Question

A wave pulse in a string is described by the equation $y_1 = \frac{5}{(3x - 4 t)^2 + 2 }$ and another wave pulse in the same string is described by $y_2 = \frac{- 5}{(3x + 4 t - 6 )^2 + 2 }$. The values of $y_1, y_2$ and $x$ are in meters and t in seconds.
Which of the following statement is correct?

• A. $y_1$ travels along -x-direction and $y_2$ along + x-direction
• B. Both $y_1$ and $y_2$ travel along -x-direction
• C. Both $y_1$ and $y_2$ travel along + x-direction
• D. At $x= 1 \, m, y_1$ and $y_2$ always cancel
• E. At time $t = 1 \, s, y_1$ and $y_2$ exactly cancel everywhere

Answer 1

Answer: (D)

$\because$ Given,
$y_{1}=\frac{5}{(3 x-4 t)^{2}+2}, y_{2}=\frac{-5}{(3 x+4 t-6)^{2}+2}$
According option (d), at $x=1$
$y_{1} =\frac{5}{(3-4 t)^{2}+2}$
$y_{2} =\frac{-5}{(3+4 t-6)^{2}+2}$
$=\frac{-5}{(3-4 t)^{2}+2}$
Both wave pulse equation are existing in same string therefore resultant equation of wave pulse.
$y=y_{1}+y_{2}=0$