Two girls generate a harmonic wave by moving a rope up and down. The wave function is described by the equation D(x,t)=(0.20m)sin(0.5πx-1.3πt). The document provides the solution to determining the wavelength, speed, effect of tension on speed, and calculating displacement at a given point using the wave function equation. It defines a harmonic wave as one undergoing simple harmonic motion or a sinusoidal wave and calculates the wavelength as 4m and speed as 2.6m/s using information from the wave function.

1. LG2
Physics101 Learning Objective
Two girls hold a piece of rope on either ends and generate a harmonic wave
by moving the rope in a smooth up and down motion. The wave function is
described as:
D(x,t)=(0.20m)sin(0.5πx-1.3πt) where x is in meters and t is in seconds.
a) What is a harmonic wave?
b) Explain what the wavelength is and determine the wavelength of the wave
c) What is the speed of the wave?
d) The girls want to move the wave as fast as they can, but in order to do this
would they have to tighten or loosen the tension in the rope? Explain using a
mathematical model.
e) Calculate the displacement of the string at x=0.5m at t=2.0s

2. LG2
Solution
a) A harmonic wave is a wave that is undergoing simple harmonic motion or in
other words a sinusoidal wave. A harmonic wave oscillates and generates a
continuous wave.
b) Wavelenth is the shortest distance in which a wave repeats itself.
λ=2π/k=2π/0.5π=4m
c) v=ω/k=1.3π/0.5π=2.6m/s
d) The speed of the wave will increase when a greater amount of tension is
exerted on the rope. We can explain this statement using the formula
v=√Ts/√μ. When looking at this formula we can see that when the tension
force increases so will the speed of the rope.
e) D(0.5, 2.0)=(0.20)sin[(0.5π*2.0)-(1.3π*2.0)]=0.19m