Question

A standing wave pattern on a string is described by
$y(x,t)=0.040\left(\sin 5\pi x\right)\left(\cos 40\pi t\right)$
where $x$ and $y$ are in meters and $t$ is in seconds. For $x\ge 0,$ What is the speed of the two traveling waves that interfere to produce this wave?

Answer 1

The nodes are located from vanishing of the spatial factor $\sin 5\pi x=0$ for which the solutions are
$5\pi x=0,\pi ,2\pi ,3\pi ,\dots \Rightarrow x=0,\frac{1}{5},\frac{2}{5},\frac{3}{5},\cdots$
we obtain
$y_1=0.020\sin (5\pi x-40\pi t)\text{and}{y}_2=0.020\sin (5\pi x+40\pi t)$
for the two traveling waves. Thus, we infer from these that the speed is $v=\omega /k=40\pi /5\pi =8.0m/s$