Deriving the Wave Equation Parameters

The general form of a sinusoidal wave travelling along the x-axis is:

$y(x, t) = A \cos(kx - \omega t + \phi)$

We need to find the values for Amplitude ($A$), Wavenumber ($k$), Angular Frequency ($\omega$), and Phase Constant ($\phi$) based on the given information.

Calculating Wave Amplitude ($A$) and Wavenumber ($k$)

The maximum displacement is given as the amplitude:

$A = 2$ cm

The wavelength is given as $\lambda = 7.5$ cm. The wavenumber $k$ is calculated using the formula:

$k = \frac{2\pi}{\lambda}$

Substituting the value of $\lambda$:

$k = \frac{2\pi}{7.5 \text{ cm}} = \frac{4\pi}{15} \text{ rad/cm} \approx 0.8377$ rad/cm

Comparing with the options, we see that Options 3 and 4 have $k \approx 0.83$. Options 1 ($k = 0.13$) and 2 ($k = 3.35$) are unlikely.

Calculating Wave Speed ($v$) and Angular Frequency ($\omega$)

The wave travels a distance of 1.2 cm in 0.3 sec. The wave speed $v$ is:

$v = \frac{\text{Distance}}{\text{Time}} = \frac{1.2 \text{ cm}}{0.3 \text{ sec}} = 4$ cm/sec

The angular frequency $\omega$ is related to the wave speed $v$ and wavenumber $k$ by $\omega = vk$:

$\omega = v \times k = 4 \text{ cm/sec} \times \frac{4\pi}{15} \text{ rad/cm} = \frac{16\pi}{15} \text{ rad/sec} \approx 3.351$ rad/sec

Comparing with the remaining Options 3 and 4:

• Option 3 uses $\omega \approx 3.5$.
• Option 4 uses $\omega \approx 3.35$.

Option 4 aligns closely with our calculated value for $\omega$.

Determining the Phase Constant ($\phi$)

We are given that the wave has a crest (maximum displacement, $y=A$) at $x=0$ at $t=0$. Plugging this condition into the general cosine wave equation:

$y(0, 0) = A \cos(k(0) - \omega(0) + \phi) = A$

$A \cos(\phi) = A \implies \cos(\phi) = 1$

The simplest solution for this condition is $\phi = 0$. Therefore, the wave equation should be of the form $y = A \cos(kx - \omega t)$.

Option 3 uses a sine function. For $y = 2\sin(0.83x - 3.5t)$, at $x=0, t=0$, $y = 2\sin(0) = 0$, which is not a crest. Thus, Option 3 is incorrect.

Final Wave Equation Selection

Based on the calculations, the wave equation should be of the form $y = A \cos(kx - \omega t)$ with:

• $A = 2$ cm
• $k \approx 0.83$ rad/cm
• $\omega \approx 3.35$ rad/sec
• $\phi = 0$

Option 4, $y = 2\cos(0.83x - 3.35t)$ cm, perfectly matches these derived parameters.

Conclusion: Option 4 correctly represents the sinusoidal wave.