First slide

Travelling wave

Question

The displacement due to a wave moving in the positive x-direction is given by $y = \frac{1}{(1 + x^{2})}$ at time t = 0 and by $y = \frac{1}{[1 + {(x - 1)}^{2}]}$ at t = 2 seconds, where x and y are in metres. The velocity of the wave in m/s is

Moderate

A

0.5
B

1
C

2
D

4

Solution

In a wave equation, x and t must be related in the form $(x - v t)$ .
We rewrite the given equations $y = \frac{1}{1 + {(x - v t)}^{2}}$
For t = 0, this becomes $y = \frac{1}{(1 + x^{2})}$ , as given
For t = 2, this becomes $y = \frac{1}{[1 + {(x - 2 v)}^{2}]} = \frac{1}{[1 + {(x - 1)}^{2}]}$
$\Rightarrow 2 v = 1 or v = 0 .5 m / s$ .

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