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 $x_o=1\text{cm}$.
The amplitudes are $A_1=2.0\text{mm}$ and $A_2=6.0\text{mm}$. Find the amplitude of the resulting wave.

• A. $2\text{mm}$
• B. $4\text{mm}$
• C. $8\text{mm}$
• D. $6\text{mm}$

Answer 1

The correct option is C $8\text{mm}$

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,
$\Delta \varphi =k{x}_0=\left(6.28{\text{cm}}^{-1}\right)\left(1.0\text{cm}\right)$
$\Rightarrow \Delta \varphi =2\pi \times 1=2\pi$
The wave satisfy the condition of constructive interference, as the phase difference is an integral multiple of $2\pi$.
The amplitude of the resultant wave is given by
$|A_1+A_2|=2.0\text{mm}+6.0\text{mm}=8.0\text{mm}$