91Ó°ÊÓ

Understanding the wave equation is foundational to grasping wave motion in physics. A wave equation describes how a wave propagates through space over time. For a traveling wave along the x-axis, the wave function might look like this:
\( y(x, t) = A \times \text{trig function}(kx - \f(x)t + \f(phase)) \),
where

• \(A\) is the amplitude of the wave,
• \(k\) is the wavenumber which relates to the wavelength,
• \(\f(x)\) is the angular frequency related to the time period,
• \(t\) is time, and
• \(x\) is the position along the x-axis.

In the given exercise, the wave equation is \( y(x, t) = 0.005 \cos (\f(alpha) x - \f(beta) t) \). By analyzing this equation, we find that \( \f(alpha) \) is related to the wavelength, and \( \f(beta) \) is linked to the time period of the wave. To find \( \f(alpha) \) and \( \f(beta) \), we utilize given information about the wavelength and the time period.

The wavelength and the time period are crucial characteristics of any periodic wave. The wavelength, denoted as \( \lambda \), is the distance between two consecutive points in phase on the wave, such as crest to crest or trough to trough. In our context, the given wavelength is 0.08m. On the other hand, the time period, denoted by \( T \), is the time it takes for one full cycle of the wave to pass a given point. For our problem, this is given as 2.0 seconds.
These values of \( \lambda \) and \( T \) can be converted into \( \f(alpha) \) and \( \f(beta) \) using the relationships \( \f(alpha) = \frac{2\pi}{\lambda} \) and \( \f(beta) = \frac{2\pi}{T} \) respectively. These equations arise from the properties of sinusoidal functions that describe waves and their cycles. By substituting the given values into these relationships, we can solve for \( \f(alpha) \) and \( \f(beta) \) to find their exact values corresponding to the options provided.

Waves are often described using trigonometric functions because of their periodic nature. In our exercise, a cosine function is used to describe the wave. The general form is \( A \cos(kx - \f(x)t + \f(phase)) \), where \( A \) is the amplitude, \( k \) is the wavenumber, \( \f(x) \) is the angular frequency, and the phase describes the initial angle at which the wave starts.
These trigonometric functions, like sine and cosine, are periodic and thus excellent at modeling the repeated oscillations of waves. The variables \( \f(alpha) \) and \( \f(beta) \) in the given wave equation correspond to the wavenumber and angular frequency, which dictate the spatial and temporal periods of the wave, representing how the wave oscillates in space and time. As such, a thorough understanding of trigonometric functions and their properties is essential for analyzing and solving wave equation problems in physics, especially for competitive exams like the JEE Main.