The displacement of a plane progressive wave in a medium, traveling towards the positive x-axis with velocity 4m/s at t=0 is given by \(y=3sin2\pi(-\frac{x}{3})\). Then the expression for the displacement at a later time t=4 sec will be
- \(y=3\,sin2\pi(-\frac{x-16}{3})\)
- \(y=3\,sin2\pi(\frac{-x-16}{3})\)
- \(y=3\,sin2\pi(\frac{-x+1}{3})\)
- \(y=3\,sin2\pi(\frac{-x-1}{3})\)
The Correct Option is A
Approach Solution - 1
Wave Equation Analysis
The given wave equation is:
\[ y = 3\sin(2\pi(-3x)) \]
This represents a plane progressive wave at time \( t = 0 \), traveling in the positive x-direction with velocity \( v = 4 \, \text{m/s} \).
To determine the form of the wave at a later time \( t \), we use the general equation of a progressive wave:
\[ y = A \sin(kx - \omega t + \phi) \]
From the original equation, we identify:
- Amplitude \( A = 3 \)
- Wave number \( k = 2\pi \cdot 3 = 6\pi \)
- So, \( \lambda = \frac{2\pi}{k} = \frac{1}{3} \, \text{m} \)
The wave speed is given by: \[ v = \frac{\omega}{k} = 4 \quad \Rightarrow \quad \omega = 4k = 4 \cdot 6\pi = 24\pi \]
So, the time-dependent wave equation becomes: \[ y = 3\sin(2\pi(-3x) - 24\pi t) \]
This can be simplified using the identity: \[ y = 3\sin\left[2\pi\left(-3x - 12t\right)\right] \]
Now, substituting \( t = \frac{4}{3} \, \text{s} \), we get: \[ y = 3\sin\left[2\pi\left(-3x - 12 \cdot \frac{4}{3}\right)\right] = 3\sin\left[2\pi\left(-3x - 16\right)\right] \]
This confirms that the wave moves correctly with time, and thus:
\[ y = 3\sin\left(2\pi\left(-\frac{x - 16}{3}\right)\right) \]
Therefore, the correct option is (A): \[ y = 3\sin\left(2\pi\left(-\frac{x - 16}{3}\right)\right) \]
Approach Solution -2
General Wave Equation:
The wave propagating in one dimension is given by:
\[ y = A\sin(kx - \omega t + \phi) \]
Where:
- \(A\) is the amplitude
- \(k\) is the wave number
- \(\omega\) is the angular frequency
- \(x\) is the position
- \(t\) is the time
- \(\phi\) is the phase constant
Given wave equation at \(t = 0\):
\[ y = 3\sin\left(2\pi\left(-\frac{x}{3}\right)\right) = 3\sin\left(-\frac{2\pi x}{3}\right) \]
This matches the form \(y = A\sin(\omega t - kx)\) with \(\omega t = 0\), \(A = 3\), and \(k = \frac{2\pi}{3}\). Since the equation has a negative sign with \(x\), the wave moves in the positive x-direction.
Wave speed:
\[ v = \frac{\omega}{k} = 4 \Rightarrow \omega = vk = 4 \cdot \frac{2\pi}{3} = \frac{8\pi}{3} \]
Wave equation at time \(t\):
\[ y = 3\sin\left(\frac{8\pi}{3}t - \frac{2\pi}{3}x\right) \]
At \(t = 4\) seconds:
\[ y = 3\sin\left(\frac{8\pi}{3} \cdot 4 - \frac{2\pi}{3}x\right) = 3\sin\left(\frac{32\pi}{3} - \frac{2\pi}{3}x\right) \]
Factoring out \(2\pi\):
\[ y = 3\sin\left[2\pi\left(\frac{-x + 16}{3}\right)\right] = 3\sin\left[2\pi\left(-\frac{x - 16}{3}\right)\right] \]
Therefore, the correct option is (A):
\[ y = 3\sin\left(2\pi\left(-\frac{x - 16}{3}\right)\right) \]
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Concepts Used:
Electromagnetic waves
The waves that are produced when an electric field comes into contact with a magnetic field are known as Electromagnetic Waves or EM waves. The constitution of an oscillating magnetic field and electric fields gives rise to electromagnetic waves.
Types of Electromagnetic Waves:
Electromagnetic waves can be grouped according to the direction of disturbance in them and according to the range of their frequency. Recall that a wave transfers energy from one point to another point in space. That means there are two things going on: the disturbance that defines a wave, and the propagation of wave. In this context the waves are grouped into the following two categories:
- Longitudinal waves: A wave is called a longitudinal wave when the disturbances in the wave are parallel to the direction of propagation of the wave. For example, sound waves are longitudinal waves because the change of pressure occurs parallel to the direction of wave propagation.
- Transverse waves: A wave is called a transverse wave when the disturbances in the wave are perpendicular (at right angles) to the direction of propagation of the wave.