# Wave Speed and Tension

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#### Trayshawn

##### Full Member
7+ Year Member

Im not quite sure why tension affects wave speed the way it does.

The explanation I've found in my studies goes something like this:

If you imagine a wave moving through a string, the molecules of the medium are displaced as it passes through. The more tension there is in the string, the more resistance it creates against that movement (almost like a hooke's law force). Thus, the tension makes the material snap back to the equilibrium position faster.

Thats all fine and dandy, BUT:

Shouldn't the tension also make it harder for the medium to be displaced upwards as the wave just begins to pass through as well?

In other words, the explanation above only considers the tensions affect in making the medium snap back faster. But, by this same rationale, it should also move up slower. Seems to me that both effects should cancel and there should be no affect on wavespeed.

#### Gauss44

##### Full Member
5+ Year Member
Im not quite sure why tension affects wave speed the way it does.

The explanation I've found in my studies goes something like this:

If you imagine a wave moving through a string, the molecules of the medium are displaced as it passes through. The more tension there is in the string, the more resistance it creates against that movement (almost like a hooke's law force). Thus, the tension makes the material snap back to the equilibrium position faster.

Thats all fine and dandy, BUT:

Shouldn't the tension also make it harder for the medium to be displaced upwards as the wave just begins to pass through as well?

In other words, the explanation above only considers the tensions affect in making the medium snap back faster. But, by this same rationale, it should also move up slower. Seems to me that both effects should cancel and there should be no affect on wavespeed.

I think density slows down waves in general, while tension speeds them up.

#### Trayshawn

##### Full Member
7+ Year Member
Yeah, I get why density slows them down. My question is why does tension speed them up? I showed my issues with the usual explanation in the first post.

##### MD
Moderator Emeritus
10+ Year Member
The tension *does* make it harder to displace. That doesn't mean it can't be. If you've ever played (seriously played) a guitar, you might have noticed that the amplitude of your strings' vibration is much greater when playing tuned down to, say, Drop C. Because at this lower tuning the tension in the strings is less, strumming with the same force you normally do produces a greater amplitude (greater displacement from equilibrium) than it would at the higher standard tuning. This means that to achieve the same amplitude as normal, less force is required at a lower tuning. You could strum more lightly in Drop C to get the same string amplitude as in standard tuning. Regardless, once your pick has released a string, the only factor affecting the speed of propagation of the wave is how quickly that displaced segment of string wants to come back to equilibrium. In the tenser, higher-tuned string the segments will experience the greater restoring force, and the wave will travel faster than in the lower tuning. In both tunings, assuming we're playing open notes in each case, the length of the string is the same, so the wavelength is the same (2L). With changed speed and same wavelength, we get a change in frequency and different pitch at different tuning.

#### Trayshawn

##### Full Member
7+ Year Member
dont play instruments, so i didnt follow much of that.

higher tension = higher restoring force. --> higher restoring force = greater speed towards equilibrium --> faster wave

BUT

higher restoring force ALSO --> lower speed away from equilibrium --> slower wave.

two effects should cancel, no?

##### MD
Moderator Emeritus
10+ Year Member
dont play instruments, so i didnt follow much of that.
Welp.
higher tension = higher restoring force. --> higher restoring force = greater speed towards equilibrium --> faster wave

BUT

higher restoring force ALSO --> lower speed away from equilibrium --> slower wave.

two effects should cancel, no?
This is where you're getting it wrong. Higher restoring force simply means greater acceleration toward equilibrium even when the medium is being displaced from equilibrium as the wave first passes through it. What the greater restoring force (tension) causes is not a decrease in wave speed (or transverse speed of displacement from equilibrium) but a decrease in wave amplitude.

#### milski

##### 1K member
5+ Year Member
None of the arguments about the restoration force towards the equilibrium is really relevant to the speed of propagation of the wave, at least not directly. They talk about how soon the return to equilibrium happens and not how fast that displacement is transferred to the next piece of string. A quick derivation of the formula for the wave speed is here: http://hyperphysics.phy-astr.gsu.edu/hbase/waves/wavsol.html#c2

For MCAT purposes, I don't think we need anything more than knowing what the relation is.

##### MD
Moderator Emeritus
10+ Year Member
None of the arguments about the restoration force towards the equilibrium is really relevant to the speed of propagation of the wave, at least not directly. They talk about how soon the return to equilibrium happens and not how fast that displacement is transferred to the next piece of string. A quick derivation of the formula for the wave speed is here: http://hyperphysics.phy-astr.gsu.edu/hbase/waves/wavsol.html#c2

For MCAT purposes, I don't think we need anything more than knowing what the relation is.

I gotta say I feel lost whenever I see that backwards 6 symbol.

In any case, the argument from restoring force makes sense at least intuitively doesn't it? The faster a point on a wave returns to the equilibrium as a wave passes through it, the faster the wave must have necessarily proceeded to the next points down the line, right? Even if this isn't a reason, it seems like an explanation of the trend.

#### milski

##### 1K member
5+ Year Member
Maybe I can make an argument for it without the math. Consider two small pieces of string next to each other. It does not matter if that part of the string is moving toward or away from equilibrium position. The piece closer to the "beginning" of the string has a slightly different displacement and is pulling the piece further away towards itself. How fast this piece will react depends on the force on it and its mass. The force with which it's being pulled increases when the tension of the string increases. As a result the string with higher tension has a vaster speed of wave propagation. (And the heavier string has a slower speed)

#### milski

##### 1K member
5+ Year Member
I gotta say I feel lost whenever I see that backwards 6 symbol.

In any case, the argument from restoring force makes sense at least intuitively doesn't it? The faster a point on a wave returns to the equilibrium as a wave passes through it, the faster the wave must have necessarily proceeded to the next points down the line, right? Even if this isn't a reason, it seems like an explanation of the trend.

Sorry, I tried to rephrase it in a saner way.

The argument leads to the correct result but by no means does that make it right. You really need to consider how the next piece of string further down reacts - how a single piece oscillates on its own just does not say anything about the propagation of the wave, just the amplitude and the frequency.

At the end, it's still pretty close since you do have to mention the tension between the two pieces, so your intuition is not misleading you.

##### MD
Moderator Emeritus
10+ Year Member
Sorry, I tried to rephrase it in a saner way.

The argument leads to the correct result but by no means does that make it right. You really need to consider how the next piece of string further down reacts - how a single piece oscillates on its own just does not say anything about the propagation of the wave, just the amplitude and the frequency.

At the end, it's still pretty close since you do have to mention the tension between the two pieces, so your intuition is not misleading you.

Your explanation does indeed make sense, thanks for that. And I totally understand, I wouldn't have known that my approach via restoring force was incorrect, even if it explained it in a way that made sense, but of course that's what physics majors like you are for right?

#### Trayshawn

##### Full Member
7+ Year Member
I still dont see how the restoring force approach works. Yes, the restoring force is greater for a more tense spring and so the string moves faster when it is already away from equilibrium to get back to equilibrium.

But what I'm saying is, when the string is AT equilibrium that same greater restoring force would make it move slower when leaving equilibrium (as would happen when the wave passes).

So, yes, it snaps back faster. But it also "snaps up?" slower. see what I'm saying??

#### Trayshawn

##### Full Member
7+ Year Member
Maybe I can make an argument for it without the math. Consider two small pieces of string next to each other. It does not matter if that part of the string is moving toward or away from equilibrium position. The piece closer to the "beginning" of the string has a slightly different displacement and is pulling the piece further away towards itself. How fast this piece will react depends on the force on it and its mass. The force with which it's being pulled increases when the tension of the string increases. As a result the string with higher tension has a vaster speed of wave propagation. (And the heavier string has a slower speed)

AHAHHHH!!!! GOTCHA!!!

But then...why does the force increase with tension? (if we're abandoning the restoring force approach, then that is no longer immediately clear)

#### milski

##### 1K member
5+ Year Member
Your explanation does indeed make sense, thanks for that. And I totally understand, I wouldn't have known that my approach via restoring force was incorrect, even if it explained it in a way that made sense, but of course that's what physics majors like you are for right?

We have to be useful for something, right?

I still dont see how the restoring force approach works. Yes, the restoring force is greater for a more tense spring and so the string moves faster when it is already away from equilibrium to get back to equilibrium.

But what I'm saying is, when the string is AT equilibrium that same greater restoring force would make it move slower when leaving equilibrium (as would happen when the wave passes).

So, yes, it snaps back faster. But it also "snaps up?" slower. see what I'm saying??

The single piece restoration force does not really matter. What matters is that when a piece moves in either direction, it pulls its neighbor piece towards itself stronger, makes it move faster and the wave ends up transmitted faster.

#### milski

##### 1K member
5+ Year Member
AHAHHHH!!!! GOTCHA!!!

But then...why does the force increase with tension? (if we're abandoning the restoring force approach, then that is no longer immediately clear)

The force pulling the two pieces towards the intermediate position between them is the component of the tension parallel to the displacement direction. For the same displacement, larger tension leads to larger force.