Question

Two plane waves of the same frequency and with vibrations in the z-direction are given by c (y, t) = (2 cm) cos a p 4 cm y - 20 s t + pb c(x, t) = (4 cm) cos a p 3 cm x - 20 s t + pb

Answer 1

The resultant wave at point (5, 2) is Ψ = 5.99 cos [ 7.15 - (20/s) t]

What are the plane waves:

• A plane wave is a special case of wave or field: whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
• plane waves are free-space modes.

Here,

Two plane waves are given:

c (5, t) = 4 cos [(8π/3) - (20/s) t]

c (2, t) = 2 cos [(3π/2) - (20/s) t]

now, the waves as imaginary exponentials,

separating the spatial parts, and then adding them together

we get The resultant:

Ψ = [  4 sin (8π/3) +  2 sin (3π/2) ]^2 +  [ 4 cos(8/3 π) +  2 cos(3/2π) ]^2

Ψ = 5.99 tan(a) = 0.747/ 5.95

a = 7.15

Ψ = 5.99 cos [ 7.15 - (20/s) t]

hence,

The resultant wave is Ψ = 5.99 cos [ 7.15 - (20/s) t]

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Your question is incomplete, but most probably the full question was:

Two plane waves with the same frequency and with vibrations (measured by Psi) in the z-direction are given by c (x, t) = (4cm.) cos [pi/3cm. x - 20/s t + pi] c (y, t) = (2cm.) cos[pi/4cm. y - 20/s t + pi]

Express the waves as imaginary exponentials, separate the spatial parts, and add them together using a phasor diagram to find the resultant at the point x = 5cm. y = 2cm