Textbook 'The Physics of Waves': Reason to force us to consider complex solution for harmonic motion?

[brettng]

TL;DR Summary: The reason to force us consider complex solution for harmonic motion.

Reference textbook “The Physics of Waves” in MIT website:
https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/

Chapter 1 - Section 1.3 (see attached file)

In (1.40), it breaks down the time translation from pi/omega to pi/2omega + pi/2omega; and concludes the square of h(pi/2omega) implying that we need to consider complex solution.

However, what prevents us use the property of irreducible solution and adopt

z(t+pi/omega) = h(pi/omega)z(t)

directly? (And this does not force us to use complex solution!)

 

[Frabjous]

You do not need to use the complex formalism. It is just easier.

 

[TSny]

TL;DR Summary: The reason to force us consider complex solution for harmonic motion.

Reference textbook “The Physics of Waves” in MIT website:
https://ocw.mit.edu/courses/8-03sc-...es-fall-2016/resources/mit8_03scf16_textbook/

In (1.40), it breaks down the time translation from pi/omega to pi/2omega + pi/2omega; and concludes the square of h(pi/2omega) implying that we need to consider complex solution.

However, what prevents us use the property of irreducible solution and adopt

z(t+pi/omega) = h(pi/omega)z(t)

directly? (And this does not force us to use complex solution!)

From page 11 of the book:

It is easy to see that a function $z(t)$ that satisfies (1.38) cannot be a real-valued function for all $t$ if there exists a value of $a$ such that $h(a)$ is complex. The author shows on page 12 of the book that if $z(t)$ obeys (1.38) and if $z(t)$ is a solution of the equation of motion for simple harmonic motion (SHM), then $h(a)$ is imaginary for $a = \pi/(2\omega)$.

 

[brettng]

Thank you so much for your help!!!!!!