Question

Two identical sinusoidal waves each of amplitude $5mm$ with a phase difference of $\frac{\pi }{2}$ are travelling in the same direction in a string. The amplitude of the resultant wave (in mm) is:-

• A. Zero
• B. $5\sqrt{2}$
• C. $\frac{5}{\sqrt{2}}$
• D. $2.5$

Answer 1

Answer: (B)

$5\sqrt{2}$

Given amplitude = 5mm

So, sinusoidal wave can be

taken as $A=5sin\omega t$

Another wave has $\frac{\pi }{2}$ phase diff-

$\Rightarrow B=5sin(\omega t+\frac{\pi }{2})$

$=5cos\omega t$

$A+B=5(sin\omega t+cos\omega t)$

$=5\sqrt{2}(\frac{1}{\sqrt{2}}sin\omega t+\frac{1}{\sqrt{2}}cos\omega t)$

$=5\sqrt{2}(sin\omega tcos{45}^{\circ }+cos\omega tsin{45}^{\circ })$

$=5\sqrt{2}\left(sin\right(\omega t+{45}^{\circ }\left)\right)$

$=5\sqrt{2}sin(\omega t+\pi /4)$