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References: Complex Numbers and Wave Superposition

  1. Complex number - Wikipedia - Introduces the imaginary unit, the real and imaginary parts, and the complex plane, the two-dimensional number system this chapter builds from before connecting it to rotation via Euler's formula.

  2. Euler's formula - Wikipedia - Establishes e^(iθ) = cos(θ) + i sin(θ), the single equation this chapter identifies as unifying rotation, the unit circle, and sine/cosine waves into one rotating complex number.

  3. Superposition principle - Wikipedia - States that the combined response of a linear system to multiple stimuli is the sum of the individual responses, the physical rule underlying this chapter's wave addition, interference, and beat frequency sections.

  4. Visual Complex Analysis (25th Anniversary Edition) - Tristan Needham - Oxford University Press - Replaced two centuries of algebraic complex-number derivations with geometric diagrams showing multiplication by e^(iθ) as pure rotation, the same rotating-point intuition this chapter builds from the unit circle to Euler's formula.

  5. The Feynman Lectures on Physics, Volume I - Richard Feynman - Addison-Wesley / California Institute of Technology - Chapter 48, "Beats," is the classic, widely cited derivation of how two waves of nearby frequency sum into a slow amplitude pulsation, exactly the beat-frequency and wave-addition intuition this chapter presents.

  6. Intuitive Understanding of Euler's Formula - BetterExplained - Presents Euler's formula as two equivalent descriptions of circular motion rather than a memorized identity, reinforcing this chapter's framing of e^(iθ) as a point rotating around the unit circle.

  7. Lecture 7: Continuous-Time Fourier Series - MIT OpenCourseWare - Covers representing periodic signals as sums of harmonically related complex exponentials, extending this chapter's closing discussion of the Fourier series and the continuous Fourier transform.

  8. Phasor and Complex Numbers - Engineering LibreTexts - Shows how a rotating phasor built from Euler's identity projects onto the real axis to produce a sinusoid, reinforcing this chapter's link between phase offset, rotation, and wave equations.

  9. Superposition and Interference - GeeksforGeeks - Worked-example tutorial on constructive and destructive interference and path-difference conditions, reinforcing this chapter's table comparing how in-phase and out-of-phase waves combine.

  10. Wave Interference and Beat Frequency - Academo - Interactive demo letting users adjust two sine wave frequencies and hear the resulting beat, directly illustrating this chapter's beat frequency formula and its audible "wah-wah-wah" pulsing.