Progressive Wave and its Equation - Physics Grade XII

Progressive Wave

A progressive wave is defined as one in which the wave profile moves forward with the wave's speed. Both transverse and longitudinal waves are progressive waves. In a progressive wave, the amplitude and frequency of particle vibrations are the same, but the phase of the vibration varies from point to point along the wave.

Progressive Wave Equation

Suppose that the wave moves from left to right with a displacement velocity of v in which each article vibrates with a simple harmonic motion. However, when compared to the motion of the particle o at the origin O (at x = 0), subsequent particles to the right have phase lag. The vibrating particle's displacement y at origin O is given by,

y = a sin ωt     ........ (1)

where 'a' is amplitude, 't' is time:
ω = 2πf, is the angular velocity, where f is the frequency of vibration.

Consider a particle P at distance x from the origin as in fig.

Φ be the phase lag of the particle P and λ be the wavelength.

At a distance λ from O, phase difference, Φ = 2π
At a distance x from O, phase difference, Φ = (2π/λ)*x

The displacement of the particle P at distance x from O is,

y = a sin(ωt - Φ)

or, y = a sin(ωt - 2πx/λ) = a sin(ωt - kx)      ........... (2)

where, k = 2π/λ is wave number or propagation constant.

or, y = a sin(2πvt/λ - 2πx/λ)  

or, y = a sin 2π/λ(vt - x)      ..........(3)

Using ω = 2π/T, equation (2) can be written as,

y = a sin(2πt/T - 2πx/λ)

or, y = a sin2x(t/T - x/λ)     .............. (4)

Equations (2), (3), and (4) are different forms of the plane-progressive wave equation, for a wave that moves from left to right so the vibration of particle P lags on that at O. If a wave is traveling from right to left then the waves arrive at P before O, therefore vibration of P leads that at O. In such a case, will be positive and wave equation is given by,

y = a sin (ωt + kx) ......... (5)

The particle velocity v for the particle at fixed distance x from O is obtained by taking the derivative of y with respect to t (taking x constant),

i.e. v = dy/dt = d/dt[a sin(ωt - kx)]

or, v = ωa cos(ωt - kx) ........... (6)

The acceleration of the particles is,

a = dy/dt = d/dt[ωa cos(ωt - kx)]

or, a = -ω²a sin(ωt - kx) ...... (7)