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 ?

The standard form of a progressive sinusoidal wave travelling in the +x-direction is
$$y = A \cos \bigl(kx - \omega t + \phi\bigr)$$ or $$y = A \sin \bigl(kx - \omega t + \phi\bigr)$$

Here
  • $$A$$ = amplitude
  • $$k = \dfrac{2\pi}{\lambda}$$ = wave number (rad cm$$^{-1}$$ when $$\lambda$$ is in cm)
  • $$\omega = 2\pi f$$ = angular frequency (rad s$$^{-1}$$)
  • Wave speed $$v = \dfrac{\omega}{k} = f\lambda$$

Given data
  • Wavelength $$\lambda = 7.5\,\text{cm}$$
  • In $$\Delta t = 0.3\,\text{s}$$ the disturbance travels $$\Delta x = 1.2\,\text{cm}$$
  • Amplitude $$A = 2\,\text{cm}$$

Step 1 - Calculate the wave speed.
$$v = \frac{\Delta x}{\Delta t} = \frac{1.2\,\text{cm}}{0.3\,\text{s}} = 4\,\text{cm s}^{-1}$$

Step 2 - Calculate the wave number $$k$$.
$$k = \frac{2\pi}{\lambda} = \frac{2\pi}{7.5} \,\text{rad cm}^{-1} \approx 0.84\,\text{rad cm}^{-1}$$

Step 3 - Calculate the angular frequency $$\omega$$ using $$\omega = vk$$.
$$\omega = (4\,\text{cm s}^{-1})(0.84\,\text{rad cm}^{-1}) \approx 3.35\,\text{rad s}^{-1}$$

Step 4 - Insert $$A$$, $$k$$, and $$\omega$$ into the standard form.
Possible cosine form (with zero phase constant):
$$y = 2 \cos\bigl(0.84\,x - 3.35\,t\bigr)\;\text{cm}$$

Step 5 - Check the given initial condition.
At $$x = 0$$ and $$t = 0$$ the crest is present, so $$y(0,0) = +2\,\text{cm}$$ (maximum upward displacement).
For the expression above:
$$y(0,0) = 2\cos(0) = 2\,\text{cm}$$✅ (condition satisfied)

Step 6 - Match with the options.
Option A: $$y = 2\cos(0.83x - 3.35t)\;\text{cm}$$ - wave number, frequency and amplitude all match the calculated values.
Option B: sine form gives zero displacement at $$t = 0$$, so it violates the initial condition.
Option C: $$k$$ and $$\omega$$ are interchanged, giving an incorrect speed $$v = \omega/k \approx 0.25\,\text{cm s}^{-1}$$.
Option D: both $$k$$ and $$\omega$$ are far from the required values.

Therefore, the correct representation of the wave is given by Option A.