Write an expression for the standing wave produced by adding the two waves (2.5 cm)sin[(3.0 m-1)x - (27 s-1 )t] and (2.5 cm)sin[(3.0 m-1)x + (27 s-1)t].
(2.5 cm)sin[(3.0 m-1)x - (27 s-1 )t] + (2.5 cm)sin[(3.0 m-1)x + (27 s-1)t] = (5cm)sin[(3.0m-1)x - (27 s-1)t]
To find the expression for the standing wave produced by adding two waves, we can use the principle of superposition. According to this principle, the resulting displacement at any point in space and time is the algebraic sum of the individual displacements caused by each wave.
Let's start by analyzing the two waves:
Wave 1:
Amplitude = 2.5 cm
Angular wave number = (3.0 m^(-1))
Angular frequency = (27 s^(-1))
Phase constant = 0
Wave 2:
Amplitude = 2.5 cm
Angular wave number = (3.0 m^(-1))
Angular frequency = -(27 s^(-1)) [negative because of opposite signs]
To find the resulting standing wave, we add the two waves together (considering their positive and negative sign conventions):
Standing wave = Wave 1 + Wave 2
= (2.5 cm)sin[(3.0 m^(-1))x - (27 s^(-1))t] + (2.5 cm)sin[(3.0 m^(-1))x + (27 s^(-1))t]
We can further simplify this expression by factoring out the common term "(2.5 cm)" and rearranging:
Standing wave = (2.5 cm)[sin((3.0 m^(-1))x - (27 s^(-1))t) + sin((3.0 m^(-1))x + (27 s^(-1))t)]
So, the expression for the standing wave produced by adding the two given waves is:
(2.5 cm)[sin((3.0 m^(-1))x - (27 s^(-1))t) + sin((3.0 m^(-1))x + (27 s^(-1))t)]