Wave motionMediumBloom L3
Question
Five waveforms moving with equal speeds on the x-axis
y1 = 8 sin (ωt + kx); y2 = 6 sin (ωt +$\frac{\pi}{2}$+ kx); y3 = 4 sin (ωt + π + kx); y4 = 2 sin (ωt +$\frac{3\pi}{2}$+kx);
y5 = $4\sqrt{2}$ sin (ωt – kx + $\frac{\pi}{4}$) are superimposed on each other. The resulting wave is :
Options
A.$8\sqrt{2}$cos kx sin (ωt + $\frac{\pi}{4}$)
B.$8\sqrt{2}$sin (ωt – kx + $\frac{\pi}{4}$)
C.$8\sqrt{2}$sin kx cos (ωt + $\frac{\pi}{4}$)
D.8 sin (ωt + kx)
Solution
Sol. y1 = 8sin(ωt + kx)
y2 = 6sin (ωt + kx + π/2)
y3 = 4sin (ωt + kx + π)
y4 = 2sin (ωt + kx + 3π/2)
$y_{5} = 4\sqrt{2}\sin(\omega t - kx + \pi/4)$
y1 = y2 + y3 + y4
y' = 4$\sqrt{2}\sin(\omega t + kx + \pi/4)$
ynet = y' = y5
=$4\sqrt{2}$ (sin(ωt – kx + π/4) + sin(ωt + kx + π/4)
= $4\sqrt{2}$ [2sin (ωt– π/4)cos kx]
= $8\sqrt{2}$sin $\left( \omega t + \frac{\pi}{4} \right)\cos kx$
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