A string is tied between two lab posts a distance L apart....

A string is tied between two lab posts a distance $L$ apart. The tension in the string and the linear mass density is such that the speed of a wave on the string is $v=343 \mathrm{m} / \mathrm{s} .$ A tube with symmetric boundary conditions has a length $L$ and the speed of sound in the tube is $v=343 \mathrm{m} / \mathrm{s} .$ What could be said about the frequencies of the harmonics in the string and the tube? What if the velocity in the string were $v=686 \mathrm{m} / \mathrm{s}$ ?

A string is tied between two lab posts a distance $L$ apart. The tension in the string and the linear mass density
is such that the speed of a wave on the string is $v=343 \mathrm{m} / \mathrm{s} .$ A tube with symmetric boundary conditions has a length $L$ and the speed of sound in the tube is $v=343 \mathrm{m} / \mathrm{s} .$ What could be said about the frequencies of the harmonics in the string and the tube? What if the velocity in the string were $v=686 \mathrm{m} / \mathrm{s}$ ?

Key Concepts

Effect of Changing Wave Speed

Altering the wave speed in a medium, such as increasing the speed from 343 m/s to 686 m/s, directly affects the resonant frequencies by scaling them with the wave speed. This means that if the wave speed is doubled, all the harmonic frequencies will also double, provided the physical dimensions of the system remain unchanged.

Harmonic Series

The harmonic series describes a set of frequencies that are integer multiples of a fundamental frequency. In both the vibrating string and the air column of a tube, the resonant frequencies occur at multiples of the lowest frequency, forming the harmonic series. This concept explains why musical instruments produce tones with characteristic timbres, as the relative amplitudes of these harmonics determine the sound quality.

Standing Waves

Standing waves are the result of the interference between waves traveling in opposite directions in a confined medium, such as a string fixed at both ends or a tube with symmetric boundary conditions. At specific frequencies, the waves align such that nodes and antinodes form at fixed positions, allowing the system to sustain vibrations at discrete resonant frequencies.

Boundary Conditions

Boundary conditions define the behavior of a wave at the endpoints of a medium. For a string tied at both ends, the displacement must be zero at the supports, whereas in the tube with symmetric boundary conditions the air pressure or displacement conditions at the boundaries dictate the possible standing wave patterns. These conditions are essential in determining the allowed resonant frequencies of the system.

Wave Speed and Frequency Relationship

The speed of a wave determines how fast energy transfers through a medium and is directly linked to the frequencies of the standing waves through the relationship f = nv/2L or f = nv/4L, depending on the specific boundary conditions. When the wave speed changes, the entire harmonic series scales proportionally, meaning that a doubling of wave speed leads to a doubling of every resonant frequency.

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