Propagating waves along a non uniform string.

[slasakai]

Homework Statement

The linear mass density of a non uniform wire under constant tension decreases gradually along the wire so that the incedent wave is transmitted without reflection. the wire is uniform for* -∞≤x≤0
In this region, a transverse wave has the form y(x,t) = 0.003cos(30x − 60t). From x=0 to x=20 the linear mass density decreases fromμ to μ/9 and is again constant as μ/9 from x=20 to x=∞
i. what is the wave velocity for large values of x(>20)?
ii.What is the amplitude of the wave for large values of a(>20m). you should get this from conservation of mechanical energy.
iii.What's y(x,t) for x=20 to x=∞.

Homework Equations

v=sqrt(F/μ)

The Attempt at a Solution

for part i. I used the above equation to equate the value of force for a wave from - infinity to 0 and 20 to infinity as the tension is uniform throughout the string and solved for the velocities in terms of each other. However I don't understand how to get a numerical answer.

for part ii and iii I am struggling to see even how to approach this question :(

any help would be most welcome.

 

[vela]

Homework Statement

The linear mass density of a non uniform wire under constant tension decreases gradually along the wire so that the incedent wave is transmitted without reflection. the wire is uniform for* -∞≤x≤0
In this region, a transverse wave has the form y(x,t) = 0.003cos(30x − 60t). From x=0 to x=20 the linear mass density decreases fromμ to μ/9 and is again constant as μ/9 from x=20 to x=∞
i. what is the wave velocity for large values of x(>20)?
ii.What is the amplitude of the wave for large values of a(>20m). you should get this from conservation of mechanical energy.
iii.What's y(x,t) for x=20 to x=∞.

Homework Equations

v=sqrt(F/μ)

The Attempt at a Solution

for part i. I used the above equation to equate the value of force for a wave from - infinity to 0 and 20 to infinity as the tension is uniform throughout the string and solved for the velocities in terms of each other. However I don't understand how to get a numerical answer.

What's the speed of the wave for x<0? You should be able to figure that out from the y(x) given to you.

for part ii and iii I am struggling to see even how to approach this question :(

any help would be most welcome.