Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

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 wave has amplitude \(A,\) period \(T,\) and wavelength \(\lambda .\) The models for two such waves are $$\begin{aligned} &y_{1}=A \cos 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right) \text { and } y_{2}=A \cos 2 \pi\left(\frac{t}{T}+\frac{x}{\lambda}\right)\\\ &\text { Show that } y_{1}+y_{2}=2 A \cos \frac{2 \pi t}{T} \cos \frac{2 \pi x}{\lambda} \end{aligned}$$

Step-by-Step Solution

Answer

By understanding the given equations, adding them together, and applying the cosine of sum and difference identities, it is shown that \(y_{1}+y_{2}=2 A \cos \frac{2 \pi t}{T} \cos \frac{2 \pi x}{\lambda}\), proving that the sum of two waves results in a standing wave equation.

Step 1: Understanding the equations

The models of the two waves are given by \(y_{1}=A \cos 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right)\) and \(y_{2}=A \cos 2 \pi\left(\frac{t}{T}+\frac{x}{\lambda}\right)\). The goal is to show that their sum equals the standing wave equation.

Step 2: Adding the equations

Add the two wave models together to get \(y_{1}+y_{2}=A \cos 2 \pi\left(\frac{t}{T}-\frac{x}{\lambda}\right) + A \cos 2 \pi\left(\frac{t}{T}+\frac{x}{\lambda}\right)\). This equation is the basis for further simplifications.

Step 3: Using the cosine of sum and difference identities

Now, use the trigonometric identities for the cosine of a sum and the cosine of a difference, i.e. \(\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B\). When applying those identities to the equation obtained in step 2, the sine parts cancel out because we obtain \(\sin\) with the same argument in both terms but with opposite signs. Ultimately the equation simplifies to \(y_{1}+y_{2}=2 A \cos \frac{2 \pi t}{T} \cos \frac{2 \pi x}{\lambda}\) which is what was asked to prove.

Key Concepts

Cosine of Sum and Difference Identities

The cosine of sum and difference identities are key mathematical tools used in trigonometry to simplify the expression of wave equations. These identities state that:

• \(\cos(A + B) = \cos A \cos B - \sin A \sin B\)
• \(\cos(A - B) = \cos A \cos B + \sin A \sin B\)

When dealing with waves moving in opposite directions, like in our exercise, these identities are applied to break down complex expressions. By adding the two wave functions from the problem, we apply the cosine difference and sum to resolve the expression.
In this case, the terms involving sine from the two expressions cancel each other out because they become opposite. Thus, we are left with a simpler expression using purely cosine terms. This manipulation is crucial for demonstrating how two traveling waves create a standing wave pattern.

Wave Superposition

Wave superposition describes the phenomenon where two or more waves overlap and combine to form a new wave pattern. This is where the equations of wave functions come into play:

• \( y_{1} = A \cos \left(2 \pi \left(\frac{t}{T} - \frac{x}{\lambda}\right) \right) \)
• \( y_{2} = A \cos \left(2 \pi \left(\frac{t}{T} + \frac{x}{\lambda}\right) \right) \)

In our exercise, the waves are traveling in opposite directions. When they meet, their displacements add up, according to the principle of superposition. By combining these wave functions, each with the same amplitude but traveling differently, they interfere to create a distinct pattern.
This leads to solutions like standing waves, illustrated by adding these two functions algebraically. Such interference can either reinforce the wave amplitude at certain points (constructive interference) or cancel them out (destructive interference).

Trigonometric Identities

Trigonometric identities are formulas that express relationships between trigonometric functions. They are instrumental for solving equations involving cyclical patterns like waves. In many exercises involving standing waves or oscillations, these identities help simplify and solve mathematical expressions.

• Aside from cosine sum and difference identities, others include Pythagorean identities, angle conversion formulas, and double angle identities.
• In our problem, we utilize trigonometric identities to decompose wave equations into forms that are easier to handle.

These transformations allow us to extract meaningful results from complex equations, such as finding a generalized standing wave formula. By mastering these identities, solving wave interference or superposition problems becomes much more straightforward.