Question

Question 10 (2 points) Two harmonic waves traveling in opposite directions interfere to produce a standing wave described by y = 3 (sin 2x) (cos 5t) where X is in meters and t is in seconds. What is the wavelength of the interfering waves? 1.00 m 2.00 m 3.14 m O 6.28 m 12.0 m

Answer 1

$y=3(\sin 2x)(\cos 5t)$

$\Rightarrow y=\frac{3}{2}(2\sin 2x\cos 5t)$

$\Rightarrow y=\frac{3}{2}[\sin (2x+5t)+\sin (2x-5t)]$

$[\because 2\sin A\cos B=\sin (A+B)+\sin (A-B)]$

$\Rightarrow y=\frac{3}{2}\sin (2x+5t)+\frac{3}{2}\sin (2x-5t)$

$\Rightarrow y=y_{1}+y_{2}$

So, interfering waves are,

$y_{1}=\frac{3}{2}\sin (2x+5t)$

$y_{2}=\frac{3}{2}\sin (2x-5t)$

Compare it with, $y=A\sin (kx\pm \omega t)$

Where, $A=$ Amplitude

            $\omega =$ Angular frequency

            Wave number

            $\lambda =$ Wavelength

In both the interfering waves,

$\Rightarrow \frac{2\pi }{\lambda }=2$

$\Rightarrow \lambda =\pi =\mathbf{3.14\: m}$