Relationship between Wave Velocity and Displacement in a String

[BareFootKing]

Homework Statement

Given the tension T and unit mass per length U of a string what is the speed of sound along the string

Homework Equations

The Attempt at a Solution

I know how to find the propagation speed of a transverse wave on a string, is that the same as the speed of sound along the string?

 

[sankalpmittal]

Homework Statement

Given the tension T and unit mass per length U of a string what is the speed of sound along the string

Homework Equations

The Attempt at a Solution

I know how to find the propagation speed of a transverse wave on a string, is that the same as the speed of sound along the string?

Yes, there is a formula. Do you have to derive it or is it just a sort of a short question ?

If you have to derive it, then there are two ways :

1. Complicated Way: You need to consider a small segment of spring and apply free body diagram etc..

2. Simpler way : Just use dimensional analysis. To what factors, you think the transverse wave velocity in a spring should depend ?

 

[BareFootKing]

Its a review question that i won't be turning, But I am able to derive the formula and in the end I get y= Asin(kx-vx) where k is the wave number and v is the propagation speed of the transverse wave where the string

and v = (T/U)^(1/2)

But I didn't know why the propagation speed of the wave was the speed of sound if sound traveled through a string.

Also thank you for giving me the simple way of doing this problem.. I always forget that If I am stuck on a problem I can use dimensional analysis. I will use that method in the future.

 

[haruspex]

If I am stuck on a problem I can use dimensional analysis.

Dimensional analysis affords powerful insights and useful checks, but it leaves you with an unknown constant.

 

[BareFootKing]

Thanks for the response.

But can someone help me understand why sound would travel at the speed of propagation. I am having trouble seeing the connection. For example if i had a large rope and held it at one end and another person held it at the other end and i sent a pulse down toward the other end how is the propagation speed of the pulse related to the speed of sound in the rope. I can't imagine that they are the same.

 

[haruspex]

But can someone help me understand why sound would travel at the speed of propagation.

How else are you going to define the speed of sound in the rope?

 

[BareFootKing]

I thought the speed of sound would a vary but be a unique constant for different mediums. constant in air, water, rope etc

But wouldn't I alter the propagation speed of the pulse by altering the speed in which I displace my end of the rope.

 

[haruspex]

I thought the speed of sound would a vary but be a unique constant for different mediums. constant in air, water, rope etc

It also depends on mechanical states such as tension.

But wouldn't I alter the propagation speed of the pulse by altering the speed in which I displace my end of the rope.

I don't believe so.

 

[sankalpmittal]

I thought the speed of sound would a vary but be a unique constant for different mediums. constant in air, water, rope etc

But wouldn't I alter the propagation speed of the pulse by altering the speed in which I displace my end of the rope.

BareFootKing, I am asking this simple question. Do you want to derive the formula or not ?

It's better not to use dimensional analysis here, as it would leave an unknown constant, as Haruspex marks. I don't know if you will like my derivation, as it is more sort of classical mechanics. If you like to derive it by my method, I am giving you this hint for a simple start off:

Hint: Consider a small arc like segment of string of length dl and divide it into two parts by a vertical line. The two parts will be similar by symmetry. Let each part subtend angle dθ radian at the centre. Take the tension on both end of segment and find out its components. What is the component that provides centripetal acceleration ?

OffTopic: Anyone having better idea of deriving are most welcomed to comment.

By OP:

But wouldn't I alter the propagation speed of the pulse by altering the speed in which I displace my end of the rope.

Since we are assuming only progressive waves in the string, we have,

Particle velocity = -wave velocity times slope of displacement curve.

What do you see from here ?