Question

The equation of two progressive waves is given by y1 = Asin(kx-wt) and y2 = Asin(kx -wt+θ). Show that
a) y= 2A sin(kx-wt)
b) y= 0

Answer 1

As per the question,

equation of progressive waves are $y_1 = A \sin(kx-wt)$ and $y_2 = A \sin(kx -wt+θ).$

When two waves are getting interfere to each other, then

$Y=y_1+y_2$

$Y=A \sin(kx-wt)+A \sin(kx -wt+θ).$

$Y=A(2\sin(\dfrac{kx-wt+kx-wt+\theta}{2})\cos(\dfrac{kx-wt-kx+wt+\theta}{2}))$

$Y=A(2\sin(\dfrac{2(kx-wt)+\theta)}{2})\cos(\dfrac{\theta}{2}))$

$Y=A(2\sin((kx-wt)+\dfrac{\theta}{2})\cos(\dfrac{\theta}{2}))$

a) If $\theta=0^\circ$

Then $Y=2A\sin(kx-wt)$

b) If $\theta=\pi$

Y=0