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 Question:
The superposition of two identical waves traveling in opposite directions creates a standing wave.
Step 2: Key Formula or Approach:
Resultant \( Y = Y_1 + Y_2 = 2A \cos(kx) \sin(\omega t) \).
The amplitude is \( R(x) = 2A \cos(kx) \).
Step 3: Detailed Explanation:
Here \( A = 1 \) and the phase part is \( 2\pi(x/4) = \pi x/2 \).
Thus, \( k = \pi/2 \).
Amplitude \( R = 2 \cos(\frac{\pi x}{2}) \).
At \( x = 0.5 \):
\[ R = 2 \cos\left(\frac{\pi \cdot 0.5}{2}\right) = 2 \cos\left(\frac{\pi}{4}\right) = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2} \text{ m} \]
Step 4: Final Answer:
The amplitude at \( x = 0.5 \) m is \( \sqrt{2} \) m.