A student is investigating the characteristics of waves in a rope. They set up an experiment where they generate a wave by shaking one end of a rope. The wave travels along the rope and is measured using a motion sensor. The student observes the following characteristics of the wave:
a) Explain how each of the observed characteristics relates to the properties of waves. b) Determine the relationship between the wavelength and frequency of a wave, and provide a mathematical equation to describe this relationship. c) Explain how the amplitude of a wave changes as it travels a longer distance along the rope and how this relates to wave energy. d) Determine the relationship between wave speed, wavelength, and frequency, and provide a mathematical equation to describe this relationship. e) Discuss whether the observed relationship between wave speed and tension in the rope is consistent with the properties of waves.
a)
The observed relationship between wavelength and frequency is related to the wave's speed. The frequency of a wave represents the number of complete wave cycles occurring per unit time. As the frequency of shaking increases, the number of wave cycles passing a given point per unit time increases. Therefore, the wavelength, which is the distance between two consecutive points in a wave cycle, decreases. This relationship follows the equation: wavelength = speed / frequency.
The decrease in amplitude as the wave travels along the rope is due to energy dissipation. Waves lose energy as they propagate through a medium. As the wave travels a longer distance, more energy is transferred to the surrounding particles or dissipated as heat, causing a decrease in amplitude.
The increase in wave speed with increased tension in the rope is explained by the wave equation. Wave speed is given by the product of wavelength and frequency. When the tension in the rope increases, the medium becomes more rigid, allowing the wave to propagate faster. This relationship follows the equation: wave speed = wavelength * frequency.
b) The relationship between wavelength (λ) and frequency (f) of a wave can be described by the equation: wavelength = speed / frequency. This relationship indicates that as the frequency increases, the wavelength decreases, and vice versa. The wavelength and frequency are inversely proportional.
c) As a wave travels a longer distance along the rope, its amplitude decreases. This is because the energy carried by the wave becomes spread out over a larger area as the wave spreads out. The energy per unit area, which corresponds to the wave's amplitude, decreases. Therefore, the amplitude of a wave decreases with distance traveled, indicating a decrease in wave energy.
d) The relationship between wave speed (v), wavelength (λ), and frequency (f) can be described by the equation: wave speed = wavelength * frequency. This equation indicates that wave speed is directly proportional to both wavelength and frequency. When wavelength or frequency increases, the wave speed also increases, and vice versa.
e) The observed relationship between wave speed and tension in the rope is consistent with the properties of waves. According to the wave equation (wave speed = wavelength * frequency), both wavelength and frequency contribute to wave speed. When tension in the rope increases, the wavelength and frequency remain constant, resulting in an increase in wave speed. This relationship aligns with the expectation that a higher tension in the medium allows waves to propagate faster.