## Sound Production and Propagation

Audible sound is any vibratory motion at frequencies between about 16 and 20,000 Hz; normally it reaches the ear through pressure waves in air. Sound is also readily transmissible through other gases, liquids, or solids; its velocity depends on the density and the elasticity of the medium, while attenuation depends largely on frictional damping. For most engineering work, adiabatic conditions are assumed.

Sound is initially produced by vibration of solid objects, by turbulent motion of fluids, by explosive expansion of gases, or by other means. The pressures, amplitudes, and velocities of the components of the sound wave within the range of hearing are quite small. Table 6.1.1 gives typical values; the sound pressures referenced are the dynamic excursions imposed on the relatively constant atmospheric pressure.

In a free field (defined as an isotropic homogeneous field with no boundary surfaces), a point source0 of sound produces spherical (Beranek 1954) sound waves (see Figure 6.1.1). If these waves are at a single frequency, the instantaneous sound pressure (Pr,t) at a distance r and a time t is

where the term v2P denotes the magnitude of peak pressure at a unit distance from the source, and the cosine term represents phase angle.

In general, instantaneous pressures are not used in noise control engineering (though peak pressures and some non-sinusoidal pulse pressures are, as is shown later), but most sound pressures are measured in root-mean-square (RMS)

valuesâ€”the square root of the arithmetic mean of the squared instantaneous values taken over a suitable period. The following description refers to RMS values.

For spherical sound waves in air, in a free field, RMS pressure values are described by

where Pr denotes RMS sound pressure at a distance r from the source, and Po is RMS pressure at unit distance from the source. (Meters in metric units, feet in English units.) Acoustic terminology is based on metric units, in general, though the English units of feet and pounds are used in engineering descriptions.

A few other terms should be defined, and their mathematical relationships noted.

Sound intensity I is defined as the acoustic power W passing through a surface having unit area; and for spherical waves (see Figure 6.1.1), this unit area is a portion of a spherical surface. Sound intensity at a distance r from a source of power W is given by

Sound intensity is also given by

where p is the adiabatic density of the medium, and c is the velocity of sound in that medium. Similarly, the following equation gives the sound pressure if the sound is radiated uniformly:

If the radiation is not uniform but has directivity, the term pc is multiplied by a directivity factor Q. To the noise-control engineer, the concept of intensity is useful principally because it leads to methods of establishing the sound power of a source.

The term pc is called the acoustic impedance of the medium; physically it represents the rate at which force can be applied per unit area or energy can be transferred per unit volume of material. Thus, acoustic impedance can be expressed as force per unit area per second (dynes/ cm2/sec) or energy per unit volume per second (ergs/cm3/ sec).

Table 6.1.1 shows the scale of mechanical magnitudes represented by sound waves. Amplitude of wave motion at normal speech levels, for example, is about 2 X 10-6 cm, or about 1 micro inch; while amplitudes in the lower part of the hearing range compare to the diameter of the hydrogen atom. Loud sounds can be emitted by a vibrat

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