find the resultant of the amplitude in the form of

Dear Student,
To have a node of 0 amplitude, the amplitudes of the two superimposing waves must be same, i.e., a
The stationary wave formed by the superposition can be of the following form:
$Y=2a\mathrm{cos}\left(kx\right)\mathrm{sin}\left(\omega t\right)\phantom{\rule{0ex}{0ex}}or\phantom{\rule{0ex}{0ex}}Y=2a\mathrm{sin}\left(kx\right)\mathrm{cos}\left(\omega t\right)\phantom{\rule{0ex}{0ex}}or\phantom{\rule{0ex}{0ex}}Y=2a\mathrm{cos}\left(kx\right)\mathrm{cos}\left(\omega t\right)\phantom{\rule{0ex}{0ex}}or\phantom{\rule{0ex}{0ex}}Y=2a\mathrm{sin}\left(kx\right)\mathrm{sin}\left(\omega t\right)$

For having a node at x = 0., the amplitude of the standing wave must be 0 at x = 0, which is possible for
$\mathrm{sin}\left(kx\right)=\mathrm{sin}\left(0\right)=0$.
So, the equation of the standing wave must be
$Y=2a\mathrm{sin}\left(kx\right)\mathrm{cos}\left(\omega t\right)\phantom{\rule{0ex}{0ex}}or\phantom{\rule{0ex}{0ex}}Y=2a\mathrm{sin}\left(kx\right)\mathrm{sin}\left(\omega t\right)$

This we get from the following superposition:

Ans (2)

Regards

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