Compound Wire - Standing Wave

In summary: Basically, it's the equation that governs how a wave behaves.That sounds like a really good intro. Thank you!
  • #1
Stealth849
38
0

Homework Statement



An aluminum wire of length l1 = 60cm, cross section area 1.00 x 10-2cm2, and density 2.60g/cm3, is joined to a steel wire of density 7.8g/cm3 and same cross sectional area. The compound wire loaded with a block of mass m = 10kg is arranged as shown so that the distance l2from the joint to the supporting pulley is 86.6cm. Transverse waves are set up in the wire using an external source of variable frequency. A node is located at the pulley.

a) find the lowest frequency of excitation for which a standing wave is observed to have a node at the joint.

b) How many nodes are observed at this frequency?

Homework Equations



f = v/λ = nv/2l

ρ = m/V

v = sqrt(FT/μ)

The Attempt at a Solution


Not too sure where to begin on this one. When it asks for the lowest frequency of excitation, I expect that means the second harmonic, because that would leave a node at the joint with the lowest frequency.

So knowing the densities of the wires, I can get v where

FT = μv2
FT = mg = 98N

μ = m/l = ρV/l = ρA where A represents the cross sectional area.

so μ1 = 2600*0.0001 = 0.26

μ2 = 7800*0.0001 = 0.78

v1 = sqrt(98/0.26) = 19.4m/s
v2 = sqrt(98/0.78) = 11.2m/s

so f = nv/2l

where n = 2 for the second harmonic

f1 = 2*0.26/(2*0.6) = 0.43Hz

f2 = 2*0.78/(2*0.866) = 0.90Hz

No idea if I'm approaching this correctly, due to the idea of reflection of waves at the joint.. Say a wave does reflect at the joint and a portion of it is sent back to the source, there would be a lot more waves thus being a much higher harmonic - not the second.

Any and all help appreciated. Thanks!
 

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  • #2
When it asks for the lowest frequency of excitation, I expect that means the second harmonic
er ... that would be the case if the wave-speed is the same in each section of wire.

Think in terms of the modes in each section of wire, if the sections were separate.
 
  • #3
Okay.. So if I know that

nv/2l = f

and am treating each section separately, except for the fact that frequency must be the same for each section, could I say that

nv1/2l1 = mv2/2l2

essentially setting frequency equal to frequency, but being able to solve for the number of nodes, or at least a ratio of nodes...?

as in

n/m = l1*v2/l2*v1

Thanks for the reply :)
 
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  • #4
Well done.
The other approach was to solve the wave equation directly :)
 
  • #5
Thanks!

I don't see how I could solve the wave equation directly as you said, however. Can you define what you mean by wave equation?

Using what I mentioned previously...

n/m = l1*v2/l2*v1
n/m = 0.6*11.2/(0.866*19.4) which is roughly 2/5

So this means that the aluminum section, l_1, has 2 nodes and the steel section has 5 nodes, yes?

and frequency can be calculated by either side,

f = nv/2l
f = 19.4/0.6 = 32.3Hz
f = 5*11.2/2*0.866 = 32.3Hz

and the total number of nodes will be 2 + 5 = 7?
 
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  • #6
Wave equation:
$$\frac{\partial^2}{\partial x^2}y(x,y) = \frac{1}{c^2}\frac{\partial^2}{\partial t^2}y(x,t)$$ ... in your case, the wave-speed, c, depends on x so it will need some modification.

Since ##c=f\lambda##, which will have the longer wavelength for the same frequency? The fast one or the slow one?
 
  • #7
Well, if c is proportional to to wavelength, the faster one should have the longer wavelength...no?

How did you get to that wave equation? Or is that a standard model for a wave?
 
  • #8
Stealth849 said:
Well, if c is proportional to to wavelength, the faster one should have the longer wavelength...no?
That's how you check your answers make sense :)
How did you get to that wave equation? Or is that a standard model for a wave?
http://en.wikipedia.org/wiki/Wave_equation
Deriving the 1D case from Hook's Law used to be a college-level exercise...
 
  • #9
Simon Bridge said:
Deriving the 1D case from Hook's Law used to be a college-level exercise...

I'm sure it still is... Looking forward to learning it eventually when college rolls around. :)

Do you mind giving me a quick introduction to how that equation works...?
 
  • #10
It's a second order partial differential equation - if we consider y(x,t) to be a transverse displacement of a point on the string at position x along the string, then it says that the way a point on a string accelerates (the RHS) depends on how the string around that point is curved (the LHS).

Any wave has to be consistent with that equation, and the external "boundary conditions" - like being fixed at both ends and having a node in a particular place.
 

Related to Compound Wire - Standing Wave

1. What is a compound wire?

A compound wire is a type of wire that is made up of two or more different materials, usually metals, that are joined together. This combination of materials allows the wire to have unique properties, such as increased strength or resistance to corrosion.

2. What is a standing wave?

A standing wave is a type of wave that appears to be standing still, rather than moving forward. This occurs when two waves of equal amplitude and frequency travel in opposite directions and interfere with each other, resulting in a wave that appears to not be moving.

3. How does a compound wire affect standing waves?

A compound wire can affect standing waves in several ways. The different materials in the wire can cause reflections and refractions of the wave, changing its amplitude and frequency. The wire's structure can also alter the standing wave's nodes and antinodes, causing changes in its overall shape and behavior.

4. What are some practical applications of compound wire and standing waves?

Compound wire and standing waves have many practical applications. In the field of telecommunications, standing waves in compound wires are used to transmit signals over long distances. They are also used in musical instruments, such as stringed instruments, to produce specific notes and tones.

5. How can we manipulate compound wire and standing waves?

Compound wire and standing waves can be manipulated in several ways. By changing the composition or structure of the wire, we can alter the properties of the standing wave. Additionally, we can vary the frequency or amplitude of the wave to produce different effects. Using external forces, such as magnets or electric fields, can also manipulate standing waves in compound wires.

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