A sinusoidal wave of wavelength 7.5 cm travels a distance of 1.2 cm along the x-direction in 0.3 sec. The crest P is at $ x = 0 $ at $ t = 0 $ sec and maximum displacement of the wave is 2 cm. Which equation correctly represents this wave?
Show Hint
- \( y = 2 \cos(0.83x - 3.35t) \, \text{cm} \)
- \( y = 2 \sin(0.83x - 3.5t) \, \text{cm} \)
- \( y = 2 \cos(3.35x - 0.83t) \, \text{cm} \)
- \( y = 2 \cos(0.13x - 0.5t) \, \text{cm} \)
The Correct Option is A
Approach Solution - 1
Approach Solution -2
We are given a sinusoidal wave traveling along +x with wavelength \( \lambda = 7.5\,\text{cm} \), it covers \( 1.2\,\text{cm} \) in \( 0.3\,\text{s} \), the crest is at \( x=0 \) at \( t=0 \), and amplitude \( A=2\,\text{cm} \). We need the correct wave equation.
Concept Used:
A right-moving sinusoidal wave can be written as \( y(x,t)=A\cos(kx-\omega t+\phi) \). Here \( k=\dfrac{2\pi}{\lambda} \), \( \omega=kv \), and the initial phase \( \phi \) is set by the initial condition. If the displacement is maximum (crest) at \( x=0,t=0 \), then \( \cos(\phi)=1 \Rightarrow \phi=0 \).
Step-by-Step Solution:
Step 1: Compute the wave speed from the given travel distance and time:
\[ v=\frac{1.2\,\text{cm}}{0.3\,\text{s}}=4\,\text{cm s}^{-1}. \]
Step 2: Compute the wave number \(k\) from the wavelength:
\[ k=\frac{2\pi}{\lambda}=\frac{2\pi}{7.5\,\text{cm}}\approx 0.8378\,\text{cm}^{-1}\ (\text{rounds to }0.83\,\text{cm}^{-1}). \]
Step 3: Compute the angular frequency \( \omega \):
\[ \omega=kv\approx(0.8378)(4)\approx 3.351\,\text{s}^{-1}\ (\text{rounds to }3.35\,\text{s}^{-1}). \]
Step 4: Use the crest-at-origin condition to fix phase:
\[ y(0,0)=A\cos(\phi)=\text{maximum}=A\ \Rightarrow\ \phi=0. \]
Step 5: Write the equation with amplitude \(A=2\,\text{cm}\):
\[ y(x,t)=2\cos(0.83\,x-3.35\,t)\ \text{cm}. \]
Final Computation & Result
The correct representation is \( y=2\cos(0.83x-3.35t)\,\text{cm} \) (first option).
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