Question

Two harmonic waves are represented in SI units by, $y_1(x,t)=0.2sin(x-3.0t)$ and $y_2(x,t)=0.2sin(x-3.0t+\varphi )$. What is the expression for the sum $y=y_1+y_{2}for\varphi =\frac{\pi }{2}$.

• A. y = 8 sin (x - 3.0t)
• B. y = 0.28 sin (x - 3.0t+$\frac{\pi }{4}$)
• C. y = 0.2 sin (x - 3.0t+$\frac{\pi }{4}$)
• D. y = 0.28 cos (x - 3.0t+$\frac{\pi }{4}$)

Answer 1

The correct option is B

y = 0.28 sin (x - 3.0t+$\frac{\pi }{4}$)

The two waves $y_1\left(x_{1}t\right)=0.2sin(x-3.0t)$

$y_2\left(x_{1}t\right)=0.2sin(x-3.0t+\varphi )$

Given,$\varphi =\frac{\pi }{2}$

Drawing a phase diagram

Resultant amplitude $A=\sqrt{(0.2{)}^2+(0.2{)}^2+2\times 0.2\times 0.2cos\left(\frac{\pi }{2}\right)}$

$A=\sqrt{2}\times 0.2=0.28m$

The new phase $\theta =ta{n}^{-1}\left(\frac{0.2}{0.2}\right)=\frac{\pi }{4}$

Sot the wave equation for $y=y_1+y_2$

$y=0.28sin(x-3.0t+\frac{\pi }{4})$