Problem 90 - Precalculus with Limits

Question

The equation of a standing wave is obtained by adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude \( A \), period \( T \), and wavelength \( \lambda \). If the models for these waves are \( y_1 = A \cos 2\pi\left(\dfrac{t}{T} - \dfrac{x}{\lambda}\right) \) and \( y_2 = A \cos 2\pi\left(\dfrac{t}{T} + \dfrac{x}{\lambda}\right) \) show that \( y_1 + y_2 = 2A \cos \dfrac{2\pi t}{T} \cos \dfrac{2\pi x}{\lambda} \)

Step-by-Step Solution

Answer

The sum of the given wave functions can be written as \( y_1 + y_2 = 2A \cos[2\pi(\dfrac{t}{T})] \cos[2\pi(\dfrac{x}{\lambda})] \), which is the the desired equation. So the equation of a standing wave is obtained by adding the displacements of the two waves.

Step 1: Analyzing the given waves

Observe that the given waves are \( y_1 = A \cos 2\pi\left(\dfrac{t}{T} - \dfrac{x}{\lambda}\right) \) and \( y_2 = A \cos 2\pi\left(\dfrac{t}{T} + \dfrac{x}{\lambda}\right) \) which depicts two waves travelling in the opposite direction.

Step 2: Calculating the sum of the waves

Express the sum of the given waves as: \( y_1 + y_2 = A \cos 2\pi\left(\dfrac{t}{T} - \dfrac{x}{\lambda}\right) + A \cos 2\pi\left(\dfrac{t}{T} + \dfrac{x}{\lambda}\right) \) \n\nThis can be further simplified, due to the identity for the sum of cosine functions, to: \n\n\( y_1 + y_2 = 2A \cos[2\pi(\dfrac{t}{T})] \cos[2\pi(\dfrac{x}{\lambda})] \)

Step 3: Compare with the desired equation

Comparing this with the desired equation, we find that both of them are the same, hence we can conclude that the sum of the given wave functions equals to the desired function.

Key Concepts

Wave Interference

Wave interference is the phenomenon that occurs when two or more waves combine to form a new wave pattern. In the exercise, we're looking at how two waves traveling in opposite directions can create a standing wave when they interfere with each other. Standing waves are special because instead of moving through space, they appear to be stationary, oscillating in place with nodes (points that don't move) and antinodes (points that move with maximum amplitude).

When wave interference occurs, the amplitude of the resulting wave at any point is the sum of the amplitudes of the interfering waves at that point. In this exercise, the two waves have the same amplitude and frequency, but they travel in opposite directions. This leads to constructive interference at some points and destructive interference at others, creating a standing wave pattern. Therefore, understanding wave interference can help us grasp how standing waves are formed from individual traveling waves.

Trigonometric Identities

Trigonometric identities are essential tools in simplifying and solving wave equations, especially when dealing with wave interference. In the given exercise, we use the identity for the sum of cosine functions to simplify the expression of the standing wave.

The cosine sum identity can be written as:

• For two angles \( u \) and \( v \), \( \cos u + \cos v = 2\cos\left(\frac{u + v}{2}\right)\cos\left(\frac{u - v}{2}\right) \)

This identity simplifies the sum of two cosines into a product of cosines, making it easier to manage algebraically. In our case, replacing \( u \) with \( 2\pi\left( \frac{t}{T} - \frac{x}{\lambda} \right) \) and \( v \) with \( 2\pi\left( \frac{t}{T} + \frac{x}{\lambda} \right) \), allows us to transform the sum of the traveling wave equations into the standard form of a standing wave.

Mastering trigonometric identities can greatly aid in manipulating and simplifying complex wave expressions, crucial for physics and engineering students.

Wave Equation

The wave equation is a vital mathematical description of how waveforms move through space and time. It generally takes the form:

• \( y(x, t) = A \cos\left(kx - \omega t\right) \)

where \( y \) is the wave function, \( A \) is the amplitude, \( k \) is the wave number, and \( \omega \) is the angular frequency.

In standing wave scenarios, the wave equation changes slightly because you're dealing with waves moving in opposite directions. This results in a wave equation that looks like the superposition of two such equations. In our exercise, the standing wave was derived from such an equation:

• \( y_1 = A \cos \left( 2\pi \left( \frac{t}{T} - \frac{x}{\lambda} \right) \right) \)
• \( y_2 = A \cos \left( 2\pi \left( \frac{t}{T} + \frac{x}{\lambda} \right) \right) \)

Adding these yields a final equation of the form typically expected for a standing wave, showcasing the real beauty of the wave equation in capturing the behavior of wave patterns across different scenarios.

Understanding how to manipulate and use the wave equation is critical not just in solving textbook problems, but also in practical applications across various scientific fields.