Question

`x(x,t)=(0.8)/([(4x+5t)^(2)+5])`, represents a moving pulse where x and y are in metre and t is in second. Find
(i) The direction of wave propagation.
(ii) The wave speed.
(iii) The maximum displacement from the mean position (i.e., the aplitude of the wave).
(iv). Whether the wave pulse is symmetric or not.

Answer 1

(i) The equation represents a travelling wave because the argument of this function is 4x+5t, which is of the form `ax+bt`. Because there is a positive sign between a and b, the wave is moving in the negative a axis direction.
(ii) The wave speed equals `(b)/(a)` and here `a=4 and b=5`. Hence the wave speed v is given by `v=(5)/(4)=1.25` unit.
(iii) Clearly y is maximum when (4x+50) is zero. hence the amplitude A is given by
`A=(0.8)/([(0)^(2)+5])=0.16` unit
(iv) At t=0, the wave pulse is represented by `y=(0.8)/([16x^(2)+5])`
As changing x to -x does not affect the value of y, wave pulse is symmeric.
the shape of this pulse is shown in the figure.
.