Question

Two waves are describe by
$y_1=0.30\sin \left[\pi \right(5x-200\left)t\right]$
and $y_2=0.30\sin \left[\pi \right(5x-200)t+\pi /3]$
where $y_1,y_2$ and $x$ are in meters and $t$ is in seconds. When these two wave are combine, a travelling wave is produced. What are the $\left(a\right)$ amplitude, $\left(b\right)$ wave speed, and $\left(c\right)$ wave length of that travelling wave?

Answer 1

Given : $y_1=0.3\sin \left(\right[\pi (5x-200)t\left]\right)$ $y_2=0.2\sin \left[\pi \right(5x-200)t+\pi /3]$

Equation of traveling wave $y=A_{net}\sin \left[\pi \right(5x-200)t+\alpha ]$

where $A_{net}=\sqrt{{0.3}^2+{0.3}^2+2\times 0.3\times 0.3\cos \pi /3}$

Or $A_{net}=0.3\sqrt{2}$

$\tan \alpha =\frac{0.3\sin \pi /3}{0.3+0.3+\cos \pi /3}=\frac{0.3\times \sqrt{3}/2}{0.3+0.3\times \sqrt{3}/2}$

$\tan \alpha =\frac{\sqrt{3}}{2+\sqrt{3}}$

So, amplitude of traveling wave $A_{net}=0.3\sqrt{2}$

Wave speed $v=\frac{200\pi }{5\pi }=40m/s$

Wavelength $\lambda =\frac{2\pi }{5\pi }=0.4m$