Radio Interference. Two radio antennas A and B radiate in...

Radio Interference. Two radio antennas $A$ and $B$ radiate in phase. Antenna $B$ is $110 \mathrm{~m}$ to the right of antenna $A$. Consider point $Q$ along the extension of the line connecting the antennas, a horizontal distance of $30 \mathrm{~m}$ to the right of antenna $B$. The frequency, and hence the wavelength, of the emitted waves can be varied. (a) What is the longest wavelength for which there will be destructive interference at point $Q$ ? (b) What is the longest wavelength for which there will be constructive interference at point $Q$ ?

Radio Interference. Two radio antennas $A$ and $B$ radiate in phase. Antenna $B$ is $110 \mathrm{~m}$ to the right of antenna $A$. Consider point $Q$ along the extension of the line connecting the antennas, a horizontal distance of $30 \mathrm{~m}$ to the right of antenna $B$. The frequency, and hence the wavelength, of the emitted waves can be varied. (a) What is the longest wavelength for which there will be destructive interference at point $Q$ ?
(b) What is the longest wavelength for which there will be constructive interference at point $Q$ ?

Key Concepts

Interference

Interference is the phenomenon that occurs when two or more waves overlap, and the resulting wave amplitude is the sum of the individual wave amplitudes. This concept is fundamental to understanding how waves combine, leading to regions where they reinforce each other (constructive interference) or cancel each other (destructive interference).

Constructive Interference

Constructive interference happens when two waves meet in phase, meaning their corresponding crests and troughs align. This results in a larger resultant amplitude. It typically occurs when the path difference between the waves is an integer multiple of the wavelength, allowing their oscillations to add together effectively.

Destructive Interference

Destructive interference occurs when two waves arrive out of phase such that the crest of one coincides with the trough of the other, leading to cancellation or reduction in amplitude. This occurs when the path difference is an odd multiple of half the wavelength, causing the waves to counteract each other.

Path Difference

Path difference is the difference in the distances that waves from different sources travel to reach a common point. This difference is crucial in determining the phase relationship between the waves when they meet, directly influencing whether their interference will be constructive or destructive.

Frequency and Wavelength Relationship

The frequency and wavelength of a wave are inversely related through the speed of the wave in a given medium. Changing the frequency alters the wavelength (since the speed of the wave remains constant) and consequently shifts the conditions for constructive and destructive interference. This relationship is key when analyzing how variations in frequency affect interference patterns.

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