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A radio broadcast antenna is 36.00 km from your house. Suppose an airplane is flying 2.200 km above the line connecting the broadcast antenna and your radio, and that waves reflected from the airplane travel 90.00 wavelengths farther than waves that travel directly from the antenna to your house. Do you observe constructive or destructive interference between the direct and reflected waves? (Hint: Does a phase change occur when the waves are reflected?). The situation just described occurs when the plane is above a point on the ground that is two-thirds of the way from the antenna to your house. What is the wavelength of the radio waves?

User SunriseM
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1 Answer

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Step-by-step explanation:

To determine whether there is constructive or destructive interference between the direct and reflected waves, we need to consider the phase change that occurs when waves are reflected.

When waves are reflected, there is a phase change of 180 degrees (π radians). This means that the reflected wave is inverted compared to the direct wave.

In this scenario, the waves reflected from the airplane travel 90.00 wavelengths farther than the waves that travel directly from the antenna to your house. Since there is a phase change of 180 degrees upon reflection, this additional distance traveled by the reflected waves corresponds to half a wavelength (λ/2).

Therefore, the condition for constructive interference is when the additional distance traveled by the reflected waves is an integer multiple of the wavelength. In this case, the additional distance is half a wavelength.

Now, let's find the wavelength of the radio waves.

The airplane is flying 2.200 km above the line connecting the broadcast antenna and your radio. Since the situation occurs when the plane is above a point on the ground that is two-thirds of the way from the antenna to your house, we can calculate the total distance from the antenna to your house:

Total distance = Distance from antenna to the airplane + Distance from the airplane to your house

Total distance = (2/3) * Distance from antenna to your house + 2.200 km

Given that the distance from the antenna to your house is 36.00 km, we can calculate the total distance:

Total distance = (2/3) * 36.00 km + 2.200 km

Total distance = 24.00 km + 2.200 km

Total distance = 26.20 km

Next, we know that the reflected waves travel an additional distance of 90.00 wavelengths compared to the direct waves. Since the total distance is the sum of the direct and reflected distances, we can write:

Total distance = Distance traveled by direct waves + Distance traveled by reflected waves

Total distance = Wavelength * Number of wavelengths traveled by direct waves + (Wavelength/2) * Number of wavelengths traveled by reflected waves

Substituting the values, we have:

26.20 km = λ * N + (λ/2) * 90

Simplifying the equation, we can divide both sides by λ:

26.20 km / λ = N + (1/2) * 90

To determine the wavelength (λ), we need to find a value for N that satisfies the equation. Since the distance traveled by the reflected waves is half a wavelength, it implies that the number of wavelengths traveled by the direct waves is an integer minus 1.

Let's try different values for N until we find a valid solution:

For N = 89:

26.20 km / λ = 89 + (1/2) * 90

26.20 km / λ = 134

For N = 88:

26.20 km / λ = 88 + (1/2) * 90

26.20 km / λ = 133

We can see that N = 88 provides a valid solution. Therefore, the wavelength of the radio waves is given by:

26.20 km / λ = 88 + (1/2) * 90

26.20 km / λ = 133

Simplifying the equation:

λ ≈ 0.196 km

So, the wavelength of the radio waves is approximately 0.196 km.

User Coup
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