Questions:

3.1. Determine the speed of the wave in the simulation using v = λ /T where wavelength and period are determined from the simulation as you did in the previous exercise using the mouse to find the wavelength and the time to find the period. What is the forward speed of this wave?

3.2. The speed of this wave is also given mathematically by v = ω/k since ω = 2π f = 2π/T and k = 2π/λ. What is the speed of this wave based on the values of ω and k in the equation? Does this match the speed you got from the simulation?

3.3. Reload the initial conditions and experiment with values of the wavenumber both smaller and larger than 2.0 rad/m keeping the angular frequency fixed. How does the wavenumber change the speed of the wave?

3.4. Reload the initial conditions and experiment with values of the angular frequency both smaller and larger than 0.8 rad/s keeping the wavenumber fixed. How does the angular frequency change the speed of the wave?

This simulation is misleading in one important way. In the simulation you can set any combination of angular frequency and wavenumber you choose and so have any speed you want for the wave. But for mechanical and accoustic waves the speed is determined by the medium in which the wave travels. As we will see, for these waves it is often the case that v = ω/k so that the angular frequency and wavenumber are inversely proportional with v a constant. Examples:

For sound waves in a fluid (for example air) the speed is determined by v = (B/ρ)^1/2 where B is the bulk modulus or compressibility of the fluid in newtons per meter squared and ρ is the density in kilograms per cubic meter.
For waves in a solid the speed is determined by v = (Y/ρ)^1/2 where Y is Young's modulus or stiffness in newtons per meter squared and and ρ is the density in kilograms per meter cubed.
For waves on a string the speed is determined by v = (T/μ)^1/2 where T is the tension in the string in newtons and μ is the mass per length in kilograms per meter.
Although electromagnetic waves do not need a medium to travel (they can travel through a vacuum) their speed in a vacuum, c = (1/μ_oε_o) ^1/2 is governed by two physical constants, the permeability μ_o and the permittivity, ε_o.
If an electromagnetic wave enters a medium (such as light going through air, or water or glass) the speed changes because the permittivity and/or permeability may be different. The amount the speed changes is given by the index of refraction n = c/v where c is the speed of light and v is the speed in the medium.

As we will see later, it is the case that speed can sometimes depend on the frequeny of the wave, a phemomenon know as dispersion.

3.5. Does sound travel slightly faster on a hot day or a cool day? Does sound travel faster or slower if the humidity is high?

3.6. Density is relatively easy to measure. What would be a clever way to measure Young's modulus, Y for a solid?

3.7. Do waves on a string travel faster or slower if the string is tighter? Do waves on a string travel faster or slower if the string is thicker?

3.8. In general the index of refraction is larger than one. What does this tell you about the speed of light in glass?

3.9. Reload the initial conditions with the 'reload' button. For a wavenumber of 4.0 rad/m experiment to find the correct angular frequency which gives the original speed of the wave you found in questions 3.1 and 3.2 (you should be able to see from the simulation when the new wave is traveling at the same speed as the original).

3.10. Calculate the wavenumber which gives the speed of the original wave for angular frequencies of 0.4, 0.6, 1.0, and 1.2 rad/s using the relationship in question 3.2. Check your answers with the simulation if you are in doubt.