The Speed of a Travelling Wave
In this section, we explore the concept of wave speed, which is the speed at which a wave propagates through a medium. The speed of a wave can be derived by observing a specific point of the wave, typically the crest, and noting its movement over time.
Key Points:
1. Wave Speed Calculation: If we observe a point moving with a fixed phase on the wave (denoted by the relationship kx - ωt = constant), we can derive the wave speed as the ratio of the distance traveled (∆x ) to the time taken ( 08∆t 09), leading to the equation
\[ v = \frac{\Delta x}{\Delta t} = \frac{\omega}{k} \]
where ω represents the angular frequency and k is the angular wave number.
- Relationship Among Wave Properties: The general relation for all progressive waves states that the speed of a wave is equal to the wavelength divided by the period of oscillation: \[ v = \lambda T = \nu \lambda \]
Here, λ is the wavelength, and ν represents the frequency of the wave.
- Mechanical Waves: The speed of a mechanical wave is determined by the mass density of the medium and its elastic properties. Specific formulas include:
- For 8A transverse wave on a stretched string: \[ v = \sqrt{\frac{T}{\mu}} \] where T is the tension and μ is the linear mass density.
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For longitudinal waves (such as sound) in a fluid: \[ v = \sqrt{\frac{B}{\rho}} \] where B is the bulk modulus and ρ is the density of the fluid.
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Estimating the Speed of Sound: The speed of sound in gases can be determined from the equation \[ v = \sqrt{\frac{\gamma P}{\rho}} \], which incorporates the bulk modulus and density for gases.
This section serves as a fundamental basis for understanding wave dynamics and their propagation characteristics.