Figure shows different standing wave patterns on a string of linear mass density \( 4.0 \times 1...

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Figure shows different standing wave patterns on a string of linear mass density \( 4.0 \times 10^{-2} \mathrm{~kg} / \mathrm{m} \) under a tension of \( 100 \mathrm{~N} \). The amplitude of antinodes is indicated in each zure. The length of the string is \( 2.0 \mathrm{~m} \).
(i) Obtain the frequencies of the modes shown in figures (a) and (b).
(ii) Write down the transverse displacement \( y \) as a function of \( x \) and \( t \) for each mode. (Take the initial configuration of the wire in each mode to be as shown by the dark lines in the figure).
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