Question

A travelling wave pulse defined as $y = \dfrac{{10}}{{5 + {{(x + 2t)}^2}}}$. In which direction and with what velocity is the pulse propagating?

Answer 1

Hint: We will observe the coefficients of x and t respectively to check that the wave is propagating in negative direction or in positive. We will equate the given equation with the equation of wave travelling in negative direction $y = f\,(x - \upsilon t)$.

Complete step by step answer:
Wave: When energy is transported from one direction to another without the actual transfer of matter is known as a wave.
In other words, it is also known as disturbance in the medium.
Wave pulse: It is a single wave which repeats itself in regular intervals. It consists of only one crest. It is also defined as the distance between two consecutive troughs.
Mathematically; wave is represented by
$y = A\,\sin \,(kx - \omega t)$
Where A is the amplitude of the wave.
k = propagation constant
$\omega $= angular velocity
t = time taken
Variables are x and y
When the coefficient of x and coefficient of t are in opposite directions, then the wave is said to be propagating in a negative direction. Whereas when coefficients are in the same direction, then the wave is said to be propagating in a positive direction.
On comparing the given equation with $y = f\,(x - \upsilon t)$, we found that the wave is propagating in a negative direction.
Coefficient of x is 1 and coefficient of t is 2 but with a negative sign.
Velocity is calculated using $v = \dfrac{{coefficient\,\,t\,\,of\,x}}{{coefficient\,\,t\,\,of\,t}} \Rightarrow v = \dfrac{2}{1}$
Velocity of wave is $2\,\dfrac{m}{{\sec }}$ and the wave is propagating in a negative direction.

Note:
If positive sign instead of negative then the direction of wave propagation would have been positive. But in the current situation, while comparing with the $y = f\,(x - \upsilon t)$ equation, it was found that wave propagation was negative. Secondly velocity cannot be negative, so while taking the ratios of coefficients, we will not consider signs. If considered then we might get $ - 2\,\dfrac{m}{{\sec }}$ as velocity which will be wrong. Therefore, the wave is moving with $2\,\dfrac{m}{{\sec }}$ in the negative direction.