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.

*Progressive wave*

*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 φ_{2} be the phase of a
particle at time *t*_{1} and *t*_{2} respectively.

φ_{1}
= 2π(t_{1}/T ? x/ λ)

φ_{2}
= 2π(t_{2}/T ? x/ λ)

φ_{2
}- φ_{1} = 2π(t_{2}/T
- t_{1}/T)

∆φ
= 2π /T ∆t

*t* = T, ∆φ = 2π. This shows that
after a time period T, the phase of a particle becomes the same.

*(ii) Variation of
phase with distance*

At a given time t
phase changes periodically with distance *x*.
Let φ_{1} and φ _{2}
be the phase of two
particles at distance
*x*_{1} and
*x*_{2} respectively from the origin at a time t.

φ_{1}
= 2π(t/T ? x_{1}/ λ)

φ_{2}
= 2π(t/T ? x_{2}/ λ)

φ_{2
}- φ_{1} = 2π/ λ (x_{2 }-
x_{1})

∆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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