The width of a particular microwave oven is exactly right to...

The width of a particular microwave oven is exactly right to support a standing-wave mode. Measurements of the temperature across the oven show that there are cold spots at each edge of the oven and at three spots in between. The wavelength of the microwaves is $12 \mathrm{cm} .$ How wide is the oven?

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Key Concepts

Standing Waves

Standing waves are the result of two waves of the same frequency and amplitude traveling in opposite directions and interfering with each other. This interference creates fixed points of no oscillation, called nodes, and points of maximum oscillation, called antinodes. The formation of standing waves is key to understanding resonant phenomena in confined spaces.

Resonant Modes and Boundary Conditions in Cavities

When a wave is confined in a space such as a cavity, only specific patterns of standing waves, known as resonant modes, can exist. These modes occur when the dimensions of the cavity allow an integer number of half wavelengths to fit exactly, with boundary conditions typically enforcing nodes at the edges. This concept is critical in determining the dimensions needed to support a standing-wave mode.

Wavelength and Node Spacing

The wavelength of a wave directly determines the spacing between nodes in a standing wave, with the distance between consecutive nodes being half of the wavelength. This relationship is fundamental when calculating the physical size of a cavity required to support a particular standing-wave pattern.

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