Question

A standing wave results from the sum of two transverse traveling waves given by
$y_1=0.050\cos (\pi x-4\pi t)$
and $y_2=0.050\cos (\pi x+4\pi t)$
where $x,y_1,$ and $y_2$ are in meters and $t$ is in seconds. (a) What is the smallest positive value of $x$ that corresponds to a node? Beginning at $t=0?$

Answer 1

Applying

$\cos \alpha +\cos \beta =2\cos \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)$

we obtain

$y^{\prime }(x,t)=2y_m\cos \left(kx\right)\cos \left(\omega t\right)$

$y^{\prime }=\left[0.10\cos \pi x\right]\cos 4\pi t,$

with SI units understood.equation for standing wave here is

For non-negative $x$, the smallest value to produce $\cos \pi x=0$ is $x=1/2$, so the answer is $x=0.50m$