Calculating Amplitude and Velocity of Mechanical Waves in Non-Uniform Wire

[struggles]

Homework Statement

The linear mass density of a non-uniform wire under constant tension gradually decreases along the wire while ensuring an incident wave is transmitted without reflection. The wire has constant density μ for x ≤ 0. In this region a transverse wave has the form y(x,t) = 0.003cos(30x -60t), where x,y are given in meters and t in seconds. From x=0 to x=20, the linear mass density decreases from μ to μ/9. For x > 20, the density remains constant at μ/9.

i) What is the wave velocity for large values of x (>20m)?
ii) What is the amplitude of the wave for large values of x? You should be able to determine this using conservation of mechanical energy,
iii)Give y(x,t) for x>20.

Homework Equations

y(x,t) = Acos(kx-wt)
w/k = v
v= sqrt(T/μ)
P = 1/2sqrt(μT)w²A²

The Attempt at a Solution

i) Comparing given wave equation and y=Acos(kx-wt) for x < 0
k = 30 w = 60
v = 60/30 =2
v²μ = T
4μ=T
Tension remains constant so
v₂₀ = √4μ/μ/9 = 6m/s

ii) E = 1/2mv². At max kinetic energy potential energy = 0. Total energy = Kinetic energy
P = ½√(μF)w²A²
Energy before = Energy After
½√(μ4μ).60²0.003² =½√(μ/9.4μ).v²k²A²
0.0027=k²A²

I have no idea if this is correct and if i can take k to be constant or not.
Any help would be really appreciated!

 

[TSny]

Welcome to PF!

What stays constant as a wave travels from one medium to another: the frequency or the wavelength?

 

[struggles]

Thank you!
The frequency stays constant and as the wave travels the wavelength would increase i think. That would mean that k wouldn't be constant.

 

[struggles]

Could I work out the frequency initially as f = v/λ = vk/2pi = 30pi and then as frequency stays the same work out k when x=20 by
k=2fpi/v = 60pi²/6 ?

Thank you for helping me out!

 

[TSny]

Yes, frequency remains constant. What can you say about ω?

How does this help when thinking about the power P?

 

[struggles]

ω = 2pif so would also be constant. This means I now know all the variables in the power equation apart from the Amplitude so by setting power before to power after I should hopefully be able to get the correct amplitude. Thank you so much!