversely as the square of the distance from a point source of spherical-waves, whence it may be inferred that the amplitude (of displacement, pressure or condensation) will vary inversely as the distance. This assumes, of course, that the amplitudes are small at all parts of the wave, and that there is no loss of energy due to viscosity, heat conduction and similar causes.
As in the case of plane-waves, it may be shown that the intensity at a point in the wave is given by 2pc.
Single and Double Sources. Energy Emission from Solid Vibra simple point source is a theoretical abstraction, but in practical cases where the source is a vibrating surface of appreciable area, each element of this area may be regarded as a simple source of spherical-waves, and the effect of the source as a whole obtained by integrating the effects of the elementary areas. Again, many vibrating bodies are not simple sources. For example a diaphragm radiating sound to air on both sides is, at any particular instant, sending out a compression pulse on one side and a rarefaction pulse on the other— thus behaving like two sources near together and in opposite phase; in other words acting like a double source. The prongs of a tuning fork each act as double sources in a similar way. Lamb (Dynamical Theory of Sound) shows that the rate of energy emission, or power, of a simple source is equal to pn'AVErc, where p and c are the density and velocity of sound in the surrounding medium, n/27 is the frequency, and A is the "strength" of the source (i.e., the maximum rate of emission of fluid at the source); the power of a double source is equal to B being the "strength" of the double source. In both cases it will be seen that the rate of energy emission increases rapidly with increasing frequency. For the same strength and frequency, the energy emitted in different media will vary directly as the density and inversely as the wave-velocity (or the cube in the case of a double source). In gaseous media, where the velocity varies inversely as Alp, the energy emission at constant frequency will vary inversely as and respec tively in the two cases. These deductions account for the ap parent feebleness of a bell or a tuning fork when vibrating in hydrogen as compared with air. The wave-velocity in hydrogen is 3.9 times that in air, therefore the energy emission from the same strength of source will be or goo times as great in air as in hydrogen. Another example:—the relative densities and wave-velocities for air and water are 1/77o and 1/4.4 respec tively—the energy emission, on these grounds, being, for sources of the same strength and frequency, about 3,400 times as great in water as in air. The loading effect of the water and the limit ations of output of sound generators must also be considered in dealing with actual cases. The quantities A and B in the above relations, denoting "strength" of source involve the area and form of the vibrating surface, amplitude and frequency. There is consequently for the same amplitude and frequency an increased rate of energy emission with increase of area in contact with the medium—exemplified by the increase of sound emitted when a tuning fork is brought into contact with a sounding board like the top of a table. The sounding board of a piano or a violin radiates practically all the vibrational energy of the strings, the damping of the latter being increased accordingly.