Imagine you are being asked to model radio waves as being driven by a sinusoidal

Question

Imagine you are being asked to model radio waves as being driven by a sinusoidal source (could mean a sine or cosine wave, depending on the following facts) with a frequency of 3000Hz where the magnitude of the vertical displacement between the crest of one wave and the trough/valley of another is 2.00m. The vertical displacement is symmetric about the origin in the y-dimension. Since these waves are radio waves they travel at nearly the speed of light (3.0x108 m/s). At t = 0 s, a wave crest, at its maximum vertical displacement, passes your testing equipment t your research location (this is the origin, x = 0m, for the system).

(b) Given the above conditions, which remain constant, what is the mathematical equation that describes the vertical position of the wave as a function of time only for neighboring testing equipment located a positive 48,270 m (30 miles) away from your original testing location. The waves move toward you from the original testing location t x=0m.

Explanation / Answer

A) Amplitude, A = 2.00 m

angular frequency, w = 2*pi*f

= 2*pi*3000

= 18849.56 rad/s

T = 1/f

= 1/3000

= 3.33*10^-3 s

lamda = v/f

= 3*10^8/3000

= 10^5 m

k = 2*pi/lamda

= 2*pi/10^5

= 6.28*10^-5 m

y = 2*cos(6.28*10^-5*x - 18849.56*t)

y = 2*cos(-18849.56*t) since(x = 0 ) <<<------Answer

B) at x = 48270 m

y = 2*cos(6.28*10^-5*48270 - 18849.56*t)

= 2*cos(3.03 - 18849.56*t) <<<------Answer

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