State the characteristics of progressive waves.

Question

Accepted Answer

The characteristics of a progressive wave, as recapped in section 6.2.1, are: (1) every particle of the medium performs the SAME type of vibration -- simple harmonic motion about its own fixed mean position; (2) all the vibrating particles share a common amplitude, period and frequency; (3) the PHASE (state of vibration) of a particle changes continuously and progressively from one particle to the next along the direction of the wave's travel; (4) no particle ever remains permanently at rest -- each one only comes to rest MOMENTARILY, at its own extreme positions of vibration; (5) particles move with their MAXIMUM velocity exactly as they pass through their own mean positions; (6) as the wave propagates, ENERGY (and momentum) is genuinely transferred along the direction of travel, while there is NO net transfer of matter -- the medium's own particles do not travel bodily with the wave; (7) the wave's speed of propagation through a given medium depends only on the properties of that medium, not on the wave's own frequency or amplitude; (8) progressive waves occur in two kinds -- TRANSVERSE waves, where particle vibration is perpendicular to the direction of propagation (producing crests and troughs), and LONGITUDINAL waves, where particle vibration is along the direction of propagation (producing compressions and rarefactions); and (9) both transverse and longitudinal mechanical waves can propagate through solids, but only LONGITUDINAL waves can propagate through fluids (liquids and gases), since a fluid cannot sustain the shearing restoring force a transverse wave needs. [!ANSWER] See the full list of characteristics in the working.

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2022Set ANNUAL1 markMCQ
$y = 8 \sin(0.02x - 4t)$
(a) 10 cm/s
(b) 20 cm/s
(c) 100 cm/s
(d) 200 cm/s
›Reveal solutionSolution

Compare y = 8sin(0.02x − 4t) with y = Asin(kx − ωt): speed = ω/k.

Comparing $y = 8\sin(0.02x - 4t)$ with the standard form $y = A\sin(kx - \omega t)$ : $k = 0.02\ \text{cm}^{-1}$ , $\omega = 4\ \text{rad/s}$ .

$v = \frac{\omega}{k} = \frac{4}{0.02} = 200\ \text{cm/s}$

✓Final answer
$(d) 200 cm/s.$
CBSE 2017Set ANNUAL1 markMCQ
$\lambda$
(a) $\lambda$
(b) $\dfrac{\lambda}{2}$
(c) $\dfrac{\lambda}{3}$
(d) $\dfrac{\lambda}{4}$
›Reveal solutionSolution

For a progressive wave $y = A\sin(kx-\omega t)$ , the particle speed is $|v| = A\omega|\cos(kx-\omega t)|$ ; two particles have equal speed at every instant only if their phase difference is $0$ or $\pi$ .

Let the wave be $y(x,t) = A\sin(kx - \omega t)$ , so the transverse particle velocity is

$v(x,t) = \frac{\partial y}{\partial t} = -A\omega\cos(kx-\omega t)$

For two particles at positions $x_1$ and $x_2 = x_1 + \Delta x$ , their velocities are $-A\omega\cos\theta$ and $-A\omega\cos(\theta+\phi)$ , where $\theta = kx_1-\omega t$ (which sweeps through all values as $t$ varies) and $\phi = k\Delta x$ is the constant phase difference between the particles.

For the two particles to have the same speed $|v|$ at every instant $t$ , we need $|\cos\theta| = |\cos(\theta+\phi)|$ for all $\theta$ . This holds only when $\phi = 0$ (particles identical, $\Delta x = 0$ or a whole wavelength) or $\phi = \pi$ (particles exactly out of phase, $\Delta x = \lambda/2$ ) — in the latter case the velocities are equal in magnitude but opposite in direction at every instant, so the speed (magnitude only) is always equal.

The smallest non-zero separation satisfying this is therefore $\Delta x = \lambda/2$ .

✓Final answer
$\lambda/2$

Q.State the characteristics of progressive waves.

Concept understanding — Progressive Waves

Q.State the characteristics of progressive waves.

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