# what is the progressive wave and what is the wave’s equation?

Yugabrat Gogoi
47 Points
9 years ago

Progressive waves distribute energy from a point source to a surrounding area. They move energy in the form of vibrating particles or fields.

There are two different types of progressive waves:

• Transverse waves - vibrations are perpendicular to the wave motion - so if the wave is travelling horizontally, the vibrations will be up and down. For example, light and water.
• Longitudinal waves - vibrations are parallel to the wave motion - so if the wave is travelling horizontally, the particles will be compressed closer together horizontally, or expanded horizontally as they go along (we call the expanded bit a rarefaction). The particle movement is a series of compressions and rarefactions. For example, sound and some earthquake waves.
(1) These waves propagate in the forward direction of medium with a finite velocity.
(2) Energy and momentum are transmitted in the direction of propagation of waves without actual transmission of matter.
(3) In progressive waves, equal changes in pressure and density occurs at all points of medium.
(4) Various forms of progressive wave function:
(i) y = A sin (ω t – kx)
(ii) y = A sin (ω t – 2π / λ  x  )
(iii) y = A sin 2π [ t / T - x / λ ]
(iv) y = A sin 2π / λ (vt – x)
(v) y = A sin ω ( t - x / v )
where y = displacement
A = amplitude
ω = angular frequency
n = frequency
k = propagation constant
T = time period
λ = wave length
v = wave velocity
t = instantaneous time
x = position of particle from origin
Important points:
(a) If the sign between t and x terms is negative the wave is propagating along positive X-axis and if the sign is positive then the wave moves in negative X-axis direction.
(b) The coefficient of sin or cos functions i.e. Argument of sin or cos function  i.e. (ω t - kx) = Phase.
(c) The coefficient of t gives angular frequency ω = 2πn = 2π / T = vk.
(d) The coefficient of x gives propagation constant or wave number k = 2π / λ = ω / V  .
(e) The ratio of coefficient of t to that of x gives wave or phase velocity. i.e. v = ω / k  .
(f) When a given wave passes from one medium to another its frequency does not change.
(g) From v = nλ ⇒  v ∝ λ   ... n = constant ⇒ v1/v2 = λ12
(5) Some terms related to progressive waves
(i) Wave number ( n¯): The number of waves present in unit length is defined as the
wave number (n¯) =  1/λ.
Unit = meter–1 ; Dimension = [L–1].
(ii) Propagation constant (k): k= / x   = Phase difference between particles / Distancebetween them
k = ω / v = Angular velocity / Wave velocity   and   k = 2π / λ = 2π λ ¯
(iii) Wave velocity (v): The velocity with which the crests and troughs or
compression and rarefaction travel in a medium,is defined as wave velocity
v = ω / k  = n λ = ωλ / 2π = λ / T .
(iv) Phase and phase difference : Phase of the wave is given by the argument of
sine or cosine in the equation of wave. It is represented by Φ(x, t ) = 2π / λ ( vt - x ) .
(v)  At a given position (for fixed value of x) phase changes with time (t).
dΦ/dt  = 2πv / λ = 2π / T ⇒  dΦ= 2π / T . dt ⇒   Phase difference = 2π / T x  Time difference.
(vi) At a given time (for fixed value of t) phase changes with position (x).
dΦ/ dt  = 2πv / λ ⇒ dΦ2π / λ x dx ⇒ Phase difference = 2π / λ x  Path difference
Time difference = T  / λ   × Path difference