11

Acoustic field of single pulsing sphere

The relation between velocity  v_r  of particles and momentary pressure produced at the studied point by the shift of medium particles is determined by the acoustic impedance of medium Z_a which in case of a single pulsing sphere is [5, p.25]

where ₀ is the medium density, P = p - p₀   is the momentary excess pressure at the studied point, and p₀ is the stationary pressure in gas.

Substituting (1) into (2), we yield

Taking the radius of non-disturbed sphere as  a  and the disturbing pressure variation as

we can easy determine the constant A :

With (5) we have fully determined the amplitude values of the main parameters of an acoustic field. But to study the wave fronts superposition, we will be further interesting in the vector characteristics of parameters _r and . To determine them, note that both in (1) and (2) we use not simply some velocity of particles but just the radial velocity _r. In case of a single pulsing sphere this difference is insufficient, because with strongly radial direction of wave propagation = _r. But in case of superposition of a few sources, the direction of resulting velocity will be not with the direction of momentary velocity of separate sources. So we will consider (1) as a scalar part of general expression for the velocity of particles of the acoustic field.

Given this, we have to present the momentary excess pressure P   as the vector value, since it is produced as a result of directed shift of medium particles radially from the field source. This is the difference of dynamical pressure from that stationary whose direction is known to be equal in all directions of the picked out region. As it follows from (2), the phase of pressure variation does not coincide with the phase of  _r  variation and has generally a complicated regularity [6, p.159]:

where (_p - _v) is the difference of phases between the momentary acoustic pressure and momentary velocity of particles of medium, R_a   is the active acoustic impedance, and X_a  is the reactive acoustic impedance.

With the help of above reasoning, we can write (6) as

then (3) will take the following form:

As we see from (8), in real variables  P_Im (r,t)   decreases asymptotically, proportionally to the first order of distance, and the phase of pressure varies with the distance in a complex way. Such combination leads to the quadratic regularity of  P_Im (r,t)  with respect to distance in the near field and causes the transition region between the near and far fields, which we will see in our further investigation. We can corroborate this quadratic regularity by a simple transformation of (11) to the following form:

In (12) the amplitude of the first term in braces stabilises at large r, and the amplitude of the second term decreases proportionally to the first degree of r. With it the phases of both regularities variation are shifted by 90^o as to each other.

Thus, to plot the pattern of an acoustic field, we can rightfully use (8). The necessity to use the regularity of pressure with respect to distance in real variables will be substantiated in the course of constructing the field pattern itself.