Engineering
Physics
Displacement and Pressure Wave Equation of Sound
Question

A plane transverse wave is propagating in a direction making an angle of 30° with positive x-axis in the x-y plane. Find phase difference between points (0, 0, 0) and (1, 1, 1). Wavelength of the wave is 1 m.

(3+1)π rad

2π rad

(2+1)π rad

none

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Solution

The phase difference Δφ between two points for a wave propagating in direction n^ is given by Δφ = (2π/λ) * (n^ · Δr). Here, λ=1 m. The propagation direction makes 30° with the x-axis in the x-y plane, so n^=(cos30°,sin30°,0)=(3/2,1/2,0). The displacement vector Δr from (0,0,0) to (1,1,1) is (1,1,1). The dot product is (√3/2)*1 + (1/2)*1 + 0*1 = (√3/2 + 1/2). Thus, Δφ = 2π * (√3/2 + 1/2) = π(√3 + 1) rad.

Final Answer: (3+1)π rad