Understanding the Relationship between Amplitude and Frequency in Waves

[rauletechuleta]

Hi everyone. The answer to the question about the relationship between amplitude and frequency is usually that they are independent of each other. Pardon the imprecision of my language, for I am not a physicist. I'll try my best to be clear. The amplitude of a wave is a measure of the distance between the wave's extremes. In light, amplitude corresponds phenomenologically with brightness. Now, I have read that amplitude measures the energy of a wave. I have also read, however, that frequency is a measure of a wave's energy, and positively correlates with it - as does amplitude. When I look at a diagram of a transversal wave, I intuitively assume that when a particle behaving as a wave moves in space, the particle -given a constant frequency- must move faster if the amplitude increases because its displacement from the resting position -the forward axis indicating direction- is greater. that is, in order to keep the frequency constant, a greater amplitude would require a greater oscillation speed. Can someone explain why this is not so, or confirm that it is so?
Thank you

 

[Drakkith]

When you say, "particle behaving as a wave", are you referring to the Quantum Mechanical wavefunction that describes the position and momentum of a particle? Or are you referring to something like atoms in a guitar string that move back and forth?

 

[rauletechuleta]

Can you explain the difference and then go on to answer the question in quantum terms?

 

[rauletechuleta]

As I understand, in quantum mechanics the so-called particle is the result of a coalescence of many waves. I was referring to “something like atoms in a guitar string that move back and forth”. Now it seems to me that ascertaining the position of a particle as a wave function would automatically render ascertaining the parameters of the waves that coalesce into the particle (amplitude, frequency, etc.) less certain.

 

[TumblingDice]

The amplitude of a wave is a measure of the distance between the wave's extremes.

Right.

in light, amplitude corresponds phenomenologically with brightness.

The amplitude of a light wave (a stream of photons) corresponds to brightness/intensity.

Now, I have read that amplitude measures the energy of a wave.

Where did you read that?

I have also read, however, that frequency is a measure of a wave's energy

Frequency corresponds directly with the energy per photon. The higher the frequency (shorter wavelength) the greater the energy of each photon.

...and positively correlates with it - as does amplitude.

There is a correlation, but 'energy' stems from frequency. The higher the frequency, the more energy per photon. The larger the amplitude of a stream of photons (a wave), the greater the intensity/brightness.

For example, if you compare two photons, one of red light, and one of blue light, the one of red light will be lower in energy. For two streams with equal number of photons, one of red light and one of blue light, the red beam of light will be of lower amplitude (intensity) than the blue beam of light.

 

[Drakkith]

Let's leave photons out of this. They will only confuse the situation.

I intuitively assume that when a particle behaving as a wave moves in space, the particle -given a constant frequency- must move faster if the amplitude increases because its displacement from the resting position -the forward axis indicating direction- is greater. that is, in order to keep the frequency constant, a greater amplitude would require a greater oscillation speed. Can someone explain why this is not so, or confirm that it is so?

It is true that the speed of the string will be greater when the amplitude is greater, as it must cover a larger distance in the same amount of time as a string of lower average amplitude if the frequencies are the same. This requires giving the string more energy as the amplitude increases.

In general, a wave of any type will have more energy when the amplitude or frequency is higher.

I don't recommend trying to get into QM wavefunctions until you understand basic, classical waves first. Otherwise you're just going to confuse yourself.

 

[rauletechuleta]

Thank you for your help and patience.

1)
“There is a correlation, but 'energy' stems from frequency. The higher the frequency, the more energy per photon. The larger the amplitude of a stream of photons (a wave), the greater the intensity/brightness.”

And so, what is the relationship between intensity/brightness and energy? For brightness is a phenomenon, but neither frequency nor energy is.

2)
“For two streams with equal number of photons, one of red light and one of blue light, the red beam of light will be of lower amplitude (intensity) than the blue beam of light.”

The photons in the blue beam also have a higher frequency. This seems to mean that the higher the frequency of the electromagnetic wave, the higher the intensity. However, will the frequency of the photons in the blue beam increase if the amplitude is increased?
Does the number of photons in the beam determine the amplitude? Is amplitude density?
How does density affect the rate of oscillation?

 

[rauletechuleta]

I assume then that if amplitude is density, both amplitude and frequency affect the total energy of the system, the first by sheer numbers, the second by dint of velocity.

 

[Drakkith]

For an EM wave, a greater amplitude means a greater energy and intensity (brightness). Bringing photons into the mix, this means that for two EM waves of equal amplitude (equal energy), the higher frequency wave will have fewer photons.

The photons in the blue beam also have a higher frequency. This seems to mean that the higher the frequency of the electromagnetic wave, the higher the intensity.

This is not true. Classically, the intensity of the EM wave depends only on its amplitude, not its frequency.

However, will the frequency of the photons in the blue beam increase if the amplitude is increased?

No, neither amplitude nor frequency affect one another. You can change either one to be any realistic amount at will.

Does the number of photons in the beam determine the amplitude? Is amplitude density?

I'd prefer to say that the combination of the amplitude and frequency of the EM wave determines the number of photons. If we look at two EM waves of the same frequency then the wave of higher amplitude has more photons, or a higher density as you put it.

How does density affect the rate of oscillation?

It doesn't.

 

[rauletechuleta]

“For an EM wave, a greater amplitude means a greater energy and intensity (brightness).”
Is this because there are more photons (i.e. higher density)? Otherwise, what accounts for the energy increase?

“Bringing photons into the mix, this means that for two EM waves of equal amplitude (equal energy), the higher frequency wave will have fewer photons.”
Is this because frequency also affects the total energy? (Isn't this what the Planck constant denotes?)
But if amplitude and energy do not affect each other, does the term 'energy' mean two different things?
In one case energy is synonymous with amplitude, and in another case it is proportional to the frequency.
What I am getting is that amplitude is proportional to the number of photons, and is not affected by the frequency of the photons, represented by a wave function. However, both amplitude and frequency determine the total energy of the system.

rauletechuleta: “This seems to mean that the higher the frequency of the electromagnetic wave, the higher the intensity.”
Drakkith: “This is not true. Classically, the intensity of the EM wave depends only on its amplitude, not its frequency.”
I see. The frequency can be modulated and the amplitude modulated without reference to it, to obtain an equal amount of energy, though 'energy' not in the sense merely of amplitude. In other words, wave A can have more energy than wave B, even if wave B has higher amplitude than wave A, if wave A has a frequency sufficiently high.

“I'd prefer to say that the combination of the amplitude and frequency of the EM wave determines the number of photons.”
How does the frequency of the EM wave determine the number of photons?

 

[Drakkith]

“For an EM wave, a greater amplitude means a greater energy and intensity (brightness).”
Is this because there are more photons (i.e. higher density)? Otherwise, what accounts for the energy increase?

Yes.

“Bringing photons into the mix, this means that for two EM waves of equal amplitude (equal energy), the higher frequency wave will have fewer photons.”
Is this because frequency also affects the total energy? (Isn't this what the Planck constant denotes?)
But if amplitude and energy do not affect each other, does the term 'energy' mean two different things?

Energy and amplitude are not the same things. Energy is the ability to perform work. Amplitude is the amount of displacement from a central point and is a property of a wave. It simply takes more energy to cause a greater amount of displacement, so that waves of greater amplitude take more energy to create.

In one case energy is synonymous with amplitude, and in another case it is proportional to the frequency. What I am getting is that amplitude is proportional to the number of photons, and is not affected by the frequency of the photons, represented by a wave function. However, both amplitude and frequency determine the total energy of the system.

Let's not talk about wave functions. They aren't even applicable here.

rauletechuleta: “This seems to mean that the higher the frequency of the electromagnetic wave, the higher the intensity.”
Drakkith: “This is not true. Classically, the intensity of the EM wave depends only on its amplitude, not its frequency.”
I see. The frequency can be modulated and the amplitude modulated without reference to it, to obtain an equal amount of energy, though 'energy' not in the sense merely of amplitude. In other words, wave A can have more energy than wave B, even if wave B has higher amplitude than wave A, if wave A has a frequency sufficiently high.

I'll be honest with you. I'm actually not that familiar with wave mechanics and I'm not exactly sure if the frequency of a wave affects the total energy (or power) of the wave. I thought I knew the answer, but after a little more looking into it, I'm not sure.

“I'd prefer to say that the combination of the amplitude and frequency of the EM wave determines the number of photons.”
How does the frequency of the EM wave determine the number of photons?

The energy of a photon is related to the frequency. A higher frequency EM wave has photons that have more energy. So if you have two waves of equal energy, and one is double the frequency of the other, the higher frequency wave has half the number of photons. Which it must, because if each photon of the higher frequency wave has double the energy of each photon of the lower energy wave, then there must be half the number in order for the waves to have the same total energy.

 

[sophiecentaur]

@rauletechuleta

You are confusing classical wave theory with quantum mechanics here. Until you know a lot more about both subjects you really need to avoid using both concepts in the same sentence (or paragraph).
The (classical) basics that you can read, everywhere, about waves and oscillations were developed long before anyone knew about photons and other elementary particles. It is all perfectly correct and mostly it doesn't need any of the 'modern' ideas in order to describe most of what we experience around us.

I would say that, as history has shown, you can get along fine with waves without knowing anything about QM, photons, etc.. But you need to know a reasonable amount about classical wave theory before you will get much out of QM. There are many apparent contradictions between classical and QM - as with your confusion about energy, amplitude and frequency. But you need to appreciate that this took many decades of work by some of the greatest minds to sort out. So don't expect to get it all at once. We all, frequently go down the wrong road whilst trying to learn this stuff.

One thing you should come to terms with very soon is that photons are nothing like 'little bullets'. The word 'particle' has its own special meaning in this context so you have to avoid leaping to conclusions based on what you already feel you know about the way 'particles' behave.

 

[rauletechuleta]

Hi. Thank you for taking the time to make these observations.

I understand, or at least I've come to terms with, that photons are nothing like “little bullets”. Photons seem to me to be something more along the line of waves, and not particles in the way that dust is a particle. As far as I have read, a photon is a quantum of light, that is, the smallest possible amount of energy, and therefore the unit of electromagnetic radiation, or in other words a discrete quantity of light (not necessarily visible light). Now, I read that the energy and the momentum of a photon depend solely on its frequency. Since a photon has no mass, how are frequency and velocity related in quantum theory?

 

[Drakkith]

Hi. Thank you for taking the time to make these observations.

I understand, or at least I've come to terms with, that photons are nothing like “little bullets”. Photons seem to me to be something more along the line of waves, and not particles in the way that dust is a particle.

Nope. Photons are neither particles nor waves. They are, as you explained in your post, the quanta of interaction of an EM wave. Instead of thinking of light in terms of photons, think of light as an EM wave first and then understand how the wave transfers energy and interacts with other objects. This will bring you much closer to understanding light. Remember that the concept of light being an EM wave is not abandoned in Quantum Physics. It is merely expanded upon.

Now, I read that the energy and the momentum of a photon depend solely on its frequency. Since a photon has no mass, how are frequency and velocity related in quantum theory?

The velocity is always c. Therefore, when the frequency changes the wavelength changes in turn. They are related by: v=c/$\lambda$, where v is the frequency, $\lambda$ is the wavelength, and c is the speed of light in a vacuum.

 

[VINAY]

I found the following equation in my book, this might help you. Total energy of a particle executing simple harmonic motion is given by the equation, $E=2∏^2mr^2√^2$ where r is amplitude, √ is frequency, m is mass of the particle.

 

[Khashishi]

It depends on the type of wave you are talking about. For electromagnetic waves (light), the energy depends on the amplitude squared and doesn't depend on the frequency. You've probably heard that the energy of a photon depends on the frequency. A photon isn't a wave. A photon does not have an amplitude. It is the particle aspect of light. The photon has the minimum amount of electromagnetic energy that can be transferred between objects.

For sound, the energy depends on the amplitude squared and sound speed, not frequency.

 

[rauletechuleta]

“Instead of thinking of light in terms of photons, think of light as an EM wave first and then understand how the wave transfers energy and interacts with other objects.”
This is what I'm going to do. I feel closer to being on the right track Drakkith. Respect.

Vinay: E=2∏^2mr^2√^2. This is a nice equation. At first glance it tells me that energy is positively correlated with both amplitude and frequency (and mass).

Khashishi: Then E=2∏^2mr^2√^2 does not refer to EM waves?

“For [LIGHT] the energy depends on the amplitude squared and doesn't depend on the frequency. You've probably heard that the energy of [THE PARTICLE ASPECT OF LIGHT] depends on the frequency. [THE PARTICLE ASPECT OF LIGHT IS NOT A WAVE]. A photon does not have an amplitude. It is the particle aspect of light (EM waves). The photon has the minimum amount of electromagnetic energy that can be transferred between objects.”
In this paragraph you differentiate between EM waves (a.k.a. light) and photons, which are not light per se, but are the particle aspect of light. You also relate wave with amplitude and photon with frequency. My first blind question: how can something with a frequency not have an amplitude?
You say “The photon HAS the...”; but should it be “The photon IS the...”. Because it seems rather that it is the wave that HAS the photon, in a rather unusual sense of 'has'.
The photon, or quantum of light, or basic unit of EM energy, is the minimum amount of EM energy THAT CAN BE TRANSFERRED BETWEEN OBJECTS. Is it ok then to consider the photon then as the unit of interaction? Is it right to think of the wave as an entity which interacts with other waves by means of photons?

 

[olivermsun]

I'm not exactly sure if the frequency of a wave affects the total energy (or power) of the wave.

The equation for a simple harmonic oscillator that's been posted in this thread has E = 2∏²mr²ω² as a function of mass m, amplitude r, and frequency ω. Notice that the restoring force (spring constant) does not appear: it's wrapped into the frequency of oscillation ω² = k / m. Fixed mass and amplitude, a higher oscillation frequency implies a "stiffer" spring constant k and hence a larger maximum potential energy at either end of the oscillation.

 

[olivermsun]

I'm not exactly sure if the frequency of a wave affects the total energy (or power) of the wave.

For the classical simple harmonic oscillator, the equation that was posted earlier in this thread has E = 2∏²mr²ω² for mass m, amplitude r, and frequency ω. Notice that the restoring force (spring constant) does not appear: it's wrapped into the frequency of oscillation ω² = k / m. Fixed mass and amplitude, a higher oscillation frequency implies a "stiffer" spring constant k and hence more work done by the restoring force throughout the motion (and larger maximum potential energy at either end of the oscillation).

 

[sophiecentaur]

. . . .Now, I read that the energy and the momentum of a photon depend solely on its frequency. Since a photon has no mass, how are frequency and velocity related in quantum theory?

You are in danger of getting into a circular argument here unless you get the whole story of photons and momentum. They have no mass so they (all) can travel at c in a vacuum. The momentum of a photon can manifest itself when light falls on an object and makes it move (sunlight on interplanetary dust, for instance). How come, with no mass? For a photon of wavelength λ, its momentum is h/λ, where h is the Planck constant. That goes along with the photon energy hf. The mass contribution to the momentum of a particle with mass and how it relates to the photon is discussed in this wiki link and this one.

 

[rauletechuleta]

Therefore, the higher the frequency, the higher the momentum. The value is always a multiple of h, since h is the minimum quantum of action.
Before I click on those links, I want to ask: an atom of an element, being composed of protons, neutrons, and electrons, has mass. An atom can be thought of as the functional synthesis of these subatomic particles. The quantum mechanical approach is to use n, l, m1, and ms to describe a particular atom of a given element by alluding to the way that electron moves, without specifying its actual location.
A photon, however, is not composed of such subatomic particles. Therefore the photon must be described differently. The wave function is a way to represent the photon as a particle.
That is a question.

 

[rauletechuleta]

Thank you for the links. Given that I don't understand everything, they ARE really helpful.

 

[rauletechuleta]

(as a side note sophiecentaur, is it correct to think that momentum depends on mass and velocity when the “particle” in question has mass, and depends solely on velocity when it has no mass? is the mass of the subatomic particles relative to one another what determines the shape of their cloud? and do photons simply travel in straight lines?)

 

[Drakkith]

(as a side note sophiecentaur, is it correct to think that momentum depends on mass and velocity when the “particle” in question has mass, and depends solely on velocity when it has no mass?

Momentum of an object with mass is governed by the following equation:

P=MV, where P is the momentum, M is the mass, and V is the velocity. Both the mass and the velocity are controllable. By that I mean we can change the mass or the velocity in the equation. So the momentum can be changed by altering either the mass of the object or the velocity.

The momentum of a photon is governed by this equation:

P=h/$\lambda$. where P is the momentum, h is the reduced plank constant, and $\lambda$ is the wavelength of the photon.

Since the reduced plank constant is, well, constant, there is only one variable on the right side of the photon equation that we can change, $\lambda$ (wavelength). This means that the momentum of a photon depends only upon the photon's wavelength.

is the mass of the subatomic particles relative to one another

Mass is not relative. It is measured from the objects own inertial frame.

what determines the shape of their cloud?

This has nothing to do with the topic. You can find plenty of information on wiki's page on atomic orbitals or you can start a new thread.

and do photons simply travel in straight lines?)

You're still thinking of photons in a way similar to "bullets". The photon is the quanta of the EM wave. The EM wave obeys wave mechanics and includes effects such as diffraction. This means that the wave can diffract around objects and be detected even when the emitter and detector are not within line of sight of each other.

 

[sophiecentaur]

(as a side note sophiecentaur, is it correct to think that momentum depends on mass and velocity when the “particle” in question has mass, and depends solely on velocity when it has no mass? is the mass of the subatomic particles relative to one another what determines the shape of their cloud? and do photons simply travel in straight lines?)

You seem to want to reduce Modern Physics just to terms that were used more than 100 years ago. It can't be done.Hundreds (tens of thousands) of very clever people have found that the old terms are just not enough. If you try to pick out bits of this subject at random and make sense of it all then you really will be disappointed.

There are many Popular Science books that present the subject of Modern Physics in an approachable way. You should visit a bookstore and skim through a few. (Be careful: many of them are not Physics -based, but the contents page can help you there. Also avoid there clearly cranky ones) I have bought many good ones in my time - and given them away to friends so I can't recommend any in particular. The first couple of chapters of many of them will deal which things at your level. The essential thing is to use someone else's structure for you learning, rather than picking here and there. It is the structure of Physics that it's all about and you need to get the wider view. Q and A is seldom a good way into Physics.