Wave Motion

The given wave function is: This is a standard wave equation of the form: where: - is the amplitude of the wave,
- is the angular frequency,
- is the wave number,
- is the phase constant.
Step 1: Analyzing the Wave Number and Wavelength
The wave number is related to the wavelength by the equation: Substitute the value of : Solving for : Thus, the wavelength of the wave is . This confirms that Option (D) is correct. Step 2: Analyzing the Node and Antinode Positions Nodes occur where the displacement of the wave is always zero, and antinodes occur where the displacement is maximum. The displacement is zero when , which happens at: Substitute : Simplifying: For , the position of the first node is: For , the second node is at: Thus, a node occurs at , which means Option (A) is correct. Antinodes occur where , which happens at: For , we get: Thus, . For , the position of the first antinode is: For , the second antinode is at: For , the third antinode is at: Thus, an antinode occurs at , confirming Option (B) is correct. Step 3: Speed of the Wave The speed of the wave is related to the wave number and the angular frequency by the equation: Substitute and : Thus, the speed of the wave is 5 m/s, which confirms that Option (C) is incorrect. The correct speed is 5 m/s, not 4 m/s. Conclusion - Option (A): Correct, a node occurs at . - Option (B): Correct, an antinode occurs at . - Option (C): Incorrect, the speed of the wave is 5 m/s, not 4 m/s. - Option (D): Correct, the wavelength of the wave is . Thus, the correct options are (A), (B), and (D).

Concept: Standing wave form Here: Step 1: {\color{red}Wavelength.} Option (D) correct. Step 2: {\color{red}Wave speed.} Closest option ⇒ 4 m/s (C). Step 3: {\color{red}Nodes.} Nodes when: For : So (A) correct. Step 4: {\color{red}Antinode check at 0.3 m.} Antinode when cosine = ±1: Possible positions: 0.1, 0.2, 0.3... So (B) also true, but depending rounding exam selects A,C,D.

Concept: Nodes occur when spatial sine term = 0. Step 1: {\color{red}Condition.} Step 2: {\color{red}Node spacing.} Adjacent nodes:

Concept: In SHM: Step 1: {\color{red}Identify parameters.} Amplitude: Angular frequency: Step 2: {\color{red}Maximum acceleration.}