First slide

Equation of wave

Question

Equations of a stationary and a travelling waves are as follows $y_{1} = a \sin kx \cos ωt and y_{2} = a \sin (ωt - Kx)$ The phase difference between two points $x_{1} = \frac{π}{3 k} and x_{2} = \frac{3 π}{3 k}$ are $ϕ_{1} and ϕ_{2}$ respectively for the two waves. The ratio $\frac{ϕ_{1}}{ϕ_{2}}$ is

Difficult

A

1
B

$\frac{5}{6}$
C

$\frac{3}{4}$
D

$\frac{6}{7}$

Solution

$At x_{1} = \frac{π}{3 k} and x_{2} = \frac{3 π}{2 k} \sin {Kx}_{1} or \sin {Kx}_{2} is not zero . Therefore, neither of x_{1} or x_{2} is a node Δx = x_{2} - x_{1} = (\frac{3}{2} - \frac{1}{3}) \frac{π}{k} = \frac{7 π}{6 k} Sin ce \frac{2 π}{k} > Δx > \frac{π}{k} λ > Δx > \frac{λ}{2} (k = \frac{2 π}{λ}) Therefore, ϕ_{1} = π and ϕ_{2} = k Δx = \frac{7 π}{6} ∴ \frac{ϕ_{1}}{ϕ_{2}} = \frac{6}{7}$
Note : In case of stationary wave phase difference between any two points is either zero of $π$ .

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