Question

If $y\left( {x,t} \right) = \dfrac{{0.8}}{{\left[ {{{\left( {4x + 5t} \right)}^2} + 5} \right]}}$, represents a moving pulse where $x$ and $y$ are in meter and $t$ is in second. Find
(i) Direction of wave propagation.
(ii) The wave speed.
(iii) The maximum displacement from the mean position (i.e., the amplitude of the wave)
(iv) Whether the wave pulse is symmetric or not.

Answer 1

For finding the answer of questions, we compare the given wave equation from the standard wave equation. On comparing, direction and wave speed is found out. The Remaining two question parts are found by maxima concept and changing $x$ by $- x$ at $t = 0$to get a wave is symmetric.

Complete step by step answer:
Moving pulse is given by
$\Rightarrow y\left( {x,t} \right) = \dfrac{{0.8}}{{\left[ {{{\left( {4x + 5t} \right)}^2} + 5} \right]}}$
Comparing $\left( {4x + 5t} \right)$ with $\left( {ax + bt} \right)$ then we get,
$\Rightarrow a = 4$ and $b = 5$ $\cdots \cdots \cdots \left( 1 \right)$
(i) When coefficients of $x$ and $t$ are positive then the wave pulse is moving negative $x$direction. So, given an equation with coefficients $a = 4$ and $b = 5$ are positive. Therefore, the direction of the given equation will be negative $x$ axis.
(ii) After getting the coefficient $a$ and $b$, we can calculate the speed by determining the value of $\dfrac{b}{a}$. And we have value of $a = 4$ and $b = 5$ from the equation $\left( 1 \right)$ .Then speed of the wave will be
$\Rightarrow \dfrac{b}{a} = \dfrac{5}{4}$
$\Rightarrow$ speed of the wave $= 1.25$ $m/s$
(iii) To get the maximum displacement from the mean position we put $\left( {4x + 5t} \right) = 0$. As the denominator is minimum, the fraction will be maximum. Therefore displacement will be
$\Rightarrow y = \dfrac{{0.8}}{5}$, putting the $\left( {4x + 5t} \right)$ is equal to zero.
After simplify we get,
$\Rightarrow y = 0.16$ m
Hence maximum displacement is $0.16m$ from the mean position,
(iv) We replace $x$ by $- x$ and also put $t = 0$ in the given equation. If the equation does not change, replacing the x by $- x$, it will be symmetric otherwise not.
Putting $t = 0$ in the given equation $y\left( {x,t} \right)$ we get,
$\Rightarrow y = \dfrac{{0.8}}{{{{\left( {4x} \right)}^2} + 5}}$
Now replace $x$ by $- x$ on the above equation we get
$\Rightarrow y = \dfrac{{0.8}}{{{{\left( {4x} \right)}^2} + 5}}$
We can notice that there will not be any change. A square of negative numbers is always positive. Therefore, the given equation is symmetric.

Note:
Maximum value of displacement can also be calculated differentiating the wave equation at time is equal to zero. Then differentiation is kept equal to zero to get the value of $x$. And we get $x = 0$. Now we put x and t are equal to zero to get the maximum displacement.