Q.A vibrating tuning fork of certain frequency is held in contact with a table top and its vibrations are noticed, and then another vibrating tuning fork of different frequency is held on the table top. Are the vibrations produced in the table top the same for both the tuning forks? Why?

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 …