Standing waves with five loops are produced on a stretched...

Standing waves with five loops are produced on a stretched string under tension. The length of the string is $50 \mathrm{~cm}$ and the frequency of vibrations is $250 \mathrm{~s}^{-1}$. Calculate the time variation of the displacement of the points which are at distances of $2 \mathrm{~cm}, 5 \mathrm{~cm}, 15 \mathrm{~cm}, 18 \mathrm{~cm}, 20 \mathrm{~cm}, 35 \mathrm{~cm}$ and $45 \mathrm{~cm}$ from one end of the string.

Key Concepts

Nodes and Antinodes

Nodes are points along a standing wave that remain at rest, where the displacement is always zero, while antinodes are the points where the displacement reaches a maximum. The distribution of nodes and antinodes along a vibrating string is determined by the wavelength and the particular harmonic mode, and this distribution is critical in describing the spatial structure of the standing wave.

Wave Equation for Vibrations

The wave equation for a standing wave describes both the time and spatial dependencies of the string's displacement. Typically, the displacement is given by a product of a spatial function (often a sine or cosine function that satisfies the boundary conditions) and a temporal function (usually a cosine or sine function oscillating with the angular frequency) to capture the oscillatory behavior of the wave over time.

Standing Waves

Standing waves are created when two waves of the same frequency and amplitude travel in opposite directions and interfere, producing a waveform that appears to be stationary. This phenomenon is particularly important on strings under tension, where the fixed ends cause waves to reflect and interfere with incoming waves, establishing patterns with fixed nodes and antinodes.

Harmonics

Harmonics are integral modes of vibration that occur when standing waves are formed on a string. Each harmonic corresponds to a specific frequency at which a string vibrates, with the fundamental frequency being the first harmonic. Higher harmonics exhibit increasing numbers of loops or segments, each reflecting a multiple of the fundamental frequency, which is intrinsic to the resonant properties of the string.

Boundary Conditions

Boundary conditions are constraints that must be satisfied by the solution to the wave equation. For a string fixed at both ends, the displacement at the endpoints must be zero. These conditions determine the allowable wavelengths and, hence, the frequencies (harmonics) at which standing waves can be established on the string. The spatial structure of the solution is directly influenced by these restrictions.

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