Mind Body Axis

Transverse Standing Waves?

A string has a linear density of 9.00 x 10^-3 kg/m and is under a tension of 310 N. The string is 1.5 m long, is fixed at both ends, and is vibrating in the standing wave pattern shown below in the drawing.
(a) Determine the speed of the traveling wave.
m/s

Answering b is easy. Because the string has a node at each end, the fundamental wavelength will be twice the length of the string, with harmonics having wavelengths at exact submultiples of the fundamental. The drawing shows the third harmonic, which will have a wavelength one third that of the fundamental. The fundamental has a wavelength of 2*1.5 m or 3.00 m, so the third harmonic has a wavelength of 3.00 m / 3 or 1.00 m.

The speed of the traveling wave, assuming that the inherent stiffness of the string is negligible, will be √(T / ρ), where T is the string tension and ρ is the mass per unit length. The tension is given as 310 N and the the mass of the string per unit length is 9.00e-3 kg/m, giving a speed of √(310 N / 9.00e-3 kg/m) or 185.6 m/s.

The frequency is the velocity divided by the wavelength, or 185.6 m/s / 1.00 m/cycle = 185.6 cycles / sec.

Vibration creates the Universal Pattern - Part 1

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