derive an expression for displacement of a plane progressive wave
derive an expression for displacement of a plane progressive wave.
Answer: The displacement of a plane progressive wave can be described by the following equation:
y(x, t) = A * sin(kx - ωt + φ)
where y is the displacement of the wave at position x and time t, A is the amplitude of the wave, k is the wave number (2π/λ) where λ is the wavelength, ω is the angular frequency (2πf) where f is the frequency, and φ is the phase constant.
The term (kx - ωt + φ) in the equation represents the phase of the wave at a given position x and time t. The phase constant φ determines the initial phase of the wave, while the term (kx - ωt) represents the phase difference between two points along the wave at position x and x+Δx at time t.
The displacement y(x, t) is periodic both in space and time, with a period of λ and 1/f, respectively. The wave travels in the positive x-direction with a speed given by v = λf = ω/k.
The displacement of a plane progressive wave describes the oscillatory motion of the wave in the direction perpendicular to the direction of propagation. The amplitude A determines the maximum displacement of the wave, while the phase constant φ determines the initial position of the wave.
In summary, the displacement of a plane progressive wave can be expressed as y(x, t) = A * sin(kx - ωt + φ), where A is the amplitude, k is the wave number, ω is the angular frequency, φ is the phase constant, and v = λf = ω/k is the speed of the wave.