A wave traveling on a Slinky⊗that is stretched to 4 m takes 2.4 s to travel the length of the Sl

step 1 As we know, velocity of a wave is given by wavelength times frequency and distance travel divide

A wave traveling on a Slinky⊗that is stretched to 4 m takes...

A wave traveling on a Slinky\otimes that is stretched to 4 $\mathrm{m}$ takes 2.4 $\mathrm{s}$ to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?

Key Concepts

Relationship between Frequency, Wavelength, and Speed

The equation v = f?, where v is the wave speed, f is the frequency, and ? is the wavelength, is central to wave theory. This fundamental relationship applies to both traveling and standing waves, enabling the calculation of one variable when the other two are known. It underscores the interconnected nature of these physical quantities in wave behavior.

Harmonic Frequencies

Harmonics are the specific frequencies at which standing waves occur in a finite medium. The fundamental frequency (first harmonic) is the lowest frequency with a standing wave pattern, and higher harmonics have wavelengths that are integer divisions of the medium’s length. The relationship between the physical length of the system, the number of nodes and antinodes, and the wavelength allows for determining the frequency required to produce a given standing wave pattern.

Wave Speed

Wave speed is a measure of how fast a wave propagates through a medium. It is computed by dividing the distance traveled by the time taken. The concept is fundamental in physics as it links the dynamics of wave propagation to the medium’s properties and is applicable to all types of waves, whether mechanical or electromagnetic.

Standing Waves

Standing waves are created by the interference of two waves traveling in opposite directions, resulting in a fixed pattern of nodes (points of zero amplitude) and antinodes (points of maximum amplitude). In systems with fixed boundaries, such as a stretched rope or Slinky, only certain discrete frequencies—corresponding to the natural harmonics of the system—will produce stable standing wave patterns.

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