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Question: A travelling wave pulse defined as \(y = \dfrac{{10}}{{5 + {{(x + 2t)}^2}}}\). In which direction an...

A travelling wave pulse defined as y=105+(x+2t)2y = \dfrac{{10}}{{5 + {{(x + 2t)}^2}}}. In which direction and with what velocity is the pulse propagating?

Explanation

Solution

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−υt)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−ωt)y = A\,\sin \,(kx - \omega t)
Where A is the amplitude of the wave.
k = propagation constant
ω\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−υt)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=coefficient  t  of xcoefficient  t  of t⇒v=21v = \dfrac{{coefficient\,\,t\,\,of\,x}}{{coefficient\,\,t\,\,of\,t}} \Rightarrow v = \dfrac{2}{1}
Velocity of wave is 2 msec⁡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−υt)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 msec⁡ - 2\,\dfrac{m}{{\sec }} as velocity which will be wrong. Therefore, the wave is moving with 2 msec⁡2\,\dfrac{m}{{\sec }} in the negative direction.