Question

Equation for a harmonic progressive wave is given by \(\mathrm{y}=\mathrm{A}\) \(\sin (15 \pi t+10 \pi x+\pi / 3)\) where \(x\) is in meter and \(t\) is in seconds. This wave is \(\ldots \ldots\) (A) Travelling along the positive \(\mathrm{x}\) direction with a speed of \(1.5 \mathrm{~ms}^{-1}\) (B) Travelling along the negative \(\mathrm{x}\) direction with a speed of \(1.5 \mathrm{~ms}^{-1} .\) (C) Has a wavelength of \(1.5 \mathrm{~m}\) along the \(-\mathrm{x}\) direction. (D) Has a wavelength of \(1.5 \mathrm{~m}\) along the positive \(\mathrm{x}\) - direction.

Short answer

The correct answer is (B) Travelling along the negative x direction with a speed of \(1.5~ms^{-1}\).

Step-by-step solution

Identify the general wave equation

A general equation for a harmonic wave traveling in the x-direction is given as: \[y(x,t) = A \sin(kx \pm \omega t + \phi)\] Here, `A` is the amplitude, `k` is the wave number (\(\frac{2\pi}{\lambda}\)), `\(\omega\)` is the angular frequency (\(2\pi f\)), and `\(\phi\)` is the phase difference.

Compare the given equation with the general wave equation

Given wave equation is: \[\mathrm{y}=\mathrm{A} \sin (15 \pi t + 10 \pi x + \pi / 3)\] Comparing it with the general wave equation, we can identify the terms in the expression: - \(k = 10\pi\) - \(\omega = 15\pi\) - \(\phi = \pi/3\)

Calculate the wavelength, frequency, and wave speed

Recall that \(k = \frac{2\pi}{\lambda}\) and \(\omega = "@/pi" \times f\). So, - Wavelength, \(\lambda = \frac{2\pi}{10\pi} = 0.2\;m\) - Frequency, \(f = \frac{15\pi}{2\pi} = 7.5\;Hz\) - Wave speed (v) can be calculated by multiplying the wavelength and frequency: \(v =\lambda \times f = 0.2 \times 7.5 = 1.5~ms^{-1}\)

Determine the direction of wave propagation

The wave is traveling in the positive x-direction if the sign between kx and \(\omega\)t is negative, and it's traveling in the negative x-direction if the sign is positive. In our case, it's a positive sign so the wave is traveling in the negative x-direction.

Analyze the answer choices

Based on our calculated parameters and identified direction, the correct answer is: (B) Travelling along the negative x direction with a speed of \(1.5~ms^{-1}\).

Key concepts

Wave Equation

In its essence, a wave equation represents the behavior of wave-like phenomena. The general wave equation for a harmonic wave traveling in the x-direction is given by \[y(x,t) = A \sin(kx \pm \omega t + \phi),\]where:

• \(A\) is the amplitude of the wave.
• \(k\) is the wave number, related to the wavelength \(\lambda\) by \(k = \frac{2\pi}{\lambda}\).
• \(\omega\) is the angular frequency, calculated with \(\omega = 2\pi f\), where \(f\) is the frequency.
• \(\phi\) represents the phase difference of the wave.

These parameters allow us to fully describe the characteristics of a harmonic progressive wave. In the context of the given exercise, the equation \[y = A \sin(15 \pi t + 10 \pi x + \pi / 3)\]can be analyzed to identify each of these terms, guiding us in understanding how this wave behaves as it travels through a medium.

Wavelength Calculation

Wavelength, denoted \(\lambda\), is the distance over which the wave's shape repeats. It is a critical parameter for understanding the spatial periodicity of waves. In wave equations, the wave number, \(k\), is related to the wavelength through the formula: \(k = \frac{2\pi}{\lambda}\). By rearranging this, the wavelength can be expressed as:\[\lambda = \frac{2\pi}{k}\]In our exercise, the wave number \(k\) is given as \(10\pi\). Substituting this into our formula results in:\[\lambda = \frac{2\pi}{10\pi} = 0.2 \text{ m}\]Understanding the wavelength helps us predict how far apart the wave peaks and troughs are, providing insight into the spread and reach of the wave in space.

Wave Speed

Wave speed is a crucial attribute of any wave, indicating how fast the wave propagates through space. The speed \(v\) of a wave is determined by the product of its wavelength \(\lambda\) and frequency \(f\):\[v = \lambda \times f\]For our harmonic wave, we have determined the following parameters:

• Wavelength \(\lambda = 0.2 \text{ m}\)
• Frequency \(f = 7.5 \text{ Hz}\)

By multiplying these together, we can calculate the wave speed:\[v = 0.2 \times 7.5 = 1.5 \text{ ms}^{-1}\]This tells us that the wave covers a distance of 1.5 meters every second as it travels through the medium. Understanding wave speed is essential for realizing how quickly the wave can transmit energy and information from one point to another.

Direction of Wave Propagation

The direction in which a wave travels is determined by the sign between the terms \(kx\) and \(\omega t\) in the wave equation. If the sign is negative (\(-\)), the wave propagates in the positive x-direction, while a positive (\(+\)) sign indicates movement in the negative x-direction.

In the given equation \(y = A \sin(15 \pi t + 10 \pi x + \pi / 3)\), the positive sign between \(k\) and \(\omega t\) implies that the wave is moving in the negative x-direction. This understanding is key, as it directly influences how we perceive the wave's movement through space.

Interpreting the direction is not only crucial for academic purposes but also for practical applications such as understanding wave behavior in different media, and in fields like acoustics and electromagnetic communications.