The equation of wave is given by Y = 10^2 2 ( (60t - 0.5x + 4)) where x and Y are in m and t in s. The speed of the wave is km h^-1.

The equation of wave is given by Y = 10^2 2 ( (60t - 0.5x + 4)) w… — Solution | JEE Physics

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The equation of wave is given by $Y = 10^2 \sin 2\pi \left( (60t - 0.5x + \frac{\pi}{4}) \right)$ where $x$ and $Y$ are in $\text{m}$ and $t$ in $\text{s}$ . The speed of the wave is _____ $\text{km h}^{-1}$ .

A
$1152 \, \text{km/h}$
B
$1000 \, \text{km/h}$
C
$800 \, \text{km/h}$
D
$1200 \, \text{km/h}$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The wave equation is

Y = 10^2 \sin 2\pi \left( 60t - 0.5x + \frac{\pi}{4} \right)

Find: The speed of the wave in $\text{km/h}$ .

Compare the given equation with the standard travelling wave form and identify the coefficients of $t$ and $x$ inside the phase term.

From the solution working:

\omega = 60, \quad k = 0.5

The wave speed is

v = \frac{\omega}{k}

Substituting the values,

v = \frac{60}{0.5} = 120 \, \text{m/s}

Now convert $\text{m/s}$ to $\text{km/h}$ :

v = 120 \times \frac{18}{5} = 1152 \, \text{km/h}

Therefore, the speed of the wave is $1152 \, \text{km/h}$ . The correct option is A.

Common mistakes

Using the coefficient of $x$ or $t$ directly as the speed is incorrect because wave speed is found from the ratio $\frac{\omega}{k}$ . First identify the time and space coefficients, then divide.
Forgetting to convert $120 \, \text{m/s}$ into $\text{km/h}$ gives a numerically incomplete answer. Multiply by $\frac{18}{5}$ to convert from $\text{m/s}$ to $\text{km/h}$ .
Confusing amplitude $10^2$ with wave speed is wrong because amplitude only gives the maximum displacement. The speed depends on the phase terms involving $t$ and $x$ .

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Answer

Step-by-step solution

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