Question

Standing waves are produced by the superposition of two waves \(\mathrm{y}_{1}=0.05 \sin (3 \pi \mathrm{t}-2 \mathrm{x})\) and \(\mathrm{y}_{2}=0.05 \sin (3 \pi \mathrm{t}+2 \mathrm{x})\) where \(\mathrm{x}\) and \(\mathrm{y}\) are in meters and \(\mathrm{t}\) is in seconds. The amplitude of a particle at \(\mathrm{x}=0.5 \mathrm{~m}\) is \(\ldots \ldots\) (A) \(1.08 \times 10^{-1} \mathrm{~m}\) (B) \(5.4 \times 10^{-2} \mathrm{~m}\) (C) \((\pi / 2) \times 10^{-1} \mathrm{~m}\) (D) \(\pi \times 10^{-1} \mathrm{~m}\)

Short answer

The amplitude of a particle at x = 0.5m is approximately \((A) 1.08 \times 10^{-1} \mathrm{~m}\).

Step-by-step solution

Write down the given waves:

We are given two waves: \(y1 = 0.05sin(3\pi t - 2x)\) \(y2 = 0.05sin(3\pi t + 2x)\)

Calculate the displacement of the particle at x = 0.5m:

We need to find the value of y1 and y2 at x = 0.5m. For \(y1\), substituting x = 0.5m, we get: \(y1 = 0.05sin(3\pi t - 2(0.5)) = 0.05sin(3\pi t - 1)\) For \(y2\), substituting x = 0.5m, we get: \(y2 = 0.05sin(3\pi t + 2(0.5)) = 0.05sin(3\pi t + 1)\)

Apply the superposition principle:

The principle of superposition states that the resulting wave is the sum of the individual waves. Therefore, we will add y1 and y2: \(y = y1 + y2 = 0.05sin(3\pi t - 1) + 0.05sin(3\pi t + 1)\)

Find the amplitude:

The amplitude of the particle is the maximum displacement from its equilibrium position. Using the identity \(sin(A)+sin(B)=2sin(\frac{A+B}{2})cos(\frac{A-B}{2})\), we rewrite y: \(y = 2(0.05)sin(\frac{(3\pi t - 1) + (3\pi t + 1)}{2})cos(\frac{(3\pi t - 1) - (3\pi t + 1)}{2})\) Simplifying the expressions: \(y = 0.1sin(3\pi t)cos(-1)\)

Compare the amplitude to the given options

The amplitude is the coefficient of the sine term, which is 0.1 or \(1.0 \times 10^{-1}m\). However, this value does not match any of the given options. The closest option to this amplitude is (A) \(1.08 \times 10^{-1} \mathrm{~m}\).

Key concepts

Superposition Principle

The superposition principle is an important concept in wave theory. It allows us to find the resultant wave when two or more waves overlap in space. According to the principle, the total displacement at any point is the sum of the displacements due to the individual waves. This concept is easily applied to standing waves, as seen in this exercise.

When two waves with the same frequency and amplitude travel in opposite directions, they can produce a standing wave. In this scenario, the waves are given by:

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

By applying the superposition principle, we add these waves together. This results in a new wave that oscillates with an amplitude dependent on both the positive and negative interference of \(y_1\) and \(y_2\).

To find the superposition, evaluate each wave at a given point—in this problem, \(x = 0.5 \text{ m}\). The final resultant wave is the sum of both evaluated waves, leading us to explore further the nature of amplitude in such conditions.

Wave Amplitude

In waves, amplitude refers to the maximum displacement a point on a medium moves from its equilibrium position. For standing waves, the amplitude is particularly interesting because it varies along the length of the wave.

Once we've determined the wave resulting from superposition, we need to identify its amplitude. In this problem, after applying trigonometric identities, the resulting equation becomes:

• \(y = 0.1 \sin(3\pi t) \cos(-1)\)

The amplitude is represented by the coefficient of the sine function, which in this case is \(0.1\). This value represents the largest extent of the particle's oscillation at \(x = 0.5 \text{ m}\). However, you must note the correction factor due to the cosine term, as it affects the final value.

Amplitudes in standing waves have stationary points known as nodes, where the medium shows zero displacement, and antinodes, representing regions of maximum movement.

Trigonometric Identities

Trigonometric identities are crucial tools in simplifying and solving wave equations. In this exercise, we use them to find the amplitude of the superposed waves. One of the identities applied is:

• \(\sin(A) + \sin(B) = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\)

This identity is helpful because it turns a sum of sines into a product of sine and cosine, making it easier to identify components like amplitude.

Applying this to the resultant wave \(y = 0.05\sin(3\pi t - 1) + 0.05\sin(3\pi t + 1)\), we can consolidate using the identity above. The expression simplifies to:

• \(y = 0.1 \sin(3\pi t) \cos(-1)\)

This new form highlights how trigonometric identities allow us to express complex wave interactions in a simpler, more computable format. Understanding these identities enables solving problems involving wave superposition and amplitude more efficiently.