Question

The equation \(\mathrm{y}=10 \sin (\pi \mathrm{x} / 4) \cos 10 \pi t\) represents a stationary wave where \(\mathrm{x}\) and \(\mathrm{y}\) are in centimeter and \(\mathrm{t}\) is in seconds. The separation between two consecutive nodes is (A) \(2 \mathrm{~cm}\) (B) \(4 \mathrm{~cm}\) (C) \(5 \mathrm{~cm}\) (D) \(8 \mathrm{~cm}\) Copyright \(\odot\) StemEZ.com. All rights reserved.

Short answer

The separation between two consecutive nodes is 4 cm. Therefore, the correct answer is (B) 4 cm.

Step-by-step solution

Identify the spatial part of the wave equation

In the given equation, y = 10 * sin(πx/4) * cos(10πt), the function of x is sin(πx/4). We will focus on this part to find the distance between nodes.

Set the spatial function equal to zero

To find the nodes, we set the displacement y to zero. This means we need to find the values of x for which sin(πx/4) = 0. We will find two consecutive x values (x1 and x2) such that sin(πx1/4) = 0 and sin(πx2/4) = 0.

Solve for the consecutive x-values

Since sin(πx/4) = 0: \( \mathrm{n} \pi = \frac{\pi x}{4} \) for any integer n. Solving for x, we get: \( \mathrm{x} = 4\mathrm{n} \) Now we need to find two consecutive x values for which sin(πx/4) = 0. For n=1, x1 = 4 For n=2, x2 = 8

Calculate the separation between consecutive nodes

The distance between two consecutive nodes is the absolute difference between their x values. \( d = \vert x_2 - x_1 \vert \) Substitute x1 = 4 and x2 = 8: \( d = \vert 8 - 4 \vert \) \( d = 4 \mathrm{~cm} \) So, the separation between two consecutive nodes is 4 cm. Therefore, the correct answer is (B) 4 cm.

Key concepts

Nodes

In the context of stationary waves, nodes are points along the medium that remain still while the wave oscillates. These are the positions where the amplitude of the wave is always zero. Nodes occur due to the destructive interference of two waves traveling in opposite directions. In our given stationary wave equation, the nodes are determined by setting the spatial part of the equation to zero, specifically:

• \( \sin\left(\frac{\pi x}{4}\right) = 0 \)

This equation becomes zero for integer multiples of \( \pi \):

• \( n\pi = \frac{\pi x}{4} \)

Solving for \( x \) gives the positions of nodes at \( x = 4n \). Therefore, the nodes in this scenario are spaced 4 cm apart. This consistent pattern of nodes is a characteristic feature of stationary waves.

Wave equation

The wave equation is a mathematical representation of the motion of a wave. In this particular problem, the equation for the stationary wave is given by: \[ y = 10 \sin \left(\frac{\pi x}{4}\right) \cos(10\pi t) \]In this equation:

• The term \( 10 \sin \left(\frac{\pi x}{4}\right) \) represents the spatial part of the wave, which describes how the wave varies in space.
• The term \( \cos(10\pi t) \) represents the temporal part, describing how the wave varies over time.

The structure of this equation indicates a stationary wave, where the spatial and temporal components are separate. The stationary nature is evidenced by the fact that the sinusoidal function \( \sin \) depends purely on \( x \), while \( \cos \) is a function of time \( t \). Thus, at any given point in space, the wave's movement is oscillating but remains fixed in place.

Sinusoidal functions

Sinusoidal functions, like sine \( (\sin) \) and cosine \( (\cos) \), are fundamental to understanding wave behavior. These functions describe periodic oscillations and are vital for representing waves in mathematics.The stationary wave in our problem is represented by:

• \( y = 10 \sin \left(\frac{\pi x}{4}\right) \cos(10\pi t) \)

Here, \( \sin \left(\frac{\pi x}{4}\right) \) defines the spatial oscillation of the wave. The argument \( \frac{\pi x}{4} \) determines the wavelength and regularity of nodes. Meanwhile, \( \cos(10\pi t) \) determines the oscillation over time. These sinusoidal functions are what make the wave "stationary" because the crest and trough of the wave do not travel along the medium. Instead, they merely oscillate in place, creating characteristic patterns of nodes and antinodes.