Question

A generator at one end of a very long string creates a wave given by
$y=\left(6.0cm\right)\cos \frac{\pi }{2}[\left(2.00m^{-1}\right)x+\left(8.00s^{-1}\right)t]$
and a generator at the other end creates the wave
$y=\left(6.0cm\right)\cos \frac{\pi }{2}[\left(2.00m^{-1}\right)x-\left(8.00s^{-1}\right)t]$
For $x\ge 0,$ what is the location of the node having the smallest value of x?

Answer 1

We need to add two cosine functions.
First convert them to sine functions using $\cos \alpha$
$=\sin (\alpha +\pi /2),\text{then apply}$
$\cos \alpha +\cos \beta =\sin (\alpha +\frac{\pi }{2})+\sin (\beta +\frac{\pi }{2})=2\sin \left(\frac{\alpha +\beta +\pi }{2}\right)\cos \left(\frac{\alpha +\beta }{2}\right)$
$=2\cos \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)$
Letting $\alpha =kx$ and $\beta =\omega t,$ we find
$y_m\cos (kx+\omega t)+y_m\cos (kx-\omega t)=2y_m\cos \left(kx\right)\cos \left(\omega t\right)$
Nodes occur where $\cos \left(kx\right)=0$ or $kx=n\pi +\pi /2,$ where $n$ is an integer (including zero).
since $k=1.0\pi m^{-1},$ this means $x=(n+\frac{1}{2})\left(1.00m\right).$
Thus, the smallest value of $x$ that corresponds to a node is $x=0.500m(n=0)$.