91Ó°ÊÓ

1. The equation of a standing wave on a string is, $y\text{'}\left(\mathrm{t}\right)=\left(3\mathrm{mm}\right)sin\left(5\mathrm{x}\right)cos\left(4\mathrm{t}\right)$
2. The equation of a standing wave on a string is,$y\text{'}\left(\mathrm{t}\right)=\left(3\mathrm{mm}\right)sin(5\mathrm{x}+\frac{\mathrm{Ï„Ï„}}{2})cos\left(4\mathrm{t}\right)$

Determine the node or antinode using the value of amplitude.

The given equation of a standing wave on a string is,

$y\text{'}\left(\mathrm{t}\right)=\left(3\mathrm{mm}\right)sin\left(5\mathrm{x}\right)cos\left(4\mathrm{t}\right)$

Now, usingin this equation,

$y\text{'}\left(\mathrm{t}\right)=\left(3\mathrm{mm}\right)sin\left(0\right)cos\left(4\mathrm{t}\right)=0$

As the amplitude of the wave becomes zero at this position, a standing wave has node at x=0.

Hence, the given standing wave has ‘node’ at x = 0.

The given equation of a standing wave on a string is,

$y\text{'}\left(\mathrm{t}\right)=\left(3\mathrm{mm}\right)sin(5\mathrm{x}+\frac{\mathrm{Ï„Ï„}}{2})cos\left(4\mathrm{t}\right)$

Now, using x =0 in this equation,

$y\text{'}\left(\mathrm{t}\right)=\left(3\mathrm{mm}\right)sin\left(\frac{\mathrm{Ï„Ï„}}{2}\right)cos\left(4\mathrm{t}\right)=\left(3mm\right)\cos \left(4t\right)$

This is the maximum amplitude possible for the wave, so at x=0 the given wave has an antinode.

Hence, the given standing wave has ‘antinode’ at x = 0

Therefore, the nodes and antinodes can be determined using the equation of amplitudes.