Question: In the equation for a stationary wave given by $y = 5 \cos \frac{\pi x}{25} \sin 100 \pi t$. Here, $...

Question

In the equation for a stationary wave given by $y = 5 \cos \frac{\pi x}{25} \sin 100 \pi t$ . Here, $x$ is in cm and $t$ in second. A node will not occur at distance $x$ is equal to

Answer 1

Solution

The stationary wave is given by

$y = 5 \cos\left(\frac{\pi x}{25}\right)\sin(100\pi t)$

Nodes occur when the amplitude $\cos\left(\frac{\pi x}{25}\right) = 0$ .

For $\cos\theta = 0$ , we have

$\frac{\pi x}{25} = \frac{(2n+1)\pi}{2}$ for $n = 0,1,2,\ldots$

Solving for $x$ :

$x = \frac{25(2n+1)}{2}$

For $n=0$ : $x = \frac{25}{2} = 12.5$ cm

For $n=1$ : $x = \frac{75}{2} = 37.5$ cm

For $n=2$ : $x = \frac{125}{2} = 62.5$ cm

Thus, nodes occur at $12.5$ cm, $37.5$ cm, and $62.5$ cm. The distance $x = 25$ cm does not match this node condition.