Exercise 36 Page 524

We are told that the equation of a standing wave can be obtained by adding the displacements of two waves traveling in opposite directions.

y = A cos (\frac{2 π t}{3} - \frac{2 π x}{5}) + A cos (\frac{2 π t}{3} + \frac{2 π x}{5})

Here,

t

represents the time and

x

is the point's position. We want to show that this formula can be rewritten as the following equation when

t

is equal to

1 .

y = - A cos (\frac{2 π x}{5})

To do so, we will start by substituting

t = 1

in the given formula.

y = A cos (\frac{2 π t}{3} - \frac{2 π x}{5}) + A cos (\frac{2 π t}{3} + \frac{2 π x}{5})

$t = 1$

y = A cos (\frac{2 π ( 1 )}{3} - \frac{2 π x}{5}) + A cos (\frac{2 π ( 1 )}{3} + \frac{2 π x}{5})

Identity Property of Multiplication

y = A cos (\frac{2 π}{3} - \frac{2 π x}{5}) + A cos (\frac{2 π}{3} + \frac{2 π x}{5})