Your firm has been hired to design a system that allows airplane pilots to make

Question

Your firm has been hired to design a system that allows airplane pilots to make instrument landings in rain or fog. You've decided to place two radio transmitters 48m apart on either side of the runway. These two transmitters will broadcast the same frequency, but out of phase with each other. This will cause a nodal line to extend straight off the end of the runway (see Figure 21.30b). As long as the airplane's receiver is silent, the pilot knows she's directly in line with the runway. If she drifts to one side or the other, the radio will pick up a signal and sound a warning beep. To have sufficient accuracy, the first intensity maxima need to be 64m on either side of the nodal line at a distance of 5.0km .

What frequency should you specify for the transmitters?

Explanation / Answer

Your firm has been hired to design a system that allows airplane pilots to make instrument landings in rain or fog. You've decided to place two radio transmitters 50 m apart on either side of the runway. These two transmitters will broadcast the same frequency, but out of phase with each other. This will cause a nodal line to extend straight off the end of the runway. As long as the airplane's receiver is silent, the pilot knows she's directly in line with the runway. If she drifts to one side or the other, the radio will pick up a signal and sound a warning beep. To have sufficient accuracy, the first intensity maxima need to be 60 m on either side of the nodal line at a distance of 3.0 km. What frequency should you specify for the transmitters?

Best Answer:  Signal emitted from point A: sin(2?f(t - sqrt((y-50)^2 + x^2))/c)
Signal emitted from point B: - sin(2?f(t - sqrt((y+50)^2 + x^2))/c)

For y = 0, these two signals will be equal in magnitude and opposite in sign: they will cancel.

In order to achieve the first antinodes, the difference in path must introduce an additional phase difference of ?:
2?f(t - sqrt((y-50)^2 + x^2))/c = (2?f(t - sqrt((y+50)^2 + x^2))/c

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