Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 81

We see that for the one‐dimensional propagation considered, the sound wave disturbances travel with a constant wave speed c, although there is no net, time‐averaged movement of the air particles. The air particles oscillate back and forth in the direction of wave propagation (x‐axis) with velocity u. We may show that for any plane wave traveling in the positive x direction at any instant

(3.12)

and for any plane wave traveling in the negative x‐direction

(3.13)

The quantity ρ c is known as the characteristic impedance of the fluid, and for air, ρ c = 428 kg s/m² at 0 °C and 415 kg s/m² at 20 °C.

The intensity of sound, I, is the time‐averaged sound energy that passes through unit cross‐sectional area in unit time. For a plane progressive wave, or far from any source of sound (in the absence of reflections):

(3.14)

where ρ = the fluid density (kg/m³) and c = speed of sound (m/s).

In the general case of sound propagation in a three‐dimensional field, the sound intensity is the (net) flow of sound energy in unit time flowing through unit cross‐sectional area. The intensity has magnitude and direction

(3.15)

where p is the total fluctuating sound pressure and u_r is the total fluctuating sound particle velocity in the r‐direction at the measurement point. The total sound pressure p and particle velocity u_r include the effects of incident and reflected sound waves.

We note, in general, for sound propagation in three dimensions that the instantaneous sound intensity I is a vector quantity equal to the product of the scalar sound pressure and the instantaneous vector particle velocity u. Thus I has magnitude and direction. The vector intensity I may be resolved into components I_x, I_y, and I_z. For a more complete discussion of sound intensity and its measurement see Chapter 8 in this book, chapters 45 and 156 in the Handbook of Acoustics [1] and the book by Fahy [13].