A transverse wave is represented by the equation $y=y_0 \sin\frac{2\pi}{\lambda}(vt-x)$ For what value of $\lambda$, is the maximum particle velocity equal to two times the wave velocity?
- $\lambda=\frac{\pi y_0}{2}$
- $\lambda=\frac{\pi y_0}{3}$
- $\lambda=2\pi y_0$
- $\lambda=\pi y_0$
The Correct Option is D
Solution and Explanation
$y=y_0 \sin\frac{2\pi}{\lambda}(vt-x)$
Particle velocity $=\frac{dy}{dt}=y_0 \cos\frac{2\pi}{\lambda}(vt-x).\frac{2\pi v}{\lambda}$
$\Big(\frac{dy}{dt}\Big)_{max}=y_0.\frac{2\pi}{\lambda}v.$
$\therefore\, \, \, \, \, y_0.\frac{2\pi}{\lambda}v=2v$ or, $\lambda=\pi y_0.$
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Concepts Used:
Waves
Waves are a disturbance through which the energy travels from one point to another. Most acquainted are surface waves that tour on the water, but sound, mild, and the movement of subatomic particles all exhibit wavelike properties. inside the most effective waves, the disturbance oscillates periodically (see periodic movement) with a set frequency and wavelength.
Types of Waves:
Transverse Waves -
Waves in which the medium moves at right angles to the direction of the wave.
Examples of transverse waves:
- Water waves (ripples of gravity waves, not sound through water)
- Light waves
- S-wave earthquake waves
- Stringed instruments
- Torsion wave
The high point of a transverse wave is a crest. The low part is a trough.
Longitudinal Wave -
A longitudinal wave has the movement of the particles in the medium in the same dimension as the direction of movement of the wave.
Examples of longitudinal waves:
- Sound waves
- P-type earthquake waves
- Compression wave