Question

A sinusoidal wave travelling in the positive x-direction has an amplitude of $10\text{cm}$, wavelength $30\text{cm}$ and frequency $6\text{Hz}$. The vertical displacement of the medium at $t=0$ and $x=0$ is also $10\text{cm}$.

• A. Wave number is $\frac{\pi }{15}{\text{cm}}^{-1}$
• B. $\text{Time period of wave is}\frac{1}{6}\text{s}$
• C. $\text{Value of phase constant is}\frac{\pi }{2}$
• D. $\text{Speed of wave is}1.8\text{m/s}$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A Wave number is $\frac{\pi }{15}{\text{cm}}^{-1}$
B $\text{Time period of wave is}\frac{1}{6}\text{s}$
C $\text{Value of phase constant is}\frac{\pi }{2}$
D $\text{Speed of wave is}1.8\text{m/s}$

Wave number $\left(K\right)=\frac{2\pi }{\lambda }=\frac{2\pi }{30}=\frac{\pi }{15}\text{rad/cm}$
Time period $\left(T\right)=\frac{1}{f}=\frac{1}{6}\text{s}$
Angular frequency $w=2\pi f=2\pi \times 6=12\pi \text{rad/s}$
Speed of wave $=f\times \lambda =6\times 0.3=1.8\text{m/s}$
Given that, amplitude $A=10\text{cm}$
and also $y=10\text{cm}$ at $x=0\&t=0$
Then, using $y=A\sin (\omega t-kx+\varphi )$
$10=10\sin \left(\varphi \right)$
or $\varphi =\frac{\pi }{2}$ is the phase constant.
Thus, options (a), (b), (c) and (d) are correct.