The equation of a simple harmonic progressive | vedclass

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(D) The given equation is $y = \frac{1}{\sqrt{a}} \sin \omega t \pm \frac{1}{\sqrt{b}} \cos \omega t$. We know that $\cos \omega t = \sin(\omega t + \

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$\sqrt{\frac{a+b}{a b}}$

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(D) The given equation is $y = \frac{1}{\sqrt{a}} \sin \omega t \pm \frac{1}{\sqrt{b}} \cos \omega t$. We know that $\cos \omega t = \sin(\omega t + \frac{\pi}{2})$. Substituting this,we get $y = \frac{1}{\sqrt{a}} \sin \omega t \pm \frac{1}{\sqrt{b}} \sin(\omega t + \frac{\pi}{2})$. This represents the superposition of two simple harmonic motions with amplitudes $A_1 = \frac{1}{\sqrt{a}}$ and $A_2 = \frac{1}{\sqrt{b}}$,and a phase difference of $\phi = \frac{\pi}{2}$. The resultant amplitude $A$ is given by $A = \sqrt{A_1^2 + A_2^2 + 2A_1 A_2 \cos \phi}$. Since $\cos \frac{\pi}{2} = 0$,the formula simplifies to $A = \sqrt{A_1^2 + A_2^2}$. Substituting the values,$A = \sqrt{(\frac{1}{\sqrt{a}})^2 + (\frac{1}{\sqrt{b}})^2} = \sqrt{\frac{1}{a} + \frac{1}{b}} = \sqrt{\frac{a+b}{ab}}$.

The equation of a simple harmonic progressive wave is given by $y = \frac{1}{\sqrt{a}} \sin \omega t \pm \frac{1}{\sqrt{b}} \cos \omega t$. The resultant amplitude of the wave is:

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