A 2.0 -m-long string vibrates at its second-harmonic...

A 2.0 -m-long string vibrates at its second-harmonic frequency with a maximum amplitude of $2.0 \mathrm{cm} .$ One end of the string is at $x=0 \mathrm{cm} .$ Find the oscillation amplitude at $x=10,20,30,40$ and $50 \mathrm{cm}$.

Key Concepts

Standing Waves

Standing waves are formed when two waves of identical frequency and amplitude travel in opposite directions and interfere. This interference creates a stationary pattern of nodes, where the displacement is always zero, and antinodes, where the displacement reaches a maximum. In the context of a vibrating string, these patterns arise due to the reflection of waves at the boundaries.

Nodes and Antinodes

Nodes are points along the standing wave where the displacement remains zero at all times, while antinodes are the points where the displacement oscillates between maximum and minimum values. The distribution of nodes and antinodes along a vibrating string is governed by the harmonic mode, with the number and positions of these points changing with each successive harmonic.

Boundary Conditions

Boundary conditions dictate how the ends of a system, such as a string, must behave. For a string fixed at one end, the boundary condition requires that the displacement at that end be zero at all times. This condition plays a critical role in determining the allowed vibrational modes and the resulting standing wave pattern along the string.

Harmonics

Harmonics refer to the integer multiples of the fundamental frequency at which a system can naturally vibrate. In a string fixed at one or both ends, the harmonic frequencies are determined by the length of the string and the speed of the wave along it. The second-harmonic frequency, for instance, implies that the string exhibits two half-wavelengths along its length.

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