A progressive wave is defined as the onward transmission of the vibratory motion of a body in an elastic medium from one particle to the successive particle.
Equation of a plane progressive wave
An equation can be formed to represent generally the displacement of a vibrating particle in a medium through which a wave passes. Thus each particle of a progressive wave executes simple harmonic motion of the same period and amplitude differing in phase from each other.
Let us assume that a progressive wave travels from the origin O along the positive direction of X axis, from left to right (Fig. 7.6). The displacement of a particle at a given instant is
y = a sin ωt ... (1)
where a is the amplitude of the vibration of the particle and ω = 2πn.
The displacement of the particle P at a distance x from O at a given instant is given by,
y = a sin (ωt - φ) ?.(2)
If the two particles are separated by a distance λ, they will differ by a phase of 2π. Therefore, the phase φ of the particle P at a distance
x is φ = 2π/λ . x
y = a sin (ωt - 2πx/ λ)
Since ω = 2πn = 2π v / λ, the equation is given by
y = a sin(2πt/ λ - 2πx/ λ)
y = a sin( 2π/ λ .(vt-x) ) ???..(4)
Since ω = 2π/ T , the eqn. (3) can also be written as
y = a sin 2π (t/T ? x/ λ)
If the wave travels in opposite direction, the equation becomes.
y = a sin 2π(t/T + x/ λ) ???(6)
(i) Variation of phase with time
The phase changes continuously with time at a constant distance.
At a given distance x from O let φ and φ2 be the phase of a particle at time t1 and t2 respectively.
φ1 = 2π(t1/T ? x/ λ)
φ2 = 2π(t2/T ? x/ λ)
φ2 - φ1 = 2π(t2/T - t1/T)
∆φ = 2π /T ∆t
(ii) Variation of phase with distance
At a given time t phase changes periodically with distance x. Let φ1 and φ 2 be the phase of two particles at distance x1 and x2 respectively from the origin at a time t.
φ1 = 2π(t/T ? x1/ λ)
φ2 = 2π(t/T ? x2/ λ)
φ2 - φ1 = 2π/ λ (x2 - x1)
∆φ = 2π / λ ∆t
The negative sign indicates that the forward points lag in phase when the wave travels from left to right.
When ∆x = λ, ∆φ = 2π, the phase difference between two particles having a path difference λ is 2π.
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