Wave pulse traveling up a hanging rope and string displacement velocity

[Pixter]

Have two questions about waves.
1. If a rope with mass "m" is suspended from the ceiling and a wave puse is produced at the lower end of the rope and will travel upwards.
Will the speed of the wave change, and if so, will it increase or decrease?
My taughts: I was trying to figure out if the tension itself will increase or not, because the speed will depend on the tension, right?. so let's say the pulse is traveling up the rope, then i taught that the mass behind the pusle will increase(because more length) and gravitational force will make the tension increasse, therefore the wavespeed should increase...right?)
2. If the displacement of a tight string is given by :
y(x,t) = A cos((2pi/constant)*(x-vt))
find a expression for vy(that is v in the y direction) at which a piece of string travels. What is the maximum value of vy? When is the maximum value greater than the wave propagation speed v?
my taught: I don't have a clue about this question though, don't knwo where to start, so if someone could give me a push in the right direction.
Thanx for the help!

 

[Galileo]

Have two questions about waves.
1. If a rope with mass "m" is suspended from the ceiling and a wave puse is produced at the lower end of the rope and will travel upwards.
Will the speed of the wave change, and if so, will it increase or decrease?
My taughts: I was trying to figure out if the tension itself will increase or not, because the speed will depend on the tension, right?. so let's say the pulse is traveling up the rope, then i taught that the mass behind the pusle will increase(because more length) and gravitational force will make the tension increasse, therefore the wavespeed should increase...right?)

That's right. Since the upper end of the rope has to carry the weight of the rope beneath it the tension in the wire will increase as you go.

2. If the displacement of a tight string is given by :
y(x,t) = A cos((2pi/constant)*(x-vt))
find a expression for vy(that is v in the y direction) at which a piece of string travels. What is the maximum value of vy? When is the maximum value greater than the wave propagation speed v?
my taught: I don't have a clue about this question though, don't knwo where to start, so if someone could give me a push in the right direction.
Thanx for the help!

You are given y(x,t) and you are asked how fast y changes with time. That's what velocity is. So.. any thoughts?

 

[Pixter]

Well i guess if a differentiate y(x,t) then I'll get vy. But I don't know how to defferentiate that when there's both x and t, only know how to defferentiate with the respects of x or t, not both at the same time.

 

[Galileo]

Well, for the velocity you need to differentiate with respect to time. Keep x constant.

 

[Pixter]

Okej, so I did it by partial differentiation and got:

((-2pi*v*A)/constant)*cos((2pi/consstant)*(x-vt))

Just want to check if this is right, and then I could probably take it from there.. or i'll just come back =)

 

[Galileo]

Okej, so I did it by partial differentiation and got:
((-2pi*v*A)/constant)*cos((2pi/consstant)*(x-vt))
Just want to check if this is right, and then I could probably take it from there.. or i'll just come back =)

Almost, but there should be a sine instead of a cosine.
If there too much terms to see what you are doing just simplify the notation.
Introduce u(t)=2pi/const.*(x-vt), so that:
$$y(x,t)=A\cos(u(t))$$
and use the chain rule.