Question

The displacement associated with a three-dimensional plane wave is given by
f (x, y, z, t) = a cos(√3/2 kx + 1/2ky - wt).
Calculate the angles made by the propagating wave with the x, y and z-axes.​

Answer 1

The angles made by the propagating wave with the x, y, and z-axes are 30°, 60°, and 90°.

Given:

f (x, y, z, t) = a cos(√3/2kx + 1/2ky - wt).

To Find:

The angles made by the propagating wave with the x, y, and z-axes are

Solution:

From the three-dimensional wave equation,

i) coefficient of kx is equal to cosθ

cosθ = √3/2

     θ = 30° for x-axis

ii) coefficient of ky is equal to cosθ

cosθ = 1/2

     θ = 60° for y-axis

iii) coefficient of kz is equal to cosθ

cosθ = 0

     θ = 90° for z-axis

Therefore, The angles made by the propagating wave with the x, y, and z-axes are 30°, 60°, and 90°.

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