The stationary wave produced on a string is represented by...

The stationary wave produced on a string is represented by the equation $y=5 \cos \left(\frac{\pi x}{3}\right) \sin (40 \pi t)$ where $x$ and $y$ are in $\mathrm{cm}$ and $t$ is in seconds. The distance between consecutive nodes is (a) $5 \mathrm{~cm}$ (b) $\pi \mathrm{cm}$ (c) $3 \mathrm{~cm}$ (d) $40 \mathrm{~cm}$

The stationary wave produced on a string is represented by the equation $y=5 \cos \left(\frac{\pi x}{3}\right) \sin (40 \pi t)$ where $x$ and $y$ are in $\mathrm{cm}$ and $t$ is in seconds. The distance between consecutive nodes is
(a) $5 \mathrm{~cm}$
(b) $\pi \mathrm{cm}$
(c) $3 \mathrm{~cm}$
(d) $40 \mathrm{~cm}$

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Key Concepts

Stationary Waves

A stationary (standing) wave is produced by the interference of two waves traveling in opposite directions with the same frequency and amplitude. This type of wave is represented as a product of a spatial function and a temporal function, resulting in fixed positions along the medium (nodes) where the amplitude is always zero, while the maximum amplitude (antinodes) occurs at other fixed points.

Nodes in Standing Waves

Nodes are the points along a stationary wave where the medium does not oscillate, meaning the amplitude is always zero. They occur at positions determined by the zeros of the spatial part of the wave equation. The distance between consecutive nodes is constant and can be found by analyzing the spatial function of the standing wave.

Wave Number

The wave number is a constant found in the argument of the spatial function (typically a cosine or sine function) in the wave equation. It represents the spatial frequency of the wave and is used to calculate the periodicity of the nodes. Specifically, understanding how the wave number affects the phase helps determine the spacing between points where the wave has zero amplitude.

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