Question

Two waves are passing simultaneously through a string.
The equation of the waves are given by,
$y_1=A_1\sin k(x-vt)$
and $y_2=A_2\sin k(x-vt+x_0)$
Where the wave number $k=6.28{\text{cm}}^{-1}$ and $x_0=1.50\text{cm}$ The amplitudes are $A_1=5.0\text{mm}$ and $A_2=4.0\text{mm}$. Find the phase difference between the waves and the amplitude of the resulting wave.

• A. $2\pi \text{and}1\text{mm}$
• B. $3\pi \text{and}1\text{mm}$
• C. $\pi \text{and}2\text{mm}$
• D. $\pi \text{and}0.5\text{mm}$

Answer 1

The correct option is B $3\pi \text{and}1\text{mm}$

The phase constant of the first wave is $k(x-vt)$ and of the second wave is $k(x-vt+x_0)$.

The phase difference between the waves is,
$\Delta \varphi =k{x}_0=\left(6.28{\text{cm}}^{-1}\right)\left(1.50\text{cm}\right)$
$\Rightarrow \Delta \varphi =2\pi \times 1.5=3\pi$
The wave satisfy the condition of destructive interference, as the phase difference is an odd multiple of $\pi$.
The amplitude of the resulting wave is given by
$|A_1-A_2|=5.0\text{mm}-4.0\text{mm}=1.0\text{mm}$