A steel guitar string of length and tension obeys the wave equation for its…

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

Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions

Problem

A steel guitar string of length

L = 0.65\,\mathrm{m}

and tension

F_{\mathrm{T}} = 80\,\mathrm{N}

obeys the wave equation for its transverse displacement

y(x,t)

, with

x=0

fixed permanently to a peg. In the small-slope approximation used to derive the wave equation, the transverse (vertical) component of the force the string exerts on whatever is tied to it at

x=L

-F_{\mathrm{T}}\,\partial y/\partial x|_{x=L}

. Three different terminations are considered at

x=L

. ((a)) The end at

x=L

is tied permanently to a second peg, so it cannot move. Write the condition there and name its type. ((b)) The end at

x=L

is instead tied to a light ring of negligible mass that slides without friction on a vertical rail, free to move up and down with no other vertical force acting on it. Using Newton's second law for the massless ring, write the condition at

x=L

and name its type. ((c)) The end at

x=L

is tied to the same light, frictionless ring, but the ring is also attached to a vertical spring, parallel to the rail, of stiffness

k = 150\,\mathrm{N\,m^{-1}}

anchored to a fixed support, exerting a restoring force

-k\,y(L,t)

on the ring when it is displaced vertically by

y(L,t)

. Derive the condition at

x=L

and name its type. ((d)) Evaluate the length scale

F_{\mathrm{T}}/k

for the spring in (c) and compare it with

L

. Comment on whether this termination behaves more like the fixed end of (a) or the free end of (b) for this string.

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