A string of length under tension

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Wave motion, the wave equation and wave energy

Conditions of Validity, Ch Sturm-Liouville theory: the spectral theorem for a differential operator

Problem

A string of length

L=1.00\ \mathrm{m}

under tension

T=50\ \mathrm{N}

, fixed at both ends, has density

\rho(x)=(1.0+2.0x)\ \mathrm{g\,m^{-1}}

for

x

in metres. A shooting code returns two of its eigenfunctions. Shape A has interior zeros at

x=0.306

0.562

and

0.790\ \mathrm{m}

; shape B has interior zeros at

x=0.395

and

0.716\ \mathrm{m}

. The program labels shape A as

y_3

and shape B as

y_4

. ((a)) Use the oscillation theorem to give the true index

n

of each shape. ((b)) Check that the zeros of the two shapes interlace, and that the single interior node of

y_2

, at

x=0.561\ \mathrm{m}

, lies between the zeros of the lower of the two modes. ((c)) Which shape has the higher frequency, and why can you say so without the program's frequencies?

Related theory

Ordinary differential equations

Browse the problem library