Section 25–2 Superposition of solutions

(Principle of superposition / Radio tuning / Two useful methods)

 

In this section, Feynman discusses the principle of superposition and how it is related to radio tuning, as well as two useful problem-solving methods (Fourier series and Green’s function).

 

1. Principle of superposition:

“This is an example of what is called the principle of superposition for linear systems, and it is very important (Feynman et al., 1963, section 25–2 Superposition of solutions).”

 

According to Feynman, the principle of superposition means that a complicated force can be broken up into a sum of separate pieces in any convenient manner. To get the complete answer, we can add the pieces of the solution together just like the total force is a sum of the pieces. However, Feynman’s figure is not simply about some simple forces that are in accordance with his explanation of the superposition principle. Essentially, he is applying the superposition of waves that are ideal sinusoidal waves. The discussion of radio tuning in the middle of the section shortly after is also based on the superposition of waves. Similarly, the Fourier series method that can be used for radio tuning is related to the superposition of waves.

 

Feynman explains that the laws of electricity (instead of laws of electromagnetism), Maxwell’s equations, which determine the electric field, turn out to be differential equations that are linear. On the other hand, there are nonlinear Maxwell’s equations and thus, Maxwell’s equations are linear because they involve idealizations and approximations. Historically, the Born–Infeld model is a field theory that is also known as nonlinear electrodynamics (Born & Infeld, 1934). In chapter 50 of Feynman’s lectures (Volume I), he clarifies that “when we discussed the transmission of light, we assumed that the induced oscillations of charges were proportional to the electric field of the light—that the response was linear. That is indeed a very good approximation (Feynman et al., 1963).”

 

2. Radio tuning:

“That is how radio tuning works; it is again the principle of superposition, combined with a resonant response (Feynman et al., 1963, section 25–2 Superposition of solutions).”

 

Feynman says that a radio station transmits an oscillating electric field of very high frequency which acts on our radio antenna. For radio tuning, we can adjust the natural frequency of a radio by changing the L or the C of its circuit. The reception of radio waves is not merely dependent on the principle of superposition and one radio-frequency. In chapter 50, Feynman adds that “… the amplitude of cos ω₁t is modulated with the frequency ω₂. We would now say that two new components have been produced, one at the sum frequency (ω₁+ω₂), another at the difference frequency (ω₁−ω₂) (Feynman et al., 1963).” To have a deeper understanding, there are at least three fundamental principles involved: generation, transmission, and reception of radio waves.

 

Feynman explains that the amplitude of the oscillating field is changed, modulated, to carry the signal of the voice, and we are not going to worry about it. Some may not understand the meaning of modulated that is used by Feynman in his explanation. This is not surprising because Feynman was able to fix radios when he was a teenager (Feynman, 1997, pp. 15-21). In addition, Feynman had some working knowledge of ham radio when he applied it in Brazil (Feynman, 1997, p. 211). Interestingly, Feynman likely had advanced knowledge of radio frequency because he was involved in a project related to radar (Feynman, 1997, p. 102). The word radar means “RAdio Detection And Ranging.”

 

3. Two useful methods:

“Out of the many possible procedures, there are two especially useful general ways that we can solve the problem (Feynman et al., 1963, section 25–2 Superposition of solutions).”

 

Feynman briefly discusses two useful problem-solving methods that are based on the principle of superposition: Fourier series and Green’s function. For the Fourier series method, he mentions that practically every curve can be obtained by adding together infinite numbers of sine waves of different frequencies. This method can also be used for radio tuning to determine the different frequencies of radio waves emitted by a radio station. In chapter 50, Feynman elaborates Fourier series in more detail and explains that this method is applicable to discontinuous curves. In a sense, our human ear-brain audio system is able to perform Fourier series to the extent that some people can distinguish the frequencies of a chord.

 

Feynman describes how a force can be likened to a succession of blows (or impulses) with a hammer and Green’s function is a method of analyzing any force by putting together the response of impulses. Note that the horizontal axis of “Fig. 25–4 A complicated force may be treated as a succession of sharp impulses” was labeled x in the first edition, but it is later revised to t that means time. Perhaps Feynman could have revealed the usefulness of Green’s function in his path integrals. In his Ph.D. thesis, Feynman has applied the Green’s function method in a forced harmonic oscillator problem from the point of view of his modified quantum mechanics. Furthermore, he simply calls it G function and one of which is G_γ(x, x′; T). This could be a reason why Fig. 25-4 was labeled x instead of t.

 

Questions for discussion:

1. How would you state the principle of superposition?

2. How would you explain the principles of radio tuning?

3. Why did Feynman discuss the Fourier Series and Green’s function method? 

 

The moral of the lesson: we can solve linear problems such as those related to radio frequencies by using Fourier Series and Green’s function.

 

References:

1. Born, M. & Infeld, L. (1933). Foundations of the new field theory. Nature, 132(3348), 1004-1004.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.