The wave equation is given as:
\[ y = 0.5 \sin\left(x\right)\left(\frac{\pi}{2} t + \frac{\pi}{3} x\right) \] This represents a transverse wave, where: - \( y \) is the displacement of the wave, - \( x \) is the position along the string, - \( t \) is the time. For a general wave equation of the form: \[ y(x, t) = A \sin(kx - \omega t) \] The velocity of the wave, \( v \), is given by: \[ v = \frac{\omega}{k} \] From the given equation, we identify: - The coefficient of \( x \) in the argument of the sine function gives \( k = \frac{\pi}{3} \), - The coefficient of \( t \) in the argument gives \( \omega = \frac{\pi}{2} \). Now, using the formula for wave velocity: \[ v = \frac{\omega}{k} = \frac{\frac{\pi}{2}}{\frac{\pi}{3}} = \frac{3}{2} = 2 \, \text{m/s} \] Thus, the velocity of the wave is \( 2 \, \text{m/s} \).
Correct Answer: (C) 2 \( \text{m/s} \)
A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 × 10–2 kg and its linear mass density is 4.0 × 10–2 kg m–1. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string?
For the travelling harmonic wave
y(x, t) = 2.0 cos 2π (10t – 0.0080 x + 0.35)
where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of
(a) 4 m,
(b) 0.5 m,
(c) \(\frac{λ}{2}\),
(d) \(\frac{3λ}{4}\)
A transverse harmonic wave on a string is described by
y(x, t) = 3.0 sin (36 t + 0.018 x + \(\frac{π}{4}\))
where x and y are in cm and t in s. The positive direction of x is from left to right.
(a) Is this a travelling wave or a stationary wave ? If it is travelling, what are the speed and direction of its propagation ?
(b) What are its amplitude and frequency ?
(c) What is the initial phase at the origin ?