Question

extra credit1

A 2-source dipole system has two in-phase sound sources located at (0, d/2) and (0, -d/2).  Given a sound detector using displacement pressure detection principle, basically a force vector measurement perpendicular to the propagation wave vector of a single sound source, find the total signal intensity expression as a function of distance along the x-axis at large distances, >> wavelength, >> d, when a 1-source system would generate displacement (b/r)sin(k(r - vt)) . Use the binomial expansion to find the first two leading terms in distance.

extra credit2

A 4-source linear quadrupole system has two in-phase sound dipoles located at (0,q/2) and (0, -q/2).  Using the above dipole result, find the total signal intensity expression as a function of distance along the x-axis at large distances, >> wavelength, >> q.  Use the binomial expansion to find the first two leading terms in distance.

if you need any other data, just use any value you want. example, you need time, you can use 0.001s or 0.1s.

Answer 1

The directional sound pattern of a multipole loudspeaker depends on the positions of the acoustic sources, their relative strengths, and their relative phase. In the case of a dipole loudspeaker, like that shown in FIG. 1, even strength acoustic sources (with opposite sign) provide a null plane half way between the sources, with a normal defined by a line connecting the sources. When the distance between the sources is very much less than a wavelength, the pressure on this null plane due to the sources is essentially zero because the pressure due to one source is cancelled by that of the other.

The dependence of the null surface on the strength of the sources and their relative phase may be illustrated for a dipole implementation with reference to FIG. 2. Here, a first source s₁ is located at (0,0,d/2) and a second source s₂ is located at (0,0,−d/2). The pressure around the sources s₁ and s₂ is rotationally symmetric about the z-axis and, therefore, only the x-z plane needs to be considered. At a given angular frequency, ω, the pressure P measured from each source s₁, s₂ at an observation point O may be defined in general as P = p 1 r   j  ( ω     t - k   r  )    equation     ( 1 )

where p₁ is the strength of the source s₁ or s₂ measured at unit distance, r is the distance from the source to the observation point O, k=ω/c is the wave number and c is the speed of sound. Allowing for a phase difference, δ, between the sources s₁and s₂, the total pressure, P_T, at point O is simply the sum of the pressures from the individual sources or P τ = p 1 r 1  j  ( ω     t - k   r 1  ) + p 2 r 2   j  ( ω     t - k   r 2  + δ ) equation     ( 2 )

For r₁,r₂>>d it is evident from FIG. 2 that

|r ₁|=r−d/2 cos θ  equation (3)

and

|r ₂|=r+d/2 cos θ  equation (4)

Substituting equations (3) and (4) into equation (2) yields a total pressure of P τ = p 1 r   j  ( ω     t - kr )  [ ( 1 + p2 p 1 ) + ( 1 - p 2 p 1 )  ( jk  d 2  cos     θ ) + j  p 2 p 1  δ ] equation     ( 5 )

For a null to exist, the real and imaginary parts of equation (5) must each be zero. Satisfying these conditions, the following relationships may be found:

p ₂=−p ₁  equation (6)

 δ=−d/cω cos θ  equation (7)

The above requirements may be used to control the direction of the null in the sound field pattern produced by the two acoustic sources of a dipole implementation. In the particular case when the null is desired in the x-z plane of FIG. 2, for example, it follows that θ=90°.

It should be noted here that the phase difference defined by equation (7) is directly proportional to ω, implying that a corresponding time delay, τ, defined by

τ=−d/c cos θ  equation (8)

may be introduced between the signals to the two acoustic sources s₁ and s₂.

The present invention applies to a two-driver dipole loudspeaker implementation as shown in FIG. 1. As mentioned, there are two acoustic sources in such an arrangement. If the sources are equal in amplitude but opposite in sign, and if there is zero phase difference (δ=0) between the sources, the amplitude measured at a distance is described by a sound directivity pattern graphically illustrated in FIGS. 3a and 3 b. This ‘figure eight’ polar pattern comprises a positive sound pressure lobe 32 and a negative sound pressure lobe 34. Each sound pressure lobe 32, 34 will extend outward from and in opposite directions from the loudspeaker i.e. axially away from the speaker. As discussed, dipoles exhibit a null zone lying in a plane perpendicular to a central longitudinal axis of the positive and negative sound pressure lobes 32, 34. If the upward direction is taken as 0 degrees, it is evident from FIG. 2 that the amplitude is maximum at 0 degrees and zero at 90 degrees.

However, by introducing a phase difference between the two sources, the null direction can be moved as shown in FIGS. 4a and 4 b. Here, a positive sound pressure lobe 42 and a negative sound pressure lobe 44 still exist. The desired null direction was θ=65°≅1.134 radians. To point the null in this direction, the amplitudes should again be equal and opposite in sign, but the phase difference between the sources should now be maintained at δ=−d/c ωcos(1.134). Note that the phase difference is a function of the frequency. In the particular example of FIG. 4, the frequency is taken as ω 2     π = 1000     Hz,

the separation of the acoustic sources is d=12 mm, the speed of sound is c=344 m/s, yielding a phase difference of

δ=−d/cω cos(1.134)≅−0.0926 radians≅−5.3°

It is apparent from FIGS. 4a and 4 b that the maximum still occurs at 0 degrees, but the zero or null now occurs at 65 degrees.