A steep cliff west of Lydia's home reflects a 1020-kHz radio signal from a station that is 74 k

step 1 First of all, we have to use wavelength equals speed of light divided by frequency new, which is

A steep cliff west of Lydia's home reflects a 1020-kHz radio...

A steep cliff west of Lydia's home reflects a $1020-\mathrm{kHz}$ radio signal from a station that is $74 \mathrm{~km}$ due east of her home. If there is destructive interference, what is the minimum distance of the cliff from her home? Assume there is a $180^{\circ}$ phase shift when the wave reflects from the cliff.

Key Concepts

Wavelength Calculation

Wavelength is the distance between successive crests or troughs of a wave, calculated as the speed of the wave divided by its frequency. Determining the wavelength is crucial in problems involving interference, as the conditions for both constructive and destructive interference are expressed in terms of multiples or fractions of the wavelength.

Wave Interference

Wave interference is the phenomenon that occurs when two or more waves meet and combine, resulting in a new wave pattern. These combinations can be constructive, where the amplitudes add, or destructive, where they cancel each other out, depending on the phase relationship between the waves.

Path Difference

Path difference refers to the difference in the distance traveled by two waves reaching the same point. This difference leads to a phase difference between the waves, which is critical in determining whether the interference will be constructive or destructive.

Phase Shift upon Reflection

A phase shift occurs when a wave reflects off a surface. In many cases, such as reflection from a hard surface or a steep cliff, the wave undergoes a 180° phase shift, effectively inverting its phase. This change is an essential component in the interference analysis since it alters the conditions required for destructive interference.

Destructive Interference Condition

Destructive interference occurs when the phase difference between two waves causes them to cancel each other out. With an additional 180° phase shift from reflection, the necessary condition for destructive interference is typically met when the path difference is an integer multiple of the wavelength, adjusted to account for the phase inversion.

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