Question

Two progressive waves \(Y_1 = \sin 2\pi \left(\frac{t}{0.4} - \frac{x}{4}\right)\) and \(Y_2 = \sin 2\pi \left(\frac{t}{0.4} + \frac{x}{4}\right)\) superpose to form a standing wave (\(x\) and \(y\) in SI units). Find the amplitude of the particle at \(x = 0.5\) m.

Hint:

In standing waves, the coefficient of the time-dependent term (\(\sin \omega t\) or \(\cos \omega t\)) is the amplitude at that point. If the result is negative, take the absolute value as amplitude is always positive.

Answer 1

Step 1: Understanding the Concept:
When two identical waves travel in opposite directions, they interfere to form a standing wave. The resultant displacement \(Y = Y_1 + Y_2\) contains a spatial part that determines the amplitude at any position \(x\).

Step 2: Key Formula or Approach:
1. Using the identity \(\sin(A-B) + \sin(A+B) = 2 \sin A \cos B\).
2. The resultant amplitude is \(A_{res} = 2A \cos(kx)\) where \(k\) is the wave number.

Step 3: Detailed Explanation:
Given \(Y_1 = \sin 2\pi \left(\frac{t}{0.4} - \frac{x}{4}\right)\) and \(Y_2 = \sin 2\pi \left(\frac{t}{0.4} + \frac{x}{4}\right)\).
Summing them:
\[ Y = 2 \sin \left( 2\pi \frac{t}{0.4} \right) \cos \left( 2\pi \frac{x}{4} \right) \] The amplitude of the stationary wave at position \(x\) is:
\[ A(x) = \left| 2 \cos \left( \frac{2\pi x}{4} \right) \right| = \left| 2 \cos \left( \frac{\pi x}{2} \right) \right| \] Substitute \(x = 0.5\) m:
\[ A(0.5) = 2 \cos \left( \frac{\pi \cdot 0.5}{2} \right) = 2 \cos \left( \frac{\pi}{4} \right) \] Since \(\cos(\pi/4) = \frac{1}{\sqrt{2}}\):
\[ A(0.5) = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2} \]
Step 4: Final Answer:
The amplitude of the particle at \(x = 0.5\) m is \(\sqrt{2}\) units.