Question

The equation of a stationary wave is:
\[y = 2a \sin\left(\frac{2\pi nt}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right)\]Which of the following is NOT correct:

• A. The dimensions of \( nt \) is \([L]\)
• B. The dimensions of \( n \) is \([LT^{-1}]\)
• C. The dimensions of \( n / \lambda \) is \([T]\)
• D. The dimensions of \( x \) is \([L]\)

Answer 1

The Correct Option is C

Comparing the given equation with the standard equation for a standing wave:
\[ \frac{2\pi nt}{\lambda} = \omega t, \quad \frac{2\pi x}{\lambda} = kx \]
where \( \omega \) is the angular frequency and \( k \) is the wave number.

Analyzing the dimensions:
\[ \left[\frac{n}{\lambda}\right] = [\omega] = [T^{-1}] \] 

For the other terms:
\[ [nt] = [\lambda] = [L], \quad [n] = [\lambda \omega] = [LT^{-1}], \quad [x] = [\lambda] = [L] \] 

Conclusion:
Hence, the dimensions of \( n / \lambda \) are [T].

Answer 2

Step 1: Given stationary wave equation.
The equation of a stationary wave is: \[ y = 2a \sin\left(\frac{2\pi n t}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right) \] where \( a \) is the amplitude, \( n \) is frequency, \( \lambda \) is wavelength, \( x \) is position, and \( t \) is time.

Step 2: Understand each term.
The general equation of a stationary wave is: \[ y = 2a \sin(\omega t) \cos(kx) \] where \( \omega \) is the angular frequency and \( k \) is the wave number.
Comparing both forms: \[ \omega = \frac{2\pi n}{\lambda} \quad \text{and} \quad k = \frac{2\pi}{\lambda}. \] But for a wave, we know that: \[ \omega = 2\pi f \] where \( f \) is the frequency (dimension \( [T^{-1}] \)).

Step 3: Dimensional analysis of \( \frac{n}{\lambda} \).
If \( n \) represents frequency, then \( [n] = [T^{-1}] \).
Also, wavelength \( [\lambda] = [L] \).
So: \[ \left[ \frac{n}{\lambda} \right] = \frac{[T^{-1}]}{[L]} = [L^{-1} T^{-1}]. \] This means the dimension of \( \frac{n}{\lambda} \) is \( [L^{-1} T^{-1}] \), not \( [T] \).

Step 4: Identify the incorrect statement.
Among the possible statements, the one saying that the dimensions of \( \frac{n}{\lambda} \) is \( [T] \) is incorrect, because its actual dimension is \( [L^{-1}T^{-1}] \).

Final Answer:
\[ \boxed{\text{The dimensions of } \frac{n}{\lambda} \text{ is } [T] \text{ is NOT correct.}} \]