Q.Have you noticed vibrations in a drill machine or in a washing machine? How do they differ from vibrations in the above two cases?

Each loop of a stationary wave is half a wavelength long, and a node–antinode gap is a quarter wavelength.

A string of length $L = 2\text{ m}$ vibrating in 2 loops means each loop has length $L/2 = 1\text{ m}$. Since each loop corresponds to half a wavelength ($\lambda/2$),

$\dfrac{\lambda}{2} = 1\text{ m} \;\Rightarrow\; \lambda = 2\text{ m}$

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Spatial phase between a node and the neighbouring antinode in a stationary wave.

For a stationary wave $y = 2A\cos(kx)\sin(\omega t)$, a node occurs where $kx=0$ and the adjacent antinode where $kx=\pi/2$ (i.e. a quarter-wavelength away, $x=\lambda/4$). The spa …

Adjacent nodes (or adjacent antinodes) are $\lambda/2$ apart; a node and its neighbouring antinode are half of that, $\lambda/4$ apart.

A stationary wave is described by $y = 2a\cos(kx)\sin(\omega t)$, where $k = 2\pi/\lambda$. Nodes occur where $\cos(kx)=0$, i.e. at $x = \lambda/4, 3\lambda/4, 5\lambda/4,\dots$, and antinodes occur where $|\cos(kx)|=1$, i.e. at $x=0,\lambda/2,\lambda,\dots$. The separation between any node an …