Standing Waves- Wave Speed

[texan2414]

Pg 243 TBR Physics Book I.

Two standing waves are plotted that are set at same frequency. However, string II is larger than string I.
The question asks which string has the highest wave speed?
a) string I
b) string II
c) wave speeds are equal
d) cannot be determined

I put C as my choice because I remember from my review that wave speeds only change when there is a change in the medium. In this context, you have the same string and it's not mentioned whether the two strings are made out of same material or not. Thus, I assumed they are made of the same material.

The correct answer however is B.

Can someone explain?

Thanks,

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[Cawolf]

f = nv/2L

(n = harmonic integer and note that if the string is free on on one end then f = nv/4L)

If both have the same frequency, but the lengths vary, you can see the relationship.

The wave on the longer string will have to have a proportionally higher velocity in order to keep the frequency constant.

 

[texan2414]

@Cawolf

I understand the velocity argument by the equation. My only source of confusion was when to use the "wave speed only changing due to medium" info for questions.
In other words, why did you not consider the medium of the string in answering this? Why was this irrelevant in standing waves?

Thanks,

 

[Cawolf]

Well, like you said, the medium is the same - and the velocity of the wave does depend on the medium. That is why the equation has the velocity term.

The thing to consider here is to think about what they are saying. In plain terms, them telling us the frequency is constant is them telling us that there are the same number of waves happening each second - on both strings. The waves on the longer string have to travel farther, so we can reason that they must travel faster - in order to occur at the same frequency of the shorter string.

I don't think that saying wave velocity only changes with medium is accurate. Maybe if a wave changes mediums, it's energy will change? Regardless, I don't think that argument is applicable in the scneario presented in this question.

 

[techfan]

Does the question say larger or longer? I thought it meant that the diameter of II was twice the diameter of I. I figured it had something to do with the v=sqrt(T/mu) where mu is the mass density and though the string is made of the same material there is more of it (like more twine in a rope, or more steel cables in the Golden Gate Bridge) so it is heavier per unit. Then using f=v/(4nL) and plugging in v gives:

f=sqrt(T/mu)*1/(4nL) which says that as mu gets larger the fundamental frequency gets smaller (or as the rope gets heavier the fundamental frequency drops). So if you keep the frequency constant v has to get larger (which I'm thinking has to do with increasing the tension in the rope).

 

[Cawolf]

The book states that the strings are of the same material and radius with one string being longer than the other.

He paraphrased so I looked it up just now to double check. It also states that the string is free on one end.