Textbook 'The Physics of Waves': Calculating Work Done by Force

[brettng]

Homework Statement

To calculate work done by external force on a simple harmonic oscillator, we need to use real force and real displacement, because the work is a nonlinear function of force. Show that the work is a nonlinear function of force.

Relevant Equations

$W=\int F \, dx$

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

Chapter 2 - Section 2.3.1 [Page 45] (see attached file)

Question: In the content, it states that we need to use real force and real displacement, because the work is a nonlinear function of force.

I understand that “nonlinear” means that a linear combination of 2 forces (i.e. real part of and imaginary part for complex solution of force) is generally not a solution, even though the real part (and the imaginary part) is individually a solution. But how to show the nonlinearity explicitly with mathematics?

In other words, could anyone prove this statement please?

Grateful if someone could help. Thank you!

 

[TSny]

I understand that “nonlinear” means that a linear combination of 2 forces (i.e. real part of and imaginary part for complex solution of force) is generally not a solution, even though the real part (and the imaginary part) is individually a solution. But how to show the nonlinearity explicitly with mathematics?

I don't think that this is the meaning of "nonlinear" here. The power associated with the driving force is equal to the product of the real part of the force and the real part of the velocity. But the amplitude of the velocity is itself proportional to the amplitude $F_0$ of the driving force. (See equations (2.19), (2.23), and (2.24). So, the power is proportional to the square of $F_0$. This means that the power is a nonlinear function of the driving force.

However, it seems to me that there is a more basic reason why you can't get the power by multiplying the complex force by the complex velocity and then taking the real part. In general, for complex numbers $z_1 = a + ib$ and $z_2 = c + id$, you can easily check that $\textrm{Re} (z_1 \cdot z_2) \neq \textrm{Re}(z_1) \textrm{Re}(z_2)$. So, to get the power of the driving force, you must first take the real parts of the complex force and the complex velocity before multiplying.

 

[brettng]

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