Derive the expression for the displacement of a transverse progressive
wave.
Answers
Answer:
In case of transverse wave displacement is given as: y(x,t) where x=propagation of the wave along x-axis, and particles oscillates along y-axis. Therefore y(x,t)= A sin(kx – ωt + φ). This is the expression for displacement.
Answer:
y = A sin(kx - ωt) is the expression for the displacement of a transverse progressive.
Explanation:
The displacement of a transverse progressive wave can be derived using the wave equation, which describes how waves propagate through a medium. The wave equation for a transverse wave is:
∂²y/∂x² = (1/v²) * ∂²y/∂t²
where y is the displacement of the wave, x is the position along the wave, t is time, and v is the velocity of the wave.
Assuming that the wave is travelling in the x direction with a constant velocity, we can separate the variables and write:
y = A sin(kx - ωt)
where A is the amplitude of the wave, k is the wave number (2π/λ), λ is the wavelength of the wave, and ω is the angular frequency (2πf), where f is the frequency of the wave.
The displacement of the wave at any position x and time t can be found by substituting the values of k, ω, x, and t into the equation:
y = A sin(kx - ωt)
For example, if we want to find the displacement of the wave at a position x = 0.5m and time t = 0.2s, we can plug in the values:
y = A sin(k0.5 - ω0.2)
The value of k and ω depend on the properties of the medium through which the wave is travelling, such as its density and elasticity. The wavelength λ and frequency f are related to the velocity of the wave v through the equation:
v = λf
Therefore, the displacement of a transverse progressive wave can be described by the equation:
y = A sin(kx - ωt)
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