A wave of angular frequency ' $ω$ ' propagates so that a certain phase of oscillation moves along x- axis, y-axis and z-axis, with speeds C₁, C₂, C₃ respectively. The propagation constant 'K' is

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$(C_{1} + C_{2} + C_{3}) (\hat{i} + \hat{j} + \hat{k})$

$\frac{ω}{\sqrt{C_{1}^{2} + C_{2}^{2} + C_{1}^{2}}} (\hat{i} + \hat{j} + \hat{k})$

$\frac{ω}{C_{1}} \hat{i} + \frac{ω}{C_{2}} \hat{j} + \frac{ω}{C_{3}} \hat{k}$

$\frac{ω (\hat{i} + \hat{j} + \hat{k})}{C}$

answer is B.

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Detailed Solution

Propagation constant is a vector quantity
$\begin{array}{l} \bar{K} = K_{x} \hat{j} + K_{y} \hat{j} + K_{z} \hat{k} \end{array}$

$K_{x} = \frac{ω}{vx} = \frac{ω}{C_{1}}; K_{y} = \frac{ω}{vy} = \frac{ω}{C_{2}}$

$K_{z} = \frac{ω}{V_{2}} = \frac{ω}{C_{2}}$

$∴ \bar{K} = \frac{ω}{C_{1}} \hat{i} + \frac{ω}{C_{2}} \hat{j} + \frac{ω}{C_{3}} \hat{k}$

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