What is the relationship between amplitude and frequency of a wave?
Some say there isn't any relationship, some say that there is, but from their answers the relationship is still unclear.
[Physics] Relationship between amplitude and frequency of a wave
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Related Solutions
Waves appear in nature and are described by wave equations,
second-order linear partial differential equation for the description of waves – as they occur in physics
All such equations have time dependent solutions and what they have in common is the oscillating behaviour in time that allows to assign a wavelength and describe the waves observed.
Now in quantum mechanics the description of the behavior of nature is dependent on differential equations of this type. The first studied is the Schrodinger equation and the link provides a good historical description of how it became evident that in the microcosm the behavior of particles was following a wave equation.
What is important to keep in mind is that in quantum mechanics the waves are probability waves, i.e. the probability if you do an experiment, like the double slit experiment, to find a particle in space at the time you look is governed by a wave solution. This is in contrast to other waves in physics which are variations in time on a medium, or in classical electromagnetism on changing fields.
So the relationship is the mathematical formulation of the differential equations describing nature in the two frameworks, not a one to one correspondence.
As I mentioned in my comment in electromagnetism, the frequency of the electromagnetic field described by classical electrodynamics appears in the energy of the photon ( the particle form of electromagnetism) in the identity E=h.nu. The interference pattern seen in the double slit experiment will display the frequency of light nu. If you continue your studies in physics you will be able to understand how the microcosm described by quantum mechanics leads to the macrocosm we call "classical physics" smoothly.
In the equation:
$$y(x,t)=Asin ~k(x-vt)$$
$A$ can be varied independently of $k$ and $v$ and hence of $f$. That is what is meant by saying that the amplitude doesn't depend on the frequency. Now, when you write the equation as:
$$A = \frac{y(x,t)}{sin ~k(x-vt)}$$
it means that the ratio of the height of the string from the mean position at some point to a function($sin ~k(x-vt)$) is always constant and equal to the amplitude. So it is basically saying that the height of the string at such $(x,t)$ where $k(x-vt) = (2n+\frac{1}{2})\pi$ is equal to the amplitude. But does this mean that you can change the ratio by changing $k$ and $v$? No you can't. And why is that? Because the numerator $y(x,t)$ also depends on $k$ and $v$.
In other words, when writing your equation for $A$ you are wrongly assuming that you can change the $k$ and $v$ in the denominator without changing the numerator. Whatever changes you make in the denominator will be reflected in the numerator and hence the ratio will remain fixed. That indeed is what the equation is saying- no matter what your $k$ and $v$ are, take this ratio and you get the amplitude.
Attempting to give more physical intuition. Imagine that you have a vibrating string and you need to find its amplitude. At what point and time will you measure its height to ensure that you got the amplitude? The crest of the wave will keep moving, making it hard to take readings.
So imagine you get another string also tied at both ends and exactly identical to the first, except that its amplitude is always 1 (which I am not saying is trivial to set up, but well this is a thought experiment). So now you just go ahead and measure the heights of both the strings at the same point simultaneously and take a ratio of the heights. You get the amplitude of the string, just as your equation says.
Now, how will you go about changing the $k$ or $v$ or $\omega$ in one string (the unit amplitude one) without changing them in the other? You have to use the same driving source so that the strings are phase matched so you cant vary the frequency of one without changing the other as well. Since they are identical you cannot change the material of one without the other. So you cannot change the denominator of your equation without also changing the numerator simultaneously and canceling out the change.
And finally a short technical answer that will make perfect sense if you are mathematically oriented.
Notice that $y(x,t)=Asin(kx-wt)$ has the $k$ and $w$ in the phase part. So no matter what you do to them, you cannot affect the amplitude part.
Specifically, the ratio of $y$ and $sin(kx-wt)$ is meant to cancel out the phase factor in the $y$ and leave the amplitude. By changing the frequency you only change the phase part that is canceled out anyways.
Best Answer
In general there is no relationship. Any combinations of frequencies and amplitudes are allowed.
There can be some relationship in certain special cases: for example if you have a source of waves which emits a specific spectrum, then the amplitudes and frequencies obey that spectrum. But spectrums can be arbitrary, so the dependency can be arbitrary.
In conclusion: generally there is no relationship.