Standing Waves (tension vs. wavelength)

In summary, the conversation discusses the attempt to decode a graph of Tension vs. λ2 of standing waves on a string and derive the equation for the line from two expressions: v=√(T/μ) and v=fλ. The OP found the correct relation T=(μf^2)λ^2 and the physical interpretation of the slope of the graph is that it represents the product of μ and f^2. However, there is confusion about the fixed wavelength in the graph, as it is not clear how it was produced from the data.
  • #1
furiouspoodle
3
0

Homework Statement



I'm trying to decode a graph of Tension vs. [tex]\lambda^2[/tex] of standing waves on a string to understand the actual meaning of the slope. I also need to derive the equation for the line from two expressions:

[tex]v= \sqrt{T/\mu} [/tex]
[tex] v= f\lambda [/tex]

Homework Equations



[tex] \lambda=2L/n [/tex]
[tex] \mu = m/L [/tex]

The Attempt at a Solution



I'm starting to think I'm completely missing something in my initial approach.

[tex] v = \sqrt {T/\mu} [/tex]

[tex] v^2 = T/\mu [/tex]

[tex]v = f\lambda[/tex]

[tex] v^2 = (f\lambda)^2[/tex]

[tex]T/\mu = f^2\lambda^2 [/tex]

[tex]T = \mu f^2\lambda^2 [/tex]

[tex]T = \frac {\mu f^2 4L^2}{n^2} [/tex]

I feel like it should lead to a clearer solution than this one, but I'm not sure what. I first thought that the slope was simply a surface tension (dynes/cm^2) on the string but I don't see how that relates to the equation or slope I'm trying to find.

I also tried this:

[tex] f = \frac {v}{\lambda} [/tex]

[tex] f = \frac {\sqrt {T/\mu}}{\lambda} [/tex]

[tex] f^2 = \frac {T/\mu}{\lambda^2} [/tex]

[tex] \mu f^2 = \frac {T}{\lambda^2} [/tex]

which seems closer, but I don't know how to apply it in a usable manner. Any help would be appreciated.
 
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  • #2
The OP found the correct relation ##T = (\mu f^2)\lambda^2##. Thus, the physical interpretation of the slope of the ##T## vs. ##\lambda^2## graph is that it represents the product of ##\mu## and ##f^2##. I think that should be an acceptable answer.
 
  • #3
I grappled with this question without coming to a satisfactory conclusion. The OP mentions a graph of tension vs. λ2 which presumably was produced from collected data. If one varies the tension and finds the frequency of the fundamental for fixed wavelength λ = 2L, the relation between the tension and the frequency is ##T=4L^2\mu f^2.## In that case, a plot of T vs. (2Lf)2 would yield a straight line with slope equal to ##\mu.## We need additional information about how this graph was produced from what data. It might elucidate the need for plotting tension vs. wavelength squared.
 
  • #4
kuruman said:
I grappled with this question without coming to a satisfactory conclusion. The OP mentions a graph of tension vs. λ2 which presumably was produced from collected data. If one varies the tension and finds the frequency of the fundamental for fixed wavelength λ = 2L, the relation between the tension and the frequency is ##T=4L^2\mu f^2.## In that case, a plot of T vs. (2Lf)2 would yield a straight line with slope equal to ##\mu.## We need additional information about how this graph was produced from what data. It might elucidate the need for plotting tension vs. wavelength squared.
Ok. Thanks.

Maybe it's just me, but if the graph plots T vs λ2 as stated by the OP, then it doesn't make sense to me to think of the wavelength as fixed.
 
  • #5
TSny said:
Ok. Thanks.

Maybe it's just me, but if the graph plots T vs λ2 as stated by the OP, then it doesn't make sense to me to think of the wavelength as fixed.
It doesn't make much sense to me either. In a standing wave experiment the wavelength can be varied by varying the driving frequency at fixed tension. I can think of an experiment in which one fixes the driving frequency, varies the tension and then finds the separation between the ends that will produce the same order harmonic. Then one can make sense of a T vs. λ2 plot.
 
  • #6
kuruman said:
I can think of an experiment in which one fixes the driving frequency, varies the tension...
Yes, that's what I had in mind. The tension can be due to some hanging weights. The weights are varied to get different standing waves (with different wavelengths) with the same fixed frequency.
 
  • #7
TSny said:
Yes, that's what I had in mind. The tension can be due to some hanging weights. The weights are varied to get different standing waves (with different wavelengths) with the same fixed frequency.
With near-zero probability of getting clarifications from the OP we can only guess ##\dots##
 
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What are standing waves?

Standing waves are a type of wave that occurs when two waves with the same frequency, amplitude, and wavelength travel in opposite directions and interfere with each other. This causes the wave to appear stationary, or "standing," rather than moving in one direction.

What is the relationship between tension and wavelength in standing waves?

The relationship between tension and wavelength in standing waves is inverse. This means that as tension increases, the wavelength decreases, and vice versa. This relationship is known as the "law of transverse standing waves."

How does tension affect the amplitude of standing waves?

Tension has no direct effect on the amplitude of standing waves. The amplitude is determined by the source of the wave and the resonance frequency of the medium in which it travels. However, changes in tension can indirectly affect the amplitude by altering the wavelength of the wave, which can then affect the amplitude at specific points along the standing wave.

What is the equation for calculating the wavelength of a standing wave?

The equation for calculating the wavelength of a standing wave is: λ = 2L/n, where λ is the wavelength, L is the length of the medium in which the wave is traveling, and n is the number of nodes or anti-nodes. This equation only applies to standing waves in a fixed medium, such as a string or air column.

Can standing waves occur in all types of waves?

No, standing waves can only occur in transverse waves, where the particles of the medium are perpendicular to the direction of the wave. This includes waves on a string, water surface waves, and electromagnetic waves. Longitudinal waves, such as sound waves, cannot form standing waves.

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