The full mathematical and computational details of how nearfield acoustic holography works are explained in the textbook by Earl Williams [1], but I will give an outline account here to bring out some of the key features. The method fits within a very broad class of mathematical problems called “boundary value problems”: we have a governing differential equation which we wish to solve within some region, plus enough conditions on the boundaries of that region to guarantee that there is only one possible solution.

Mathematicians have devoted a lot of effort over the years to establishing just what is needed in terms of boundary conditions to go with particular differential equations, in order to ensure a unique solution. We will make use of one of those results — but we won’t prove the underlying uniqueness theorem here: for that, the curious reader will have to consult a textbook such as Morse and Feshbach [2].

An easily visualised example of a boundary value problem is to do with heat flow. Suppose we take a block of metal, and we sit it in a bath of icy water so that the bottom and the sides are all held at $0^\circ$ C. But the top surface is left out of the water, so that it is at room temperature, $20^\circ$ C. We have fixed the temperature on all the surfaces of our block, and that is enough information to determine, uniquely, the distribution of temperature inside the block after it has been left to sit for long enough for the temperature to settle to its equilibrium pattern.

Essentially the same boundary conditions are appropriate for our acoustics problem. If we have a region containing no sound sources, the sound field within the region must satisfy the wave equation (see section 4.1.1). If the sound pressure is known at every point on the boundaries of our region, that is enough information to determine the complete internal sound field (these are called “Dirichlet boundary conditions” in the mathematician’s jargon).

Figure 1 shows the region of interest for NAH reconstruction. Measurements are made on a plane, shown as the solid line and labelled as the hologram plane. For the purposes of theory we will imagine that the sound pressure has been measured over this entire plane. The sound pressure distribution over this plane has been caused by our vibrating guitar top (or whatever other object is being tested). We can idealise this as a *source plane*, shown as the dashed line.

We can see straight away that there is a potential ambiguity. Exactly the same sound pressure distribution would be created at the hologram plane if the source plane were swapped to be at the same distance on the opposite side. For an unambiguous reconstruction, the reconstruction method needs to know ahead of time which side the source plane is on. This knowledge is “hard-wired” into the algorithm: the two sides of the hologram plane need to be treated in very different ways.

If we were only interested in reconstructing the sound field above the hologram plane in Fig. 1, things would be easy. The boundary value problem to be solved has the known (measured) sound pressure over the hologram plane. To define the rest of our region for the purposes of the uniqueness theorem stated earlier, we augment the plane with a very large hemisphere, shown dashed, which we imagine to be pushed out “to infinity”. On that hemisphere, we know two things: the sound pressure tends to zero, and it consists entirely of *outgoing waves*. There are no sound sources out there, so no waves can be coming inwards “from infinity”. (In the jargon, this is sometimes called the “Sommerfeld radiation condition”.)

We have now satisfied the conditions of the theorem, and the sound field within this region is uniquely determined. In fact, there is a straightforward way to compute it, which is a close relative to something we have already seen, back in section 4.3.2. Then, we were looking at the radiation of sound by a vibrating plate. Although we didn’t mention it at the time, this was also a boundary value problem which was formally completed by a hemisphere at infinity on which the Sommerfeld radiation condition was satisfied. We showed that the resulting sound field pressure $p(\mathbf{x})$ at position $\mathbf{x}$ was given by the Rayleigh integral

$$p(\mathbf{x}) = e^{i \omega t} \int_S{G(\mathbf{x},\mathbf{x’}) v(\mathbf{x’}) d^2 \mathbf{x’}} \tag{1}$$

where the integral is over the plane $S$ of the vibrating source plate with velocity $v(\mathbf{x’})e^{i \omega t}$ at position $\mathbf{x’}$, $d^2 \mathbf{x’}$ is the element of area in that plane, and the function $G$ is defined by

$$G(\mathbf{x},\mathbf{x’})= \dfrac{i \omega \rho_0}{2 \pi}\dfrac{e^{i k |\mathbf{x}-\mathbf{x’}|}}{|\mathbf{x}-\mathbf{x’}|} . \tag{2}$$

where $\rho_0$ is the density of air and $k=\omega/c$ is the wavenumber of sound waves in air, where $c$ is the speed of sound. Equation (1) takes the form of a *convolution integral*, and the function $G$ is an example of something called called a *Green’s function*. (These are named after George Green, an English mathematician of the early 19th century.)

Our hologram reconstruction problem is very similar, and it can be solved by a convolution integral of exactly the same form as equation (1), with $v(\mathbf{x’})$ replaced by $p_m(\mathbf{x’})$, the measured pressure at position $\mathbf{x’}$ in the hologram plane:

$$p(\mathbf{x}) = e^{i \omega t} \int_H{F(\mathbf{x},\mathbf{x’}) p_m(\mathbf{x’}) d^2 \mathbf{x’}} \tag{3}$$

where now the integral is over the hologram plane $H$. We need a different Green’s function, $F$, which is closely related to the “free space Green’s function” $G$ of equation (2):

$$F(\mathbf{x},\mathbf{x’})=- \dfrac{1}{2 \pi} \dfrac{\partial}{\partial z’}\left[ \dfrac{e^{i k |\mathbf{x}-\mathbf{x’}|}}{|\mathbf{x}-\mathbf{x’}|}\right] \tag{4}$$

where $z’$ denotes the component of $\mathbf{x’}$ in the direction perpendicular to the hologram plane.

To reconstruct the sound field on the other side of the hologram plane, between there and the source plane, is more difficult. The source of sound waves is now on the lower plane, so we can still apply equation (3) to calculate the sound field at any position above that plane in terms of the pressure $p_s$ on the source plane:

$$p(\mathbf{x}) = e^{i \omega t} \int_S{F(\mathbf{x},\mathbf{x’}) p_s(\mathbf{x’}) d^2 \mathbf{x’}} \tag{5}$$

where the integral is now over the source plane $S$. But we don’t know $p_s$! Our measurement is of $p_m$, which is inside the new region. They are, of course, related by

$$p_m(\mathbf{x}) = \int_S{F(\mathbf{x},\mathbf{x’}) p_s(\mathbf{x’}) d^2 \mathbf{x’}} \tag{6}$$

where both $p_s$ and $p_m$ are assumed to be sinusoidal at frequency $\omega$.

We need to invert equation (6) to obtain $p_s$ in terms of $p_m$. The key to that is to Fourier transform everything in sight, and take advantage of a mathematical result called the *convolution theorem* (we proved the basic version of this theorem back in section 2.2.8). That theorem states that the Fourier transform of a convolution integral is the product of the separate Fourier transforms of the input function ($p_s$ in our case) and the Green’s function $F$. There may be a numerical factor as well, depending on the particular convention in use for Fourier transforms: for my convention (see section 2.2.1) there is a factor $4 \pi^2$ for the case we are interested in, which involves a two-dimensional spatial Fourier transform in the two in-plane directions (denoted $x$ and $y$), without affecting the dependence on the perpendicular direction $z$. If we denote Fourier transformed functions by a hat, then for example

$$\hat{F}(k_1,k_2,z)=\dfrac{1}{4 \pi^2}\int_{-\infty}^\infty{\int_{-\infty}^\infty{F(x,y,z)e^{i k_1 x+ik_2 y} dx}dy} \tag{7}$$

where $k_1$ and $k_2$ are the wavenumbers in the $x$ and $y$ directions.

By virtue of the convolution theorem, equation (6) then becomes

$$\hat{p}_m=4 \pi^2 \hat{F} \hat{p}_s \tag{8}$$

so that we can accomplish the inversion we need trivially:

$$\hat{p}_s=\dfrac{\hat{p}_m }{4 \pi^2 \hat{F}} . \tag{9}$$

An inverse Fourier transform then gives the pressure distribution we need. However, there is a snag to be considered. Because of the division by $\hat{F}$, the result may become seriously inaccurate if that function becomes small for some values of the wavenumbers $k_1$ and $k_2$. This is indeed a danger, for reasons that relate to another topic we investigated earlier.

The physical interpretation of $\hat{F}$ for particular values of $k_1$ and $k_2$ is that it describes the amplitude and phase of sound pressure at a distance from the source plane, in response to a *plane wave* of pressure in that source plane. The plane wave has a resultant wavenumber $\sqrt{k_1^2+k_2^2}$. This is a similar question to the one we looked at in section 4.3, with details in section 4.3.3: the sound radiated by a plane wave travelling in a plate in contact with the air.

What we found in section 4.3 was that when the resultant wavenumber on the surface is smaller than the wavenumber $k$ for sound in air, propagating waves are generated so that on average the amplitude of sound is the same at all distances from the surface. But when the resultant wavenumber on the surface is greater than $k$, *evanescent waves* are generated which decay exponentially away from the surface. Figure 2 reminds us what these evanescent waves look like: it is a repeat of Fig. 13 from section 4.3.

We learn three important things from this. First, high wavenumbers in the Fourier transform are indeed likely to lead to very small values of $\hat{F}$ because of the exponential decay, which gets faster as the wavenumber gets bigger. The consequence for practical implementations of NAH is that it is necessary to impose a low-pass filter on the wavenumbers used in the Fourier decomposition, before the inverse transform is performed. Second, we see why it is useful to do the measurement with the hologram plane as close as possible to the source plane. The shorter the distance over which $\hat{F}$ operates, the less scope there is for the exponential decay to produce very small values. Finally, we see why the word “nearfield” gets into the name of this method. To obtain maximum resolution for features in the source plane, the hologram plane needs to be deep in the nearfield of the radiated sound, so that the evanescent field components can be sampled without being overwhelmed by measurement noise.

[1] Earl G. Williams, “Fourier Acoustics”, Academic Press (1999).

[2] P. M. Morse and H. Feshbach, “Methods of Theoretical Physics”, Volume I, McGraw-Hill (1953).