A stretched string of length lies along the -axis

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Superposition, boundaries and interference

Mathematical Methods for Physics, Ch Separation of variables in Cartesian, cylindrical and spherical geometry

Problem

A stretched string of length

L

lies along the

x

-axis. It is clamped at

x=0

, so

y(0,t)=0

for all

t

, and its other end at

x=L

is attached to a light ring that slides without friction on a smooth vertical rail, so the string exerts no transverse restoring force there:

\partial y/\partial x = 0

x=L

for all

t

. Its transverse displacement

y(x,t)

obeys the wave equation

\partial^2y/\partial t^2 = c^2\,\partial^2y/\partial x^2

with wave speed

c

. (a) Substitute

y(x,t)=X(x)G(t)

, separate the wave equation into two ordinary equations joined by a separation constant

\lambda

, and give the argument for why a function of

x

alone that equals a function of

t

alone for every

x

and

t

must be a constant. (b) Apply the two boundary conditions to each sign of

\lambda

in turn, and show that only

\lambda_n=\left(\dfrac{(2n-1)\pi}{2L}\right)^2

n=1,2,3,\dots

, gives a standing wave that is not identically zero, with

X_n(x)=\sin\!\left(\dfrac{(2n-1)\pi x}{2L}\right)

. (c) The string has

L=1.2\,\mathrm{m}

and

c=180\,\mathrm{m\,s^{-1}}

. Give the frequencies, in

\mathrm{Hz}

, of the three lowest modes.

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