Question

If a wave travelling in positive x-direction with $A=0.2$ m, velocity $=360$ m/s and $λ=60$ m, then correct expression for the wave is:

• A. $y=0.2 \sin[2π(6t + \fracx60)]$
• B. $y=0.2 \sin[π(6t + \fracx60)]$
• C. $y=0.2 \sin[2π(6t - \fracx60)]$
• D. $y=0.2 \sin[π(6t - \fracx60)]$

Hint:

For wave travelling in +x direction: $y = A \sin(kx - ω t)$.

Answer 1

The Correct Option is C

To find the correct expression for the wave, we need to use the standard wave equation and match it with the given parameters. The general equation for a wave traveling in the positive x-direction is:

\(y(x, t) = A \sin\left(2\pi\left(\frac{t}{T} \pm \frac{x}{\lambda}\right)\right)\)

where:

• \(A\) is the amplitude of the wave, given as 0.2 m.
• \(\lambda\) is the wavelength, given as 60 m.
• \(v\) is the velocity of the wave, given as 360 m/s.

First, we'll calculate the frequency \(f\). The relationship between velocity, frequency, and wavelength is:

\(v = f\lambda\)

Rearranging for \(f\) gives:

\(f = \frac{v}{\lambda} = \frac{360 \, \text{m/s}}{60 \, \text{m}} = 6 \, \text{Hz}\)

Therefore, the angular frequency \(\omega\) is:

\(\omega = 2\pi f = 2\pi \times 6 = 12\pi \, \text{rad/s}\)

And the wave number \(k\) is:

\(k = \frac{2\pi}{\lambda} = \frac{2\pi}{60} \, \text{rad/m}\)

Substitute the calculated values into the wave equation:

\(y(x, t) = 0.2 \sin\left(2\pi \cdot 6t - \frac{2\pi}{60} x\right)\)

Simplifying the expression inside the sine function, we have:

\(y(x, t) = 0.2 \sin\left(12\pi t - \frac{\pi}{30} x\right)\)

This matches the given option:

$y=0.2 \sin[2π(6t - \frac{x}{60})]$

This is the correct expression because:

• The wave is traveling in the positive x-direction, indicated by the minus sign in the phase term opposite to \(x\).
• The amplitude, frequency, and phase values align with the problem's parameters.