First slide

Stationary waves

Question

Two waveforms, $y_{1} (x, t) = A \sin (k x - ω t)$ and $y_{2} (x, t) = A \sin (k x + ω t + \frac{π}{3})$ , travelling along x- axis are superposed. The position of nodes is given by $(k = \frac{2 π}{λ}; ω = 2 π f)$

Difficult

A

$x_{n} = (n + \frac{1}{6}) \frac{λ}{2}$
B

$x_{n} = (n - \frac{1}{6}) \frac{λ}{2}$
C

$x_{n} = n \frac{λ}{2}$
D

$x_{n} = (2 n + 1) \frac{λ}{4}$

Solution

$\begin{array}{l} Given y_{1} (x, t) = A \sin (k x - ω t) \dots . (1) \\ y_{2} (x, t) = A \sin (k x + ω t + \frac{π}{3}) \dots . (2) \\ From principle of superpostion \\ y = y_{1} + y_{2} \\ y = A \sin (k x - ω t) + A \sin (k x + ω t + \frac{π}{3}) \\ (∵ \sin C + \sin D = 2 \sin \frac{C + D}{2} \cdot \cos \frac{C - D}{2}) \\ = 2 A \sin (k x + \frac{π}{6}) \cos (ω t + \frac{π}{6}) \end{array}$
$\begin{array}{l} Nodes are given by k x + \frac{π}{6} = n π \\ k x + \frac{π}{6} = n π (∵ k = \frac{2 π}{λ}) \\ \Rightarrow \frac{2 π x}{λ} + \frac{π}{6} = n π \\ \Rightarrow x = (n π - \frac{π}{6}) \frac{λ}{2 π} \\ ∴ x_{n} = (n - \frac{1}{6}) \frac{λ}{2} \end{array}$

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