Subjective Type

For a particular transverse standing wave on a long string, one of the antinodes is at $$ x=0 $$ and an adjacent node is at $$ x=0.10 \mathrm{m} . $$ The displacement $$ y(t) $$ of the string particle at $$ x=0 $$ is shown in Fig. $$ 16-40 $$ where the scale of the $$ y $$ axis is set by $$ y_{s}=4.0 \mathrm{cm} . $$ When $$ t=0.50 \mathrm{s}, $$ what is the displacement of the string particle at $$ x=0.20 \mathrm{m} $$

Solution

From the $$ x=0 $$ plot (and the requirement of an anti-node at $$ x=0 $$ ), we infer a standing wave function of the form
$$y(x, t)=-(0.04) \cos (k x) \sin (\omega t)$$
where $$ \omega=2 \pi / T=\pi \mathrm{rad} / \mathrm{s}, $$ with length in meters and time in seconds.
The parameter $$ k $$ is determined by the existence of the node at $$ x=0.10 $$ (presumably the first node that one encounters as one moves from the origin in the positive $$ x $$ direction).
This implies $$ k(0.10) =\pi / 2 $$ so that $$ k=5 \pi \mathrm{rad} / \mathrm{m} $$
With the parameters determined as discussed above and $$ t=0.50 \mathrm{s} $$, we find
$$y(0.20 \mathrm{m}, 0.50 \mathrm{s})=-0.04 \cos (k x) \sin (\omega t)=0.040 \mathrm{m}$$

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