# Contradictory Sources: Trying to understand waves on a rope being reflected at a boundary

• AsemEzzat
AsemEzzat
Homework Statement
I apologize for putting this here, but I don't know where to put it.
Relevant Equations
None
My physics book says the following:
"If the waves in the smaller spring have a greater speed because the spring is stiffer, then the reflected wave will be inverted"
This indicates that a stiffer/denser spring results in a greater speed and vice versa
Now Google, ChatGPT, and an online source (Alef) say that
"What happens if you send the same wave pulse through the rope, but it is now attached to a stiffer, heavier rope?

As you can see, some of the energy is transmitted through the second rope. This is refraction. The wave slows down at the boundary. The rest of the energy in the wave reflects back."
And Google says that denser springs slow the wave

Did I understand the physics book wrong or is the book wrong?

Once again sorry for posting it here.

First, you can't learn physics from Google or ChatGPT.

In general, a sound or pressure wave will move faster through a stiffer material. Compare a slinky with an iron bar, as extreme examples.

That said, it's not clear to me precisely which scenario you are considering.

AsemEzzat
There are plenty of videos out there showing this in action.
Here's one:

Which of the scenarios you mention matches the real world demos?

AsemEzzat
The scenario I am referring to is transmission. I want to know how speed is affected when going from heavy to light and opposite.

Plenty of intuitive real world examples too.

Speed of vibrations in water compared to air...
Speed of vibrations in ocean waves when transmitted to a ship's steel hull...
Speed of seismic activity through Earth compared to through air...

AsemEzzat
So, transmission to denser materials, that is stiffer, slows down the wave. So, when the wave is going from heavy to light, it will go faster.

AsemEzzat said:
So, transmission to denser materials, that is stiffer, slows down the wave. So, when the wave is going from heavy to light, it will go faster.
On what evidence do you base that conclusion?

AsemEzzat
Well, firstly air is less dense than water and sound can hardly be heard underwater, so this means the incident wave has greater speed than transmitted wave. So, we can say the opposite is true, a wave going from a denser medium to a less dense medium will be faster. Am I wrong?

AsemEzzat said:
Well, firstly air is less dense than water and sound can hardly be heard underwater,
Whether or not it can hardly be heard does not bear upon how fast it moves.

Do you know why it is much more difficult for a person to tell where a sound is coming from underwater compared to on land? It's because humans can't echolocate underwater. We are able echo locate on land because our brains can tell the very slight difference in arrival times between our two ears. In water, the sounds arrives at both our ears at almost exactly the same time, so sounds seems to be coming from everywhere.What about the vibrations of a steel freighter in ocean waves? When a wave hits the bow of the ship, which arrives at the pilothouse first? The ocean wave? Or the vibrations through the hull?

AsemEzzat
Sound arrives at the almost exactly the same time, while on air there is a slightly larger difference? Waves in denser mediums travel faster then.
So, the vibrations will arrive first. So, a wave going to a denser material will have greater speed and when denser to less denser, it slows down. When denser to lighter, the reflected wave will also be inverted.

AsemEzzat said:
Well, firstly air is less dense than water and sound can hardly be heard underwater, so this means the incident wave has greater speed than transmitted wave.
For information, the speed of sound in air (under typical conditions) is around 340m/s. In water it is around 1500m/s and in iron is around 5100 m/s.

It's important to distingish between the amplitude of a wave and the speed of a wave. In simple terms, it is the amplitude (not the speed) of a sound wave which controls the 'loudness'.

AsemEzzat
When lighter to denser, the reflected wave will be inverted*

Steve4Physics said:
For information, the speed of sound in air (under typical conditions) is around 340m/s. In water it is around 1500m/s and in iron is around 5100 m/s.
And to bookend this list, the speed of sound in diamond - one of the hardest substances known - is 12 kilometres per second.

AsemEzzat
Thank you all for the help, greatly appreciated.

AsemEzzat said:
Thank you all for the help, greatly appreciated.
So, bringing this back to your OP, can you determine which source is right and which is wrong?

AsemEzzat
AsemEzzat said:
stiffer, heavier rope
Stiffer or heavier?
The square of the speed of the wave in a stretched string is proportional to the tension (analogous to stiffness in extensive media) but inversely proportional to the density. So the wave could be faster because the rope is stiffer or slower because it is denser.

nasu and AsemEzzat
haruspex said:
Stiffer or heavier?
The square of the speed of the wave in a stretched string is proportional to the tension (analogous to stiffness in extensive media) but inversely proportional to the density. So the wave could be faster because the rope is stiffer or slower because it is denser.
Are you saying that waves travel slower in denser mediums?

DaveC426913 said:
So, bringing this back to your OP, can you determine which source is right and which is wrong?
Google, ChatGPT, and Alef are wrong. The physics book is right I just didn't understand it correctly. By saying "If the waves in smaller spring have a greater speed because the spring is stiffer, then the reflected wave will be inverted" it means by smaller denser and it is correct because denser mediums have greater speeds. This is also reinforced by the video you sent as I saw that when the wave travels from the lighter (the bigger spring) to the denser (the smaller spring), the reflected wave is inverted. But, does it really mean by smaller denser? Because if so, that would contradict with the rope being stiff. The other person said that velocity is proportional to tension and inversely proportional to density. Wouldn't be wrong to say "The denser rope has a greater speed because it is stiffer."

AsemEzzat said:
Are you saying that waves travel slower in denser mediums?
For a given stiffness, yes. Water is denser than air but far stiffer, so the sound is faster.

AsemEzzat
So travelling from heavier (denser) to lighter (less dense), increases speed and vice versa? And from a less stiff rope to a stiffer rope, speed increases and reflected wave is inverted. Does this mean that DaveC426913 is wrong or is density inversely proportional to velocity only in a stretched string?

#### ​

AsemEzzat said:
Does this mean that DaveC426913 is wrong or is density inversely proportional to velocity only in a stretched string?
At no time did I invoke the words density or stiffness.

AsemEzzat
I'll write everything I understood:
Freighter's steel hull is stiffer (more tension) than water, thus wave travel faster in it
When a wave is transmitted from a less stiff medium to a stiffer medium, transmitted wave's speed increases and the reflected wave is inverted.
When a wave is transmitted from a denser medium to a less dense medium, the same thing happens (Transmitted wave's speed increases and the reflected wave is inverted)

Now if the above is true, the only thing left I want to know is
By heavier/lighter, does it mean density? Just to confirm

AsemEzzat said:
I'll write everything I understood:
Freighter's steel hull is stiffer (more tension) than water, thus wave travel faster in it
When a wave is transmitted from a less stiff medium to a stiffer medium, transmitted wave's speed increases and the reflected wave is inverted.
When a wave is transmitted from a denser medium to a less dense medium, the same thing happens (Transmitted wave's speed increases and the reflected wave is inverted)

Now if the above is true, the only thing left I want to know is
By heavier/lighter, does it mean density? Just to confirm
You are still not getting that these two considerations must be taken together.

When a wave is transmitted from a less stiff medium to a stiffer medium of the same or lower density, transmitted wave's speed increases.

When a wave is transmitted from a denser medium to a less dense medium of the same or higher stiffness, the same thing happens (Transmitted wave's speed increases

If the wave is transmitted from a denser medium to a less dense medium of lower stiffness, or from less dense to denser of higher stiffness, it could speed up or slow down.

I am unsure wrt inversion. I would think that only depends on stiffness.

nasu and PeroK

## 1. Why do waves on a rope invert when reflected from a fixed boundary?

When a wave on a rope encounters a fixed boundary, the point at the boundary cannot move. The wave exerts a force on this point, and according to Newton's third law, the boundary exerts an equal and opposite force on the wave. This results in the wave inverting, or flipping upside down, upon reflection.

## 2. How does the reflection of a wave differ at a free boundary compared to a fixed boundary?

At a free boundary, the end of the rope is free to move. When the wave reaches this boundary, it reflects without inverting because the boundary can move to accommodate the wave's motion. In contrast, at a fixed boundary, the wave inverts upon reflection because the boundary cannot move.

## 3. What happens to the amplitude of a wave when it is reflected at a boundary?

The amplitude of a wave can change upon reflection depending on the nature of the boundary and the medium. At a fixed boundary, the amplitude of the reflected wave is typically the same as the incident wave but inverted. At a free boundary, the amplitude remains the same but without inversion. However, if the boundary involves a different medium, part of the wave energy may be transmitted, resulting in a reduced amplitude of the reflected wave.

## 4. Why do some waves get partially transmitted and partially reflected at a boundary?

When a wave encounters a boundary between two different media, part of the wave's energy can be transmitted into the second medium while the remainder is reflected back into the original medium. The proportion of energy that is transmitted versus reflected depends on the impedance mismatch between the two media. A greater difference in impedance results in more reflection and less transmission.

## 5. How can we mathematically describe the reflection and transmission of waves at a boundary?

The reflection and transmission of waves at a boundary can be described using the principles of wave impedance and boundary conditions. The reflection coefficient (R) and transmission coefficient (T) are used to quantify the respective amplitudes of reflected and transmitted waves. These coefficients can be derived from the continuity of displacement and force (or their equivalents) at the boundary, leading to equations that depend on the properties of the two media involved.

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