In water of uniform depth, a wide pier is supported on...

In water of uniform depth, a wide pier is supported on pilings in several parallel rows $2.80 \mathrm{m}$ apart. Ocean waves of uniform wavelength roll in, moving in a direction that makes an angle of $80.0^{\circ}$ with the rows of pilings. Find the three longest wavelengths of waves that are strongly reflected by the pilings.

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Key Concepts

Bragg Reflection

This is the phenomenon where waves are strongly reflected from a periodic structure when the spacing between elements causes the reflected waves from successive elements to interfere constructively. The condition for strong reflection is derived by matching the added path length between scatterers to an integral multiple of the wavelength, which amplifies the reflected wave intensity.

Constructive Interference

Constructive interference occurs when waves, after scattering from different elements of a periodic array, arrive in phase such that their amplitudes add. In contexts like wave reflections by pilings, only specific wavelengths will meet the interference condition, leading to enhanced reflection at those wavelengths.

Geometric Projection of Periodicity

When waves approach a periodic structure at an angle, the effective spacing between the scatterers in the direction of wave propagation is modified. This projection changes the interference condition, meaning that the spacing relevant for reflection is the component of the physical spacing along the wave’s propagation direction. This concept is crucial in determining which wavelengths will be strongly reflected.

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