Question

Standing waves are produced by the superposition of two waves \(y_{1}=0.05 \sin (3 \pi t-2 x)\) and \(y_{2}=0.05 \sin (3 \pi t+2 x)\) where \(\mathrm{x}\) and \(\mathrm{y}\) are in meters and \(\mathrm{t}\) is in seconds. The velocity (in \(\mathrm{ms}^{-1}\) ) of a particle at \(\mathrm{x}=0.25 \mathrm{~m}\) at \(\mathrm{t}=0.5 \mathrm{~s}\) is \(\ldots \ldots\) (A) \(0.1 \pi\) (B) \(0.3 \pi\) (C) zero (D) \(0.3\)

Short answer

The magnitude of the velocity of a particle at x = 0.25 m and t = 0.5 s is 0.3π ms^{-1} (Option B).

Step-by-step solution

Find the equation of the resulting standing wave

To find the equation of the standing wave formed by the superposition of two waves, y1 and y2, we simply add the two equations. The given equations for the waves are: \( y_{1} = 0.05 \sin(3 \pi t - 2x) \) \( y_{2} = 0.05 \sin(3 \pi t + 2x) \) The combined wave is: \( y = y_{1} + y_{2} = 0.05 \sin(3 \pi t - 2x) + 0.05 \sin(3 \pi t + 2x) \) Now that we have the equation of the combined wave, we can proceed to the next step.

Calculate the partial derivative with respect to time

In order to find the velocity of a particle in the wave, we need to calculate the partial derivative of the wave equation, y, with respect to time, t. Taking the derivative with respect to time, we get: \( v = \frac{\partial y}{\partial t} = 0.05 \cdot 3 \pi \cos(3 \pi t - 2x) - 0.05 \cdot 3 \pi \cos(3 \pi t + 2x) \) Simplifying the equation: \( v = 0.15 \pi (\cos(3 \pi t - 2x) - \cos(3 \pi t + 2x)) \) Now that we have the equation for the velocity of the particle, we can proceed to find the velocity at the specific point.

Plug in values of x and t

We are given x = 0.25 meters and t = 0.5 seconds. Plug these values into the velocity equation to find the velocity at the specified point. \( v = 0.15 \pi (\cos(3\pi \cdot 0.5 - 2 \cdot 0.25) - \cos(3\pi \cdot 0.5 + 2 \cdot 0.25)) \) Calculating the value of the cosine functions: \( v = 0.15 \pi (\cos(\pi) - \cos(2 \pi)) \) The values of the cosine functions are: \( \cos(\pi) = -1 \) \( \cos(2 \pi) = 1 \) Substitute these values into the velocity equation: \( v = 0.15 \pi ((-1) - 1) \) Computing the final value for the velocity: \( v = 0.15 \pi (-2) \) \( v = -0.3\pi \) However, since we are looking for the magnitude of the velocity, we will disregard the negative sign: The final answer is: (A) 0.1π (B) 0.3π ✅ (C) zero (D) 0.3

Key concepts

Superposition of Waves

In the study of wave physics, understanding the concept of superposition is fundamental. When two or more waves traverse the same medium, they can overlap in such a way that their effects combine. This phenomenon is called 'superposition'.
When dealing with standing waves, as in this exercise, the superposition leads to characteristic patterns of nodes and antinodes because the waves are of the same frequency and amplitude, but travel in opposite directions. This interaction creates a standing wave that appears to be stationary.
The key here is that, by adding the two individual wave functions as seen in the exercise, we obtain a new wave function that represents the standing wave. This combined function showcases how waves can interfere constructively and destructively. Such insights are crucial for understanding various physical phenomena, from musical instruments to electromagnetic fields.

Wave Equation

The wave equation describes the behavior of waves in a given medium. In this exercise, two sinusoidal waves are described by their equations. A wave equation generally has the form: \[ y(x,t) = A \, \sin(kx - \omega t + \phi) \]Where:

• \( A \): Amplitude of the wave
• \( k \): Wavenumber, related to wavelength
• \( \omega \): Angular frequency, related to time period
• \( \phi \): Phase angle

For the waves in this exercise, we have specific equations for each wave, which we combine to find the standing wave equation. Understanding this is crucial because it helps break down how different properties such as amplitude, frequency, and velocity of waves play a role in overall wave behavior. This kind of wave equation is widely used in a variety of scientific fields, including acoustics, optics, and quantum mechanics.

Partial Derivative

When analyzing waves, partial derivatives become very useful, especially if we want to find rates of change with respect to one variable while keeping others constant. In this case, we use the partial derivative of the wave function with respect to time, \( \frac{\partial y}{\partial t} \), to find the velocity of a particle in the medium.
Performing the partial derivative helps us understand how a wave changes at any given point in time. This is because the derivative with respect to time gives us the temporal rate of change, which directly correlates to velocity – how fast a particular point on the wave is moving at a specific time and position.
A solid grasp of partial derivatives and their applications in the wave equation can lead students to deeper insights in fields involving wave dynamics, including but not limited to, fluid dynamics and electromagnetism.

Wave Velocity

Understanding wave velocity is integral when studying the motion of waves. In the exercise, we need to determine the velocity at a particular point in time and space. Wave velocity describes how fast a wave propagates through a medium, whereas particle velocity describes how fast particles in the medium are moving as the wave passes.
To find the velocity at a specific point, the partial derivative with respect to time is used on the standing wave equation. This gives us a velocity equation. By substituting given values of \( x \) and \( t \), we can calculate the specific velocity at those coordinates.
The resulting calculation shows how standing waves possess unique characteristics. Despite having parts that appear motionless (nodes), the waves themselves have dynamic properties we can measure and analyze. Understanding this distinction between wave velocity and particle velocity is crucial for students to conceptualize how energy and information travel through different media.