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 distance (in meters) between two consecutive nodes is (A) \(\pi / 2\) (B) \(\pi\) (C) \(0.5\) (D) \(1.0\)

Short answer

The distance between two consecutive nodes in the standing wave produced by the given waves is \(\frac{\pi}{2}\) meters.

Step-by-step solution

Write down the given wave expressions

Given that two waves are: \(y_1 = 0.05\sin(3\pi t - 2x)\) and \(y_2 = 0.05\sin(3\pi t + 2x)\)

Find the resultant wave

The standing wave is formed by the superposition of two waves, so we need to add them: \(y = y_1 + y_2 = 0.05\sin(3\pi t - 2x) + 0.05\sin(3\pi t + 2x)\)

Simplify the combined wave equation

To simplify, we can use the trigonometric identity: \(\sin(a) + \sin(b) = 2\cos(\frac{a-b}{2})\sin(\frac{a+b}{2})\) In this case, \(a = 3\pi t - 2x\) and \(b = 3\pi t + 2x\). Using the identity: \(y = 0.05 [2\cos(\frac{(3\pi t - 2x) - (3\pi t + 2x)}{2})\sin(\frac{(3\pi t - 2x) + (3\pi t + 2x)}{2})]\) \(y = 0.1\cos(-2x)\sin(3\pi t)\)

Find the node positions

Nodes are points of zero amplitude. So, we need to find the values of x for which \(y = 0\). Setting the equation to zero: \(0 = 0.1\cos(-2x)\sin(3\pi t)\) As we are looking for the positions of nodes, we are concerned with the spatial term \(\cos(-2x)\) which becomes zero, and not the time-dependent term \(\sin(3\pi t)\). So, we have: \(\cos(-2x) = 0\)

Calculate the distance between consecutive nodes

The cosine term is zero when the argument is equal to odd multiples of \(\frac{\pi}{2}\), i.e., \(-2x = (2n + 1) \frac{\pi}{2}\) where \(n = 0, 1, 2, ...\) Solving for x, we have: \(x = -(2n + 1) \frac{\pi}{4}\) The distance between consecutive nodes is the difference between the positions of two consecutive nodes (when n and n+1): \[\begin{aligned} d &= x(n+1) - x(n) \\ &= - (2(n+1) + 1)\frac{\pi}{4} - (- (2n + 1)\frac{\pi}{4}) \\ &= - \frac{2\pi}{4} \\ &= - \frac{\pi}{2} \end{aligned}\] Since distance cannot be negative, we take the absolute value, so the distance between two consecutive nodes is \(\frac{\pi}{2}\) meters. Therefore, the correct answer is (A) \(\frac{\pi}{2}\).

Key concepts

Node Separation

Standing waves have unique points called nodes, where the wave has zero amplitude. These nodes are crucial as they represent locations where there is no movement. In a standing wave, nodes are evenly spaced due to the periodic nature of the wave.

To find the distance between consecutive nodes, we focus on the spatial part of the wave equation. For our example, the standing wave equation is simplified to:

• \( y = 0.1\cos(-2x)\sin(3\pi t) \).

The node positions are determined where the cosine component equals zero.

• Solving \( \cos(-2x) = 0 \) gives us the positions of the nodes:

• \( -2x = (2n + 1)\frac{\pi}{2} \), where \( n \) is an integer.

By solving for \( x \), we find the spatial positions. The separation between nodes is calculated by taking the difference between two successive node positions. For instance, with two nodes at

• \( x_n = -\frac{(2n+1)\pi}{4} \) and \( x_{n+1} = -\frac{(2(n+1)+1)\pi}{4} \).

Hence, the distance \( d \) is:

• \( d = \frac{\pi}{2} \)

This shows that in our given wave, nodes are separated by \( \frac{\pi}{2} \) meters.

Wave Superposition

Superposition is a fundamental concept explaining how standing waves form. It is the process where two or more waves overlap, resulting in a new wave pattern. With standing waves, constructive and destructive interference occur at various points along the wave.

In our scenario, the standing wave is formed when two waves intersect. Each of these waves has the same amplitude but moves in opposite directions, leading to a unique combined wave. For our given equations:

• \( y_1 = 0.05\sin(3\pi t - 2x) \)
• \( y_2 = 0.05\sin(3\pi t + 2x) \)

When these waves superimpose, their equations add to form a standing wave:

• \( y = y_1 + y_2 \)

The result shows alternating nodes, where the amplitude is zero, and antinodes, where it is maximum. This standing wave pattern remains stationary, creating a phenomenon where energy is stored rather than transmitted.

Trigonometric Identities

Trigonometric identities are essential to simplify wave equations, especially in the context of waves and standing waves. These identities help in combining and breaking down trigonometric expressions that have various uses in physics and engineering.

In the case of standing waves, when combining the two waves, the trigonometric identity used is:

• \( \sin(a) + \sin(b) = 2\cos\left(\frac{a-b}{2}\right)\sin\left(\frac{a+b}{2}\right) \)

For our exercise wave equations:

• \( a = 3\pi t - 2x \)
• \( b = 3\pi t + 2x \)

This identity aids in transforming the sum of two sinusoidal waves into the form:

• \( y = 0.1\cos(-2x)\sin(3\pi t) \)

The resulting expression shows how standing waves are analytically described, detailing spatial and temporal components separately. Understanding this identity is crucial for solving problems involving standing waves and simplifies the process of analyzing wave interactions.