Plane-Wave Propagation
9.2 The Bobbing Boats
Imagine a large swimming pool in which there are several toy boats arrayed in a straight line. Although we use Figure 9.1 for a slightly different purpose elsewhere, for now we can imagine that the black dots in that figure represent our toy boats. Far across the pool, someone is slapping the water repeatedly, generating waves that proceed outward, in essentially concentric circles, across the pool. By the time the waves reach the boats, the circular shape has flattened out so that the wavefronts are essentially straight lines.
The straight lines in Figure 9.1 can represent these wavefronts.
As the wavefronts reach the boats, the boats bob up and down. If the lines of the wavefronts were oriented parallel to the line of the boats, then the boats would bob up and down in unison. When the wavefronts come in at some angle, as shown in the figure, the boats will bob up and down out of syncwith one another, generally. By measuring the time it takes for the peak to travel from one boat to the next, we can estimate the angle of arrival of the wavefronts. This leads to two questions:
1. Is it possible to get the boats to bob up and down in unison, even though the wavefronts arrive at an angle, as shown in the figure?
2. Is it possible for wavefronts corresponding to two different angles of arrival to affect the boats in the same way, so that we cannot tell which of the two angles is the real one?
We need a bit of mathematical notation. We let the distance from each boat to the ones on both sides be a constant distance Δ. We assume that the water is slapped f times per second, sof is thefrequency, in units of cycles per second. As the wavefronts move out across the pool, the distance from one peak to the next is called thewavelength, denotedλ. The product λf is the speed of propagation c; so λf =c. As the frequency changes, so does the wavelength, while the speed of propagation, which depends solely on the depth of the pool, remains constant. The angleθ measures the tilt between the line of the wavefronts and the line of the boats, so thatθ= 0 indicates that these wavefront lines are parallel to the line of the boats, whileθ=π2 indicates that the wavefront lines are perpendicular to the line of the boats.
FIGURE 9.1: A uniform line array sensing a plane-wave field.
Ex. 9.1 Let the angleθ be arbitrary, but fixed, and let Δbe fixed. Can we select the frequency f in such a way that we can make all the boats bob up and down in unison?
Ex. 9.2 Suppose now that the frequencyf is fixed, but we are free to alter the spacingΔ. Can we choose Δso that we can always determine the true angle of arrival?
9.3 Transmission and Remote Sensing
For pedagogical reasons, we shall discuss separately what we shall call the transmission and the remote-sensing problems, although the two prob-lems are opposite sides of the same coin, in a sense. In the one-dimensional transmission problem, it is convenient to imagine the transmitters located at points (x,0) within a bounded interval [−A, A] of the x-axis, and the measurements taken at pointsP lying on a circle of radiusD, centered at the origin. The radius D is large, with respect to A. It may well be the case that no actual sensing is to be performed, but rather, we are simply interested in what the received signal pattern is at points P distant from the transmitters. Such would be the case, for example, if we were analyzing or constructing a transmission pattern of radio broadcasts. In the remote-sensing problem, in contrast, we imagine, in the one-dimensional case, that our sensors occupy a bounded interval of thex-axis, and the transmitters or reflectors are points of a circle whose radius is large, with respect to the size of the bounded interval. The actual size of the radius does not matter and we are interested in determining the amplitudes of the trans-mitted or reflected signals, as a function of angle only. Such is the case in astronomy, far-field sonar or radar, and the like. Both the transmission and remote-sensing problems illustrate the important role played by the Fourier transform.
9.4 The Transmission Problem
We identify two distinct transmission problems: the direct problem and the inverse problem. In the direct transmission problem, we wish to deter-mine the far-field pattern, given the complex amplitudes of the transmitted signals. In the inverse transmission problem, the array of transmitters or
reflectors is the object of interest; we are given, or we measure, the far-field pattern and wish to determine the amplitudes. For simplicity, we consider only single-frequency signals.
We suppose that each point x in the interval [−A, A] transmits the signal f(x)eiωt, where f(x) is the complex amplitude of the signal and ω >0 is the common fixed frequency of the signals. LetD >0 be large, with respect toA, and consider the signal received at each pointPgiven in polar coordinates byP= (D, θ). The distance from (x,0) toP is approximately D−xcosθ, so that, at time t, the pointP receives from (x,0) the signal f(x)eiω(t−(D−xcosθ)/c), where c is the propagation speed. Therefore, the combined signal received atP is
B(P, t) =eiωte−iωD/c A
−A
f(x)eixωcosc θdx.
The integral term, which gives the far-field pattern of the transmission, is F(ωcosθ
c ) =
A
−A
f(x)eixωcosc θdx, whereF(γ) is the Fourier transform off(x), given by
F(γ) = A
−A
f(x)eixγdx.
HowF(ωcosc θ) behaves, as a function ofθ, as we changeA and ω, is dis-cussed in some detail in the chapter on direct transmission.
Consider, for example, the functionf(x) = 1, for|x| ≤A, andf(x) = 0, otherwise. The Fourier transform off(x) is
F(γ) = 2Asinc(Aγ), where sinc(t) is defined to be
sinc(t) =sin(t)
t ,
fort= 0, and sinc(0) = 1. Then F(ωcosc θ) = 2A when cosθ= 0, so when θ = π2 and θ = 32π. We will have F(ωcosc θ) = 0 when Aωcosc θ = π, or cosθ = Aωπc. Therefore, the transmission pattern has no nulls if Aωπc >1.
In order for the transmission pattern to have nulls, we needA > λ2, where λ=2πcω is the wavelength. This rather counterintuitive fact, namely that we need more signals transmitted in order to receive less at certain locations, illustrates the phenomenon of destructive interference.
9.5 Reciprocity
For certain remote-sensing applications, such as sonar and radar array processing and astronomy, it is convenient to switch the roles of sender and receiver. Imagine that superimposed plane-wave fields are sensed at points within some bounded region of the interior of the sphere, having been transmitted or reflected from the pointsP on the surface of a sphere whose radiusDis large with respect to the bounded region. Thereciprocity principletells us that the same mathematical relation holds between points P and (x,0), regardless of which is the sender and which the receiver.
Consequently, the data obtained at the points (x,0) are then values of the inverse Fourier transform of the function describing the amplitude of the signal sent from each pointP.
9.6 Remote Sensing
A basic problem in remote sensing is to determine the nature of a distant object by measuring signals transmitted by or reflected from that object.
If the object of interest is sufficiently remote, that is, is in the far field, the data we obtain by sampling the propagating spatio-temporal field is related, approximately, to what we want byFourier transformation. The problem is then to estimate a function from finitely many (usually noisy) values of its Fourier transform. The application we consider here is a common one of remote-sensing of transmitted or reflected waves propagating from distant sources. Examples include optical imaging of planets and asteroids using reflected sunlight, radio-astronomy imaging of distant sources of radio waves, active and passive sonar, and radar imaging.