Oscillation of a 600 Hz tuning fork sets up standing waves...

Oscillation of a 600 Hz tuning fork sets up standing waves in a string clamped at both ends. The wave speed for the string is 400 $\mathrm{m} / \mathrm{s}$ . The standing wave has four loops and an amplitude of 2.0 $\mathrm{mm}$ (a) What is the length of the string? (b) Write an equation for the displacement of the string as a function of position and time.

Oscillation of a 600 Hz tuning fork sets up standing waves in a string clamped at both ends. The wave speed for the string is
400 $\mathrm{m} / \mathrm{s}$ . The standing wave has four loops and an amplitude of
2.0 $\mathrm{mm}$ (a) What is the length of the string? (b) Write an equation
for the displacement of the string as a function of position and time.

Key Concepts

Standing Waves

Standing waves are the result of two waves of identical frequency and amplitude traveling in opposite directions interfering with one another. In a string that is fixed at both ends, this interference produces points that appear to be stationary (nodes) and points that oscillate with maximum amplitude (antinodes). The pattern remains fixed in space, even though individual particles of the medium oscillate with time.

Harmonics

Harmonics are the natural modes of vibration that occur at discrete frequencies in a system with fixed boundaries. For a string clamped at both ends, only specific wavelengths that accommodate an integral number of half-wavelengths are allowed. Each configuration corresponds to a harmonic, and the number of loops or antinodes in the standing wave pattern is directly related to the specific harmonic excited.

Wave Speed, Frequency, and Wavelength Relationship

The relationship between a wave's speed, frequency, and wavelength is given by the equation v = f?, where v is the speed of the wave, f is its frequency, and ? is its wavelength. This equation is key in understanding how changes in any one quantity affect the others, and it is particularly important in determining the characteristics of standing waves on a string.

Boundary Conditions

Boundary conditions specify the behavior of a wave at the boundaries of the medium—in this case, the fixed ends of the string where displacement must be zero. These conditions restrict the possible wave modes and ensure that only wavelengths that result in nodes at the ends can be sustained, directly influencing the formation of standing waves and the determination of harmonics.

Wave Equation Representation

The wave equation for a vibrating string is often expressed as a sinusoidal function that describes the displacement of the string in both space and time. This representation incorporates the amplitude, angular frequency, wave number, and phase of the wave. It models the oscillatory behavior of the string and is essential for analyzing how the displacement varies along the string and with time.

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