Computer Algebra Recipes

168 CHAPTER 4. NONLINEAR ODE MODELS
4.2.2 Oh What Sounds We Hear!
A man falls in love through his eyes, a woman through her ears.
Woodrow Wyatt, British journalist, To the Point,\The Ears Have It," 1981
Wyattarguesthatwhatissaidtoawomanbyaman,andwhatshebelieves
about his status, is usually more important than the super¯ciality of good looks.
At the end of the nineteenth century, the famous German scientist Heinrich
Helmholtz knew that the ear perceives frequencies that are not present in the
incident acoustic waves. He attributed this to an asymmetric nonlinear response
of the eardrum's tympanic membrane and developed a simple mechanical model
for the eardrum's vibrations.
In the present century, a nervous young mathematics major by the name of
Mike has an hour or so to fret over his upcoming date with Vectoria. Why is
Mike nervous? This is the night that he plans to reveal his serious intentions for
her hand, and he is worried about possible rejection. To calm him down, let's
try to take his mind o® Vectoria by asking him to reproduce Helmholtz's derivation.
Helmholtz's model involves applying Newton's second law to generate a
phenomenological nonlinear ODE describing the one-dimensional displacement
x(t) of the freely vibrating eardrum about the equilibrium position x =0.
Reluctantly acquiescing to our request, Mike begins by loading the plots
library package, and then formally Taylor expanding the restoring force per
unit mass, F (x), about x = 0 out to third order in x.
> restart: with(plots):
> Force:=taylor(F(x),x=0,3);
Force := F (0) + D(F )(0) x + 1
2 (D(2) )(F )(0) x2 +O(x3 )
The coe±cients D(F )(0) and D (2) (F )(0) in the above output stand for the ¯rst
and second derivatives of F with respect to x, evaluated at x = 0, respectively.
Mike then assumes that the displacement x is su±ciently small that only terms
to second order in x have to be retained. So he removes the O(x3 )term.
> Force:=convert(Force,polynom);
Force := F (0) + D(F )(0) x + 1
2 (D(2) )(F )(0) x2 Why keep the second-order term? The term that is linear in x corresponds to
Hooke's law, while the quadratic term is the ¯rst nonlinear correction to Hooke's
law and is responsible for the asymmetric response noted by Helmholtz. As the
eardrum vibrates about the equilibrium position x = 0, the linear (x) termin
the force changes sign with x, but the quadratic (x2 ) term does not change sign.
In equilibrium the restoring force must vanish, so Mike sets F (0) = 0. He
also makes the symbolic substitutions D(F )(0) = ¡! 2 0 and D(2) (F )(0) = ¡2 ¯,
> Force:=subs(fF(0)=0,D(F)(0)=-omega[0]^2,(D@@2)(F)(0)=
-2*betag,%);
Force := ¡!0 2 x ¡ ¯x 2