Question

Does the superposition \(\psi(x, t)=A \sin (k x-\omega t)+\) \(2 A \sin (k x+\omega t)\) generate a standing wave? Answer this question by using trigonometric identities to combine the two terms.

Short answer

The given superposition is \(\psi(x, t) = A\sin(kx - \omega t) + 2A \sin(kx + \omega t)\). Using the sum-to-product trigonometric identity, we rewrite it as \(\psi(x, t) = 2\sin{(kx)}\cos{(-\omega t)}\). Since this expression represents a product of a spatial component and a temporal component, we can conclude that the superposition generates a standing wave.

Step-by-step solution

1. Rewrite given superposition in terms of sine functions

Given superposition: \[\psi(x, t) = A\sin(kx - \omega t) + 2A \sin(kx + \omega t)\]

2. Apply the sum-to-product trigonometric identity

The sum-to-product identity states that for any angles α and β: \[\sin(\alpha) + \sin(\beta) = 2\sin{\frac{(\alpha + \beta)}{2}}\cos{\frac{(\alpha - \beta)}{2}}\] Here, we can consider: \[\alpha = kx - \omega t\] and \[\beta = kx + \omega t\] Applying the sum-to-product identity to the given superposition: \[\psi(x, t) = 2\left[\sin{\frac{(kx - \omega t + kx + \omega t)}{2}}\cos{\frac{(kx - \omega t - (kx + \omega t))}{2}}\right]\] This simplifies to: \[\psi(x, t) = 2\left[\sin{(\frac{2kx}{2})}\cos{(\frac{-2\omega t}{2})}\right]\]

3. Simplify the expression and check for standing wave form

The simplified form of ψ(x, t) is: \[\psi(x, t) = 2\sin{(kx)}\cos{(-\omega t)}\] Since the expression is in the form of a product of a spatial component (a function of x) and a temporal component (a function of t), we can conclude that the superposition generates a standing wave.

Key concepts

Superposition Principle

The Superposition Principle is a fundamental concept in wave theory. It explains how two or more waves can combine in space to form a new wave pattern. According to this principle, when two waves meet, their displacements add together at each point they coincide. The beauty of superposition is that each wave maintains its shape and speed; they don't alter each other permanently. Instead, they create an interference pattern that can be constructive or destructive, depending on their phase relationship.In the exercise, we see the superposition of two sine waves with different amplitudes: \( \psi(x, t) = A\sin(kx - \omega t) + 2A \sin(kx + \omega t) \). This equation demonstrates the principle by adding two wave functions. Such composition can lead to various phenomena, including standing waves. The solution applies the superposition principle to analyze whether these waves form a standing wave, focusing on the sum-to-product identities to combine the terms.

Trigonometric Identities

Trigonometric identities are mathematical tools that help simplify expressions involving trigonometric functions like sine and cosine. These identities enable us to transform and manipulate equations into more workable forms. In wave analysis, these identities become particularly useful in simplifying complex wave superpositions to determine resulting waveforms.In this exercise, the sum-to-product identity is vital. It allows combining the two sine functions into a form that can be used to verify the standing wave nature of the superposition:

• Sum-to-product: \( \sin(\alpha) + \sin(\beta) = 2\sin\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right) \)

Here, with \(\alpha = kx - \omega t\) and \(\beta = kx + \omega t\), the identity transforms the addition of two sine waves into a single term involving both sine and cosine functions. This transformation simplifies the expression, making it clearer that the original wave superposition indeed leads to a standing wave pattern.

Wave Functions

Wave functions are mathematical descriptions of waves. They represent oscillating quantities, such as displacement in space and time, typically using trigonometric functions like sine and cosine. The general wave function can be written as \( A\sin(kx - \omega t) \), where:

• \(A\) is the amplitude, the maximum displacement from the rest position.
• \(k\) is the wave number, related to the wavelength.
• \(\omega\) is the angular frequency, related to the period of the wave.
• \(t\) represents time, while \(x\) represents position.

In a standing wave, represented by a function like \( \psi(x, t) = 2\sin(kx)\cos(-\omega t) \), the wave appears to be stationary, with nodes and antinodes. This is due to the interplay between spatial and temporal components in the wave function. Standing waves form from the superposition of two waves traveling in opposite directions, sharing similar frequency and amplitude, creating fixed points of zero displacement (nodes) and maximum displacement (antinodes). The exercise concludes with the realization that the simplified wave function exhibits characteristics of standing waves, as evidenced by the separation of time-dependent and space-dependent parts.