Question

A transverse harmonic wave on a string is described by $y\left(x, t\right) = 3\,sin \left(36t + 0.018x+\frac{\pi}{4}\right)$ where $x$ and $y$ are in $cm$ and $t$ is in $s$. Which of the following statements is incorrect?

• A. The wave is travelling in negative $x$-direction
• B. The amplitude of the wave is $3 \,cm$
• C. The speed of the wave is $20 \,ms^{-1}$
• D. The frequency of the wave is $\frac{9}{\pi} Hz$

Answer 1

Answer: (D)

The given transverse harmonic wave equation is
$y = 3\,sin \left(36t + 0.018x+\frac{\pi}{4}\right)\quad\ldots\left(i\right)$
As there is positive sign between $t$ and $x$ terms therefore the given wave is travelling in the negative $x$ direction. The standard transverse harmonic wave equation is
$y = asin\left(\omega t+kx+\phi\right)\quad\ldots\left(ii\right)$
Comparing $\left(i\right)$ and $\left(ii\right)$, we get
$a = 3\, cm$, $\omega = 36 \,rad \,s^{-1}, k = 0.018\, rad \,cm^{-1}$
$\therefore $ Amplitude of wave, $a = 3 \,cm$
Frequency of the wave, $\upsilon = \frac{\omega}{2\pi} = \frac{36}{2\pi} = \frac{18}{\pi}Hz$
Velocity of the wave
$v = \frac{\omega}{k} = \frac{36\,rad\,s^{-1}}{0.018\,rad\,cm^{-1}}$
$= 2000\,cm\,s^{-1} = 20\,ms^{-1}$