T1.1 Wave Motion (Mathematical Expression of 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

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