Question

The equations of two waves acting in perpendicular directions are given as$x=a \cos(\omega t+\delta)$ and $y=a \cos(\omega t+\alpha),$ where $\delta=\alpha+\frac{\pi}{2}, $ the resultant wave represents

• A. a circle (c.w)
• B. a circle (a.c.w)
• C. an Ellipse (c.w)
• D. an ellipse (a.c.w)

Answer 1

The Correct Option is B

Given : $ x = a \cos(\omega t+\delta)$ and $y= a \cos (\omega t+ \alpha)\hspace25mm ...(i)$ where, $\delta=\alpha+\pi/2$ $\therefore\, \, \, \, \, x=a \cos(\omega t+\alpha+\pi/2)$ $=- a \sin (\omega t+\alpha)\hspace15mm ...(ii)$ Given the two waves are acting in perpendicular direction with the same frequency and phase difference $\pi/2$ From equations (i) and (ii), $x^2 + y^2 = a^2$ which represents the equation of a circle.