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Figure 6.8 The Geometry of a Doublet Source

Receiver

Figure 6.8 The Geometry of a Doublet Source

Receiver

The total power radiated by the doublet can be calculated by integrating the square of the pressure over all angles

Uf p

Po r2 sin 9 d9

to obtain

sin k d kd

where the plus sign refers to an in-phase doublet and the minus sign to an out-of-phase doublet.

Now there is a great deal of important information in these few equations. Plotting Eq. 6.41 in Fig. 6.9, we can examine the relative power of a doublet compared to a simple source. The top curve shows the power of an in-phase pair as a function of k d. When the two sources are close (compared with a wavelength) together, k d is less than 1.

Therefore, when k d is less than 1, the separation distance d between sources is less than a sixth of a wavelength. For this configuration, the acoustic pressure effectively doubles, the combined source power increases by a factor of four, and the sound power level increases by 6 dB.

Figure 6.9 Total Power Radiated from a Doublet Source (Rossing and Fletcher, 1995)

The total power W of two omnidirectional sources having the same f+ +) or the opposite (+ -) polarity and separation distance d, as a Function of the Frequency parameter k.d. The power radiated by a single source is .

The total power W of two omnidirectional sources having the same f+ +) or the opposite (+ -) polarity and separation distance d, as a Function of the Frequency parameter k.d. The power radiated by a single source is .

kd

One way this can occur in buildings is when a source is placed close to a hard surface such as a concrete floor or wall. The surface acts like an acoustic mirror and the original source energy is reflected, as if the source were displaced by a distance d/2 behind the surface of the mirror. If the distances are small enough and the frequency low enough, the pressure radiates from the original and reflected source in phase in all directions. When sound level measurements are made very close to a reflecting surface, a 6 dB increase in level can be expected due to pressure doubling.

As the distance d between each source increases, the angular patterns become more complicated. The radiated power for a doublet at higher values of k d, also shown in Fig. 6.9, is about twice the power of a single source whether or not the sources are in phase. This is the same result that we found for incoherent (random phase) sources—a 3 dB (10 log N) increase for two sources combined.

Although the overall power of a doublet has a relatively simple behavior, the directivity pattern is more complicated. Typically, these directivity patterns are displayed in the form of polar plots. The front of the source, which is usually the loudest direction, is shown pointing toward the top of the diagram and the decrease in level with angle is plotted in increments around the source. A uniform directivity pattern is a perfect circle. The directional characteristic R9 is one commonly encountered descriptor. It is the terms in the parentheses in Eq. 6.37 and represents the directional pattern of the sound pressure for a doublet. The directivity is its square and the directivity index is 10 log of that. The directional characteristic pattern produced by an in-phase doublet at various frequencies is shown in Fig. 6.10. Note that the direction of greatest level, when 9 is 0, is at right angles to the line between the sources. The directivity patterns vary with frequency. The half-beamwidth angle, which is defined as the angle between on axis and the first zero, occurs when (k d/2) sin 9 = n/2. The half beamwidth for an in-phase doublet is f = sin-1 (X/2d) (6.43)

Dipole Sources and Noise Cancellation

If we have a doublet, where the two sources have opposite polarities, the configuration is called a dipole. A practical example of this type of source is an unbaffled loudspeaker. Since sound radiates from the rear of the loudspeaker cone as well as from the front, and the two signals are out of phase, the signal from the rear can combine with the front signal and produce a null pattern at right angles to the axis of the cone.

If the dipole sources are close together, when k d < 1, from Fig. 6.9 we see that the total power radiated approaches zero. This is the basis for the field of active noise cancellation in a three-dimensional acoustic space. Two sources of opposite polarity, when positioned close enough together, radiate a combined null signal. In practice, this can be accomplished by generating a cancellation signal quite close (d less than A/6) to the source or to the receiver. A microphone can be used to sense the primary noise signal and by appropriate processing a similar signal having the opposite phase can be produced. Active noise cancellation systems are available in the form of headphones, which suppress sounds having a frequency of less than about 300-400 Hz. Source-cancellation systems for environmental noise control are less common, but have been applied successfully to large transformers and exhaust stacks. Noise cancellation systems have not been applied to the general run of noise problems, due

Figure 6.10 Directional Characteristics in Terms of R of an In-Phase Doublet Source as a Function of the Distance between the Sources and the Wavelength (Olson, 1957)

The direction corresponding to the angle 0 is measured relative to the perpendicular to the line connecting the two sources. The three dimensional polar plots are surfaces of revolution about the line joining the two sources.

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