Spirituality, Quantum Physics & Life: Progressive Wave

Equation of Progressive Wave:

(This is a mathematical language for analysis of wave motion).

Suppose a progressive wave traveling from left to right along x - axis as shown in figure. Let us consider a particle vibrate with simple harmonically at the origin $o$.

The displacement '$y$' of the vibrating particle at origin '$O$', at any instant '$t$' is given by:

$y = a Sin \omega t$ (i)

Where, $a$ = amplitude of the particle; $\omega$ = angular velocity; $t$ = time

Again, Consider a particle '$P$' at distance '$x$' from the origin '$O$' as in figure. Let $\phi$ be the phase lag of the particle $P$. We know that for a distance of $\lambda$, Phase difference = $2\pi$.

So the Phase Difference '$\phi$' at '$P$' at a distance '$x$' from '$O$' is $(x/\lambda)2\pi$ Then the displacement of a particle '$P$' at distance '$x$' from '$O$' is,

$y = aSin(\omega t - \phi)$ (ii)

Substituting this value of '$(\phi)$' in $eq^n.$ (ii), we get

$y = a Sin(\omega t - \frac{2\pi x}{\lambda})$

The quantity $2\pi/\lambda = k$ is called wave number or propagation constant. Then

$y = aSin(\omega t - k x)$ (iii)

Since $\omega = 2\pi/T$; Then $eq^n$ (iii) becomes,

$y = a Sin(\frac{2\pi}{T} t - \frac{2\pi}{\lambda}x)$

or, $y = a Sin 2\pi(\frac{t}{T} - \frac{x}{\lambda})$ (iv)

Again, $\omega = 2\pi f = 2\pi v/\lambda$; Where '$v$' be the velocity of the wave, we have

$y = a Sin(\omega t - \frac{2\pi}{\lambda}x)$

or, $y = a Sin(\frac{2\pi v}{\lambda} t - \frac{2\pi}{\lambda}x)$

or, $y = a Sin \frac{2\pi}{\lambda} (vt - x)$ (v)

$Eq^n$ (iv) or $Eq^n$ (v) represents the Plane Progressive Wave. If the wave travels from right to left i.e. negative $x- axis$. So equation of the wave in this case is,

$y = a Sin \frac{2\pi}{\lambda} (vt + x)$ (vi)

Differential Equation of Wave Motion:

The equation of wave is (from $eq^n$ (v) )

$y = a Sin \frac{2\pi}{\lambda} (vt - x)$ (vii)

Differentiating $Eq^n$ (vii) with respect to $t$ is,

$\frac{\mathrm{d} y}{\mathrm{d} t} = \frac{2\pi v}{\lambda}a Cos (vt - x)$ (viii)

Again differentiating $eq^n$ (viii) with respect to $t$, we get

$\frac{\mathrm{d} ^2y}{\mathrm{d} t^2} = -\frac{4\pi^2 v^2}{\lambda^2}a Sin \frac{2\pi}{\lambda} (vt - x)$ (ix)

When the $eq^n$ (vii) is differentiated with repect to $x$, we get

$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{2\pi}{\lambda}a Cos \frac{2\pi}{\lambda}(vt - x)$

$\frac{\mathrm{d} ^2y}{\mathrm{d} x^2} = -\frac{4\pi^2}{\lambda^2}a Sin \frac{2\pi}{\lambda} (vt - x)$ (x)

From $Eq^n$ (ix) and (x), we have

$\frac{\mathrm{d} ^2y}{\mathrm{d} t^2} = v^2\frac{\mathrm{d} ^2y}{\mathrm{d} x^2}$ (xi)

Which is the differential wave equation.

Note: This is adapted from [Principle of PHYSICS - XII]

Click here for: Mathematical Expression of Stationary Wave

Return to Main Menu

"Wave that moves in a medium."

Progressive (Traveling) Waves:

A wave, which travels (moves) from one point to another point in a medium.

Progressive wave

All particle which are in motion are in same amplitude and frequency but phase of the vibration ranges point to point along the waves. No particles are permanently at rest but they vibrate with different phase.

Both Transverse wave and Longitudinal waves are Progressive wave.

For Mathematical description of Progressive wave - Click here.

Stationary (Standing) Waves:

A wave, that remains in a constant position. This is because, the medium is moving in the opposite direction to the wave (i.e. two progressive waves of same amplitude and frequency travel with the same speed in opposite direction to each other).