By Leo L. Beranek
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Extra resources for Acoustical Measurements
C. Plane Waves The pressure wavefronts that radiate from a monopole source are spherical in shape. However, a receiver far away from the source does not notice this curvature of the wavefront. The wavefront seems to be a flat surface, just as the surface of the earth seems to be flat because we are so far away from the center. The flat wavefront is known as a plane wave. A plane wave is characterized only by its propagation direction. ) The pressure does not depend upon the other two directions of space.
If the signal is propagating as a plane wave (more about which later) then the rms velocity is related to the pressure by a simple proportionality: u = p/z, (18) where z is the specific acoustical impedance. This impedance is equal to the product of the air density and the speed of sound: z = av. 21 k g / m 3, and with a speed of sound of v = 344 m/s, the specific impedance is 415 kg/m2s or 415 rayls. 8 x 10 -s m/s. This is enormously slower than the rms velocity of air molecules due to the kinetic energy of thermal motion at room temperature, which is about 500 m/s.
We assert here that the frequency ~o(t) is the time derivative of the instantaneous phase of a sine. If the signal is x(t) = sin[O(t)], (99) then the instantaneous phase is O. ) It follows that the phase is the integral of the time-dependent frequency: O(t) = f t dt' co(t'). (100) In an FM signal, the instantaneous frequency consists of a constant carrier frequency, coc, plus a variation. If the variation is sinusoidal, with frequency OJm, then co(t) = oJ, + Ao~ COS((Omt+ ~). (101) Therefore, the phase is given by O(t) = coct + Aco sin(cOmt + cb).
Acoustical Measurements by Leo L. Beranek