Waves on a String linear density?

In summary, the waves on a string are affected by the linear density of the strings. If you try solving the problem you'll find the lengths cancel.
  • #1
just.karl
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Waves on a String "linear density?"

Two steel guitar strings have the same length. String A has a diameter of .5mm and is under 410N of tension. String B has a diameter of 1.0mm and is under a tension of 820N. Find the ratio of the wave speeds, v_a/v_b, in these two strings.

Linear density u=m/L and v=(F/u)^1/2 to find the wave velocity then I would just divide the two to find the ratio.


What I'm confused about is how to I relate the diameter to the linear density equation or if I do at all.
Please help!
 
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  • #2
The mass of the steel strings will be their volume multiplied by their density. The mass per unit length is just their total mass divided by their length. The one way we vary linear density in guitar strings is to use different diameter strings rather than the same diameter but different density material (which is more difficult).
 
  • #3
So since they have the same length, could I then just say that String A is 1/2 and string B is 1 for u? In short?
 
  • #4
just.karl said:
So since they have the same length, could I then just say that String A is 1/2 and string B is 1 for u? In short?

No, because [itex]\mu[/itex] will be have an r squared dependence.
 
  • #5
I'm not sure what r squared dependence is and what it's referring to.
 
  • #6
just.karl said:
I'm not sure what r squared dependence is and what it's referring to.

The mass of the string is the string volume times the density. The volume is [itex]\pi r^2 l[/itex] where [itex] r[/itex] is the radius of the string. Since the linear mass density is the mass divided by the length then [itex]\mu=...[/itex] fill in the rest. :smile:
 
  • #7
I should have know that... Thanks

But for the length can I give it any value great than zero and still have it work out alright?
 
  • #8
just.karl said:
I should have know that... Thanks

But for the length can I give it any value great than zero and still have it work out alright?

If you try solving the problem you'll find the lengths cancel.
 
  • #9
I realized that right after... sorry

Thanks for all your help. I really appreciate it.
 

1. What is the definition of linear density in relation to waves on a string?

Linear density, also known as mass per unit length, refers to the amount of mass present in a given length of a string. In the context of waves on a string, it is an important factor in determining the speed and frequency of the waves.

2. How does linear density affect the behavior of waves on a string?

The linear density of a string affects the speed and frequency of the waves that travel through it. A higher linear density means that the waves will move slower and have a lower frequency, while a lower linear density will result in faster waves with a higher frequency.

3. How is linear density measured in waves on a string?

Linear density is typically measured in units of mass per unit length, such as kilograms per meter (kg/m) or grams per centimeter (g/cm). This can be determined by measuring the mass of the string and dividing it by its length.

4. What factors can affect the linear density of a string?

The linear density of a string can be affected by its material composition, thickness, and tension. Materials with a higher density, such as steel, will have a higher linear density compared to materials with a lower density, such as nylon. Thicker strings will also have a higher linear density than thinner strings, assuming the same material. Higher tension in a string can also increase its linear density.

5. How does the linear density of a string impact the quality of sound produced by waves?

The linear density of a string is directly related to the pitch and tone of the sound produced by waves traveling through it. A higher linear density will result in lower-pitched and deeper sounds, while a lower linear density will produce higher-pitched and brighter sounds. This plays a significant role in the sound quality of musical instruments such as guitars and violins.

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