Fixed Rope and Wave Solution

Classical Mechanics Level 3

A rope of length 1 is fixed to a wall at x = 0 x=0 and shaken at the other end so that

x ( 1 , t ) = sin ⁡ ω t . x(1,t) = \sin \omega t.

Which of the following is a possible displacement of the rope as a function of x x and t t consistent with these boundary conditions, assuming the waves of the rope propagate with velocity v = 1 v=1 ?

sin ⁡ ω x sin ⁡ ω cos ⁡ ω t \frac{\sin \omega x}{\sin \omega} \cos \omega t sin ⁡ ω x cos ⁡ ω sin ⁡ ω t \frac{\sin \omega x}{\cos \omega} \sin \omega t sin ⁡ ω x sin ⁡ ω sin ⁡ ω t \frac{\sin \omega x}{\sin \omega} \sin \omega t cos ⁡ ω x sin ⁡ ω t \cos \omega x \sin \omega t

Matt DeCross by Matt DeCross , 27, USA

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1 solution

Matt DeCross
May 10, 2016

The forcing term means that at x = 1 x=1 the displacement must always be given by sin ⁡ ω t \sin \omega t . Only the correct answer above gives sin ⁡ ω t \sin \omega t at all times at x = 1 x=1 (by plugging in), regardless of any other conditions.

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