A ripple on water is modelled as with amplitude and wavelength

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Scientific programming for physicists · Scientific programming for physicists

Computational Physics, Ch Letting the computer do the algebra

Problem

A ripple on water is modelled as

y = a\sin(kx)

with amplitude

a = 0.50\ \mathrm{m}

and wavelength

\lambda = 2.0\ \mathrm{m}

, so

k = 2\pi/\lambda = \pi\ \mathrm{m^{-1}}

. The length of the surface over one wavelength is

L = \int_0^{\lambda}\sqrt{1 + a^2k^2\cos^2(kx)}\,\mathrm{d}x.

Handed this definite integral, SymPy

1.14.0

pulls the constant

\tfrac{1}{2}

out of the square root and returns the integral itself, unevaluated. ((a)) State what that return does and does not tell us about

L

, and name the two things worth trying next. ((b)) The integrand is bounded, smooth, and periodic over exactly the interval of integration. Choose a rule from Chapter 7 on those two grounds and justify the choice in one sentence. ((c)) Apply the chosen rule at

N=4

8

16

and

32

panels, quote each estimate, and use the successive differences to say how many digits of

L

the table supports.

Browse the problem library