Question

Two travelling waves produces a standing wave represented by equation. \(y = 1.0 \text{ mm} \cos(1.57 \text{ cm}^{-1})x \sin(78.5 \text{ s}^{-1})t\). The node closest to the origin in the region \(x>0\) will be at \(x = \)_________ cm.

Hint:

Always identify if the spatial part is \(\sin(kx)\) or \(\cos(kx)\). If it's \(\sin(kx)\), the origin itself (\(x=0\)) is a node. If it's \(\cos(kx)\), \(x=0\) is an antinode.

Answer 1

Correct Answer: 1

Step 1: Understanding the Concept:
Nodes in a standing wave are positions where the amplitude is permanently zero. In the given equation, the spatial part \(\cos(kx)\) determines the nodes.
Step 2: Key Formula or Approach: The equation is of the form \(y = A \cos(kx) \sin(\omega t)\). Nodes occur when the amplitude part is zero: \(\cos(kx) = 0\).
Step 3: Detailed Explanation: Given: \(k = 1.57 \text{ cm}^{-1}\).
We know that \(\cos \theta = 0\) when \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots\)
For the node closest to the origin in the region \(x>0\):
\[ kx = \frac{\pi}{2} \]
\[ 1.57 \times x = \frac{3.14159}{2} \]
\[ 1.57 \times x \approx 1.5708 \]
\[ x \approx \frac{1.5708}{1.57} \approx 1 \text{ cm} \]
Step 4: Final Answer: The node closest to the origin is at \(x = 1\) cm.