For a more careful mathematical description of monopoles, dipoles and so on, we first examine the pulsating sphere result from section 4.1.2 for the case of very small Helmholtz number $\omega a/c$. The amplitude factor $A$ from eq. (8) then takes the approximate form

$$A \approx -\omega^2 a’ a^2 \rho_0 = i \omega a^2 v_a \rho_0 \tag{1}$$

where $v_a$ is the velocity on the surface of the pulsating sphere. We can write this in terms of the net volume flux $q=4 \pi a^2 v_a$, and thus obtain an expression for the far-field pressure

$$p=\dfrac{i \omega \rho_0 q}{4 \pi r} e^{i \omega(t-r/c)} . \tag{2}$$

Using the same approximation, we can obtain the radial velocity of air particles, $v(r)$, using eq. (7) of section 4.1.2:

$$v \approx \dfrac{q}{4 \pi r} \left( \dfrac{1}{r} + \dfrac{i \omega}{c} \right) e^{i \omega(t-r/c)} . \tag{3}$$

For positions $r$ within the near field, the first term in the brackets dominates, and the velocity decays like $1/r^2$. Further out, the second term takes over and the far field decays like $1/r$.

The sphere radius $a$ no longer appears explicitly in either of eqs. (2) or (3). There is a limiting case, in which a vanishingly small sphere produces a volume flux $q$. This will not describe a physical system because the pressure and velocity are singular as $r \rightarrow 0$, but it is a result of great mathematical importance. Strictly, it is this limiting solution that is called an *acoustic monopole*.

Equation (2) is the pressure field produced by a volume source described by a *Dirac delta function*. Such solutions, in which the source term in a governing equation is replaced by a delta function, are known as *Green’s functions*. They can be used to assemble the solution for a more general source distribution by a process of linear superposition known as *convolution*. For the case of sound pressure, this will lead to a result called the *Rayleigh integral*: we will return to it in section 4.3.2.

For the present, we are interested in a different question. We want to obtain a similar approximation for a pair of monopoles with opposite phase, to obtain the idealised acoustic dipole. We can use a trick. Rather than calculating the pressure at a single point due to two monopoles with flux $q$ and $-q$ a short distance $h$ apart, we can calculate the *difference* of pressure in the field from eq. (2), at two *points* that are a distance $h$ apart. Suppose the line joining these two points is parallel to the $z$ axis. In the limiting case of small $h$, the result can be expressed in terms of a derivative of the pressure field:

$$p=-\dfrac{i \omega \rho_0 }{4 \pi} (hq) \dfrac{\partial}{\partial z} \left( \dfrac{e^{i \omega(t-r/c)}}{r} \right) . \tag{4}$$

Now a small movement $dz$ is related to a change $dr$ via the polar angle $\theta$: $dr=dv \cos \theta$, so

$$p=-\dfrac{i \omega \rho_0}{4 \pi} (hq) \dfrac{\partial}{\partial r} \left( \dfrac{e^{i \omega(t-r/c)}}{r} \right) \cos \theta \tag{5}$$

$$=-\dfrac{c \rho_0 }{4 \pi} (hq) \dfrac{\omega^2}{c^2} \left( 1+ \dfrac{c}{i \omega r}\right) \dfrac{e^{i \omega(t-r/c)}}{r} \cos \theta .\tag{6}$$

For this case the pressure has a near-field term decaying like $1/r^2$, given by the second term in the brackets, as well as a far-field term decaying like $1/r$ given by the first term in the brackets. The source strength $q$ and the separation $h$ only appear in the combination $qh$, which we can define as the *dipole moment* $M$.

For this problem, because of the dependence on $\theta$, the particle velocity is not purely radial. However, we learn something interesting if we calculate the radial component of this velocity, by the same method used in section 4.1.2. The result is

$$v_r = – \dfrac{1}{i \omega \rho_0} \dfrac{\partial p}{\partial r} = -\dfrac{M}{4 \pi} \left(\dfrac{\omega^2}{c^2 r} -\dfrac{2 i \omega}{c r^2} -\dfrac{2}{r^3} \right) e^{i \omega(t-r/c)} \cos \theta . \tag{7}$$

On a surface with $r=\mathrm{constant}$, the radial velocity simply varies proportional to $\cos \theta$. But this is the radial velocity produced by rigid motion of a sphere in the direction of the axis from which $\theta$ is defined. So, as claimed in section 4.3, the ideal dipole field is exactly the same as the field generated by rigid oscillation of a sphere.

The same derivative trick can be used to derive the corresponding expression for an ideal quadrupole source: instead of the difference of two dipoles a short distance apart, we can calculate the difference of pressure at two nearby points in a single dipole field. We do not need to work through the details, but one qualitative fact about the solution can be noted. So far, we have seen that the pressure field of a monopole source decays like $1/r$ at all distances, near field or far field. For a dipole, the far field still decays like $1/r$ but the near field decays more rapidly, like $1/r^2$. For a quadrupole, this trend continues: the far field still decays as $1/r$, but the near field decays very rapidly, proportional to $1/r^3$. This is the underlying explanation of the behaviour of sound from a tuning fork, described in section 4.3.