Question

Derive an Expression for displacement of progressive wave?
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Answer 1

y(x, t) = a sin (kx + ωt + φ ), The above equation represents a transverse wave moving along the negative direction of the X-axis. ... Here, the plot shows a wave travelling in the positive X direction

Answer 2

Progressive wave: A wave in which the disturbance propagates continuously is called progressive wave.

• Consider a long string stretched in a horizontal direction along x -axis.

• if this string is given up and down,then the waves travel from the left end of the string to the right end with speed v.

• let at t=0, then wave on the string is represented by a sine curve as shown in figure.

[refer to attachment ]

The displacement 'y' of the element of the string at position x (at 't') is given by,

$\implies{y=a sin(wt+Φ)}$

$\implies{y=a sin(w(0)+Φ)}$

$\implies{y=a sin(0+Φ)}$

$\implies{y=a sin Φ}$ .... ( 1 )

The relation between path difference and phase difference is given by,

$\implies{Φ = \dfrac{2 \pi}{\lambda} (x1 - x2) ....( 2 )}$

now, substitute equation ( 2 ) in ( 1 )

the equation becomes,

$\implies{y = a \sin\frac{2 \pi}{\lambda} (x1 - x2)}$

where, a is the maximum displacement (or amplitude) of the oscillation and lamda is the wavelength of the wave.

The disturbance or the wave travels a distance 'vt' will oscillate and the displacement of this element at time 't' is given by,

$\implies{y = a \sin \dfrac{2\pi}{ \ \:\lambda }( x - vt)}$

$\implies{y = a \sin 2\pi (\dfrac{x}{\lambda} - \dfrac{vt}{\lambda} )}$

since, $\dfrac{v}{\lambda} = f$

$\implies{y = a \sin 2\pi (\dfrac{x}{\lambda} - ft)}$

$\implies{y = a \sin ( \dfrac{2\pi \: x}{\lambda} - 2\pi \: ft)}$

Since, $2\pi \: f =\omega \: and \: \frac{2\pi}{ \lambda } = k$

$\implies\boxed{y = a \sin(kx - \omega \: t)}$

This equation represents a travelling wave or a progressive wave propagating along positive X- Axis.

Note:-

If the progressive wave travels along negative x- axis, then this equation is represented by

$\implies{y = a \sin(kx + \omega \: t)}$

Thanks!