Question

# The following four waves are sent along strings with the same linear densities (x is in meters and t is in seconds). Rank the waves according to the tension in the strings along which they travel, greatest first:(1) $y_1 = (3 \text{~mm}) \sin(x - 3t)$, (2) $y_2 = (6 \text{~mm}) \sin(2x - t)$,(3) $y_3 = (1 \text{~mm}) \sin(4x - t)$, (4) $y_4 = (2 \text{~mm}) \sin(x - 2t)$

Solution

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Introduction

Knowing the four waves, precisely their unique mathematical forms, which we can observe below,

$\diamond \quad y_1(x,t)=(3~\mathrm{mm})\cdot\sin{(x-3\cdot t)}\\[5pt] \diamond \quad y_2(x,t)=(6~\mathrm{mm})\cdot\sin{(2\cdot x-t)}\\[5pt] \diamond \quad y_3(x,t)=(1~\mathrm{mm})\cdot\sin{(4\cdot x-t)}\\[5pt] \diamond \quad y_4(x,t)=(2~\mathrm{mm})\cdot\sin{(x-2\cdot t)}\\[5pt]$

and the fact that the linear densities $\mu$ of the strings along which they travel are identical, we need to rank them with respect to the corresponding values of the tension $\tau$ in the strings.