Computer Algebra Recipes

5.1. CHECKING SOLUTIONS 215
Neglecting sti®ness and assuming Ã(x; 0) = 0,
4 vd
Ã(x; t) =
¼2 1X 1 sin(n¼x0=a)cos(n¼d=(2 a))
c n (1 ¡ (nd=a) 2 sin(n¼x=a)sin(n¼ct=a)
)
n=1
istheshapeofthestringattimet. Verify that this series solution is correct
and animate it for parameter values of your own choosing.
5.1.3 Three Easy Pieces
I would advise you Sir, to study algebra, if you are not already an
adept in it: your head would be less muddy ...
Samuel Johnson, English writer and lexicographer (1709{1784)
Spurred by her earlier success, and feeling happier after her long phone conversation
last night with Mike, Vectoria is pursuing another vibrating string
example, which she has entitled Three Easy Pieces. Thisdoesnotrefertothe
old Jack Nicholson movie with a similar title (cf. Five Easy Pieces), but to
the fact that the algebraic manipulations involved in dealing with plane-wave
propagation along a three-piece string are easy if one uses computer algebra.
When an intermediate section of a very long horizontal string has a greater
mass density ² than the remaining two identical portions of the string, a transverse
plane wave incident on that section will in general experience partial
re°ection and transmission. Recall that the velocity of the transverse wave is
given by c = p T =², whereT is the tension in the string. The wave number is
k = !=c =2¼=¸, where! is the angular frequency and ¸ is the wavelength.
Since the frequency of the wave and the tension must remain the same in each
region, the ratio r of wave numbers in two di®erent regions of mass density ²2
and ²1 is given by r = k2=k1 = p ²2=²1. Vectoria reads in a certain physics
text that for the case k1 = K, theratior = 3, and the more massive segment
(labeled 2) stretches from x =0tox = L, the energy transmission (T )and
re°ection (R) coe±cientsaregivenby,
9
8 ¡ 8cos(6KL)
T =
; R =
17 ¡ 8cos(6KL) 17 ¡ 8cos(6KL) :
Here T and R measure the fraction of the incident power that is transmitted and
re°ected. The power here is proportional to the square of the string amplitude.
Vectoria's objective is to verify the cited re°ection and transmission coe±cients
and plot them for K =1asafunctionofL. Hermethodofattackisto
write down plane-wave expressions in each region, determine the coe±cients by
matching the solutions at x =0andx = L, and then calculate T and R.
She begins by entering r =3,k1 = K, andk2 = rk1.
> restart: r:=3: k[1]:=K; k[2]:=r*k[1];
k1 := K k2 := 3 K