What are analogies in physics?

• In physics, analogy is a powerful tool for studying different systems that obey laws which are identical from a mathematical point of view.
• Feynman summed up this principle with the phrase "the same equations have the same solutions". It means that if two physical quantities, even belonging to completely different physical systems, obey the same equations, they will behave in the same way over time, that is, they will follow the same dynamics.
• Analogy becomes an important tool because it allows the language, formalism and results developed in one area of physics (such as the theory of electrical circuits) to be applied to another, even a completely different one (such as acoustics).
• On the basis of analogy, all linear systems, whatever their nature, have electrical dynamic equivalents. This is very convenient, because electrical signals are easy to build, manipulate and study (compared, for example, with mechanical ones). This suggests that mechanical, acoustic or thermal systems can be studied almost exactly like electrical circuits. (See necessary cautions below.)

An example

• a simple electrical circuit made of a battery and a resistance (for example a light bulb) satisfies Ohm's law at every instant

${\displaystyle V=RI}$

where V is the potential difference produced by the battery and applied across the bulb at a given instant, I the current flowing through the bulb at the same instant, and R the resistance of the bulb.

• Since, to a first approximation, the resistance is a physical property of the bulb, which we assume to be constant, Ohm's law simply says that the current flowing through it at every instant is directly proportional to the applied potential difference.
• Let us now consider a simple hydraulic system made of a pump and a pipe in which water flows (slowly). If P is the pressure developed by the pump at a given instant, and Q the water flow rate in the pipe at the same moment, to a first approximation we can describe the flow in the circuit by the law

${\displaystyle P=R_{i}Q}$,

where ${\displaystyle R_i}$ is a property of the pipe in which the water flows.

• This law (which can be derived rigorously) simply tells us that the water flow rate is directly proportional to the pressure difference between the ends of the pipe. It is the Hagen-Poiseuille law, valid when the flow is slow and regular (laminar): for a cylindrical pipe of radius r and length l one finds

${\displaystyle R_i=\frac{8\eta l}{\pi r^4}}$,

where ${\displaystyle \eta }$ is the viscosity of the liquid.

The analogy between the electrical system and the hydraulic system is evident, so much so that we can study the hydraulic system as if it were an electrical system, and vice versa, and call the constant ${\displaystyle R_i}$ the hydraulic resistance of the pipe.

Necessary cautions

Analogy is a powerful tool, but it must be used with some caution.

1. All analogies generally hold in a specific regime, or for a limited range of parameters.
   • For example, there are conductors to which Ohm's law does not apply, and there are turbulent regimes in which the hydraulic equation changes form. Whether a simple resistive equation of motion is enough to describe the dynamics of the system under study must be assessed case by case.
2. The analogy concerns the form of the solutions of the equations of motion, and not the nature of the quantities involved.
   • For example, from the electrical-hydraulic analogy one can deduce that, in the linear regime, the current in an ohmic circuit behaves like an ideal fluid in a pipe, but not that electric current is a fluid. The physical nature of the two quantities is very different, even though their behaviour in certain situations may be the same.
3. The fact that two systems are dynamically analogous does not mean that all the physical laws valid for one must hold by analogy for the other
   • For example, except at very high frequencies, electrical circuits obey Kirchhoff's first law (or junction rule), according to which the algebraic sum of the currents entering and leaving a node is zero. In an acoustic system, instead, the analogous law for the airflow does not hold for an arbitrary region: air is compressible, so part of the air flowing into a region can accumulate there by being compressed, without flowing out instantly. The junction rule holds again in lumped acoustic circuits if the compressibility of each volume of air is represented by a separate element, the acoustic compliance, analogous to a capacitor: the flow "entering" the compliance is the flow that compresses the air, just as the current entering a capacitor is the one that accumulates charge on its plates. This is precisely why an electrical circuit can be built that behaves like a given acoustic system.

Finally, an important clarification, which must be kept in mind especially when the subject of study is the waves passing through a physical system.

Just as electrical circuits are broken down into elementary components, each associated with a particular impedance element, can "acoustic circuits" likewise be drawn on the basis of the electro-acoustic analogy?

The answer is yes, but it requires an important clarification: representing a complex system of oscillators as a network of many elementary oscillators (perhaps harmonic ones) is a standard technique in physics. This technique is used both to describe mechanical structures (see normal modes and the wave equation in the chain of oscillators) and for electrical circuits (see oscillating circuits, radio). However, when the fundamental quantities needed for the oscillation are "lumped" into single "point-like" elements, we are assuming that the wavelength of the waves that can travel through the system is much larger than the size of the elementary components. If a wave had a wavelength smaller than or equal to the size of the oscillating element, this element could in turn be represented as a network of smaller elements, which would mean that it is not an "elementary" component. Equivalently, we can say that when we represent a physical system carrying waves as a network of elementary oscillators, the minimum wavelength we can describe in that system is equal to (twice) the minimum distance between two consecutive oscillators. An analogue of this rule in time rather than in space is of great importance for sound sampling (see also the guided tour of the Fourier app on analogue sound and digital sound)

For electrical circuits this observation becomes important in two cases:

1. when extremely miniaturised circuits operate at very high frequency
2. in high-voltage transmission networks, which can be very extended and can therefore propagate "waves" of large wavelength.

For mechanical systems the observation is essentially relevant when simulating the structure of matter at the atomic level. In this case, of course, there can be no elementary oscillator smaller than a single atom.

In acoustic systems, on the other hand, it is common for the emitted waves to have a wavelength much smaller than the size of the resonator that emits them (this generally holds for the higher harmonics of a given fundamental, which instead usually has a wavelength comparable to the size of the resonator). In this case the division into elementary impedance components outlined below is not possible.

Analogy for linear lumped-element systems

Assuming, then, that the waves involved in a system have a wavelength much larger than the components of the system, it is possible to establish a close analogy between the elementary components of each system, in each of which a single dynamic property is imagined to be concentrated.

Tables

Some physical quantities belonging to different types of systems related by dynamic analogy.

Dynamic equations for the three main types of analogous quantities

Meaning of the symbols

Thermal systems have no analogue of mass: heat has no inertia and always flows from the hotter body to the colder one. For this reason a thermal system, unlike mechanical, acoustic and electrical ones, cannot oscillate freely.

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