# Difference between a continuously differentiable function and a wave

• B
• redtree
In summary, an absolutely continuously differentiable function and a wave are distinct concepts. A linear function is continuously differentiable but not a wave. Every wave function can be expressed as a superposition of plane waves, which are absolutely continuously differentiable functions. The set of wave functions is a subset of the group of absolutely continuously differentiable functions, but the group structure is only additive and does not have multiplicative inverses. Additionally, a function can be a weak solution to the wave equation without being continuously differentiable.
redtree
TL;DR Summary
What is the difference between an absolutely continuously differentiable function and a wave?
What is the difference between an absolutely continuously differentiable function and a wave? Are all absolutely continuously differentiable equations waves?

A linear function is certainly smooth, in particular continuously differentiable, and it is not a wave function. These are distinct concepts. For approximations with wave functions, see https://en.wikipedia.org/wiki/Fourier_series

Is a wave function a subset of the group of absolutely continuously differentiable functions?

Every wave function can be considered as a superposition of plane waves, where each plane wave is absolutely continuously differentiable.

redtree said:
Is a wave function a subset of the group of absolutely continuously differentiable functions?
The set of wave functions is. A single wave function is an element of the two sets. The group structure is only additive, which does not contain much information. So group is the wrong word here. We do not have multiplicative inverses. It is an algebra (I think. Depending on how wave function is defined. The continuously differentiable functions are.)

Right. My mistake on the terminology

To perplex things just a bit, the following little lemma might be of interest:

If ##f(x)## is a two times differentiable function of one real variable then $$f(\vec{k}\cdot\vec{x}-\omega t)$$ is a wave function of n+1 variables, ##\vec{k},\vec{x}\in \mathbb{R}^n##, ##\omega,t\in \mathbb{R}## for any ##n\in\mathbb{N}##

I did not understand the question but answer:)
A function ##H(x-at)## is a solution of the wave equation as well but in the generalized sense
here ##H## is the Heaviside step function
##(x\in\mathbb{R})##

Delta2 said:
To perplex things just a bit, the following little lemma might be of interest:

If ##f(x)## is a two times differentiable function of one real variable then $$f(\vec{k}\cdot\vec{x}-\omega t)$$ is a wave function of n+1 variables, ##\vec{k},\vec{x}\in \mathbb{R}^n##, ##\omega,t\in \mathbb{R}## for any ##n\in\mathbb{N}##
As pointed out, you can drop the differentiability and still have a weak solution to the wave equation.

martinbn said:
As pointed out, you can drop the differentiability and still have a weak solution to the wave equation.
Sorry I don't understand, what do we mean by weak solution?

Delta2 said:
Sorry I don't understand, what do we mean by weak solution?
For example, we shall say that a function ##u(t,x)\in L^1_{loc}(\mathbb{R}_+\times\mathbb{R})## is a weak solution to the equation ##u_{tt}=u_{xx}## if for any ##\psi\in\mathcal{D}(\mathbb{R}_+\times\mathbb{R}),\quad \mathbb{R}_+=\{\xi>0\mid\xi\in \mathbb{R}\}## it follows that
$$\int_{\mathbb{R}_+\times\mathbb{R}}u(t,x)(\psi_{tt}-\psi_{xx})dtdx=0$$

shock waves meet this definition

Delta2
Thanks @wrobel for making me remember that time is usually taken to be positive , as well as the concept of weak solution to a PDE.

## 1. What is the difference between a continuously differentiable function and a wave?

A continuously differentiable function is a function that has a derivative at every point in its domain. A wave, on the other hand, is a disturbance that travels through a medium, characterized by periodic oscillations.

## 2. Can a continuously differentiable function also be a wave?

Yes, a continuously differentiable function can also exhibit wave-like behavior. For example, a sine or cosine function is continuously differentiable and can be used to model waves.

## 3. How are the graphs of a continuously differentiable function and a wave different?

The graph of a continuously differentiable function will typically be a smooth curve, while the graph of a wave will have a more oscillatory or periodic shape.

## 4. What are some real-life examples of continuously differentiable functions and waves?

A continuously differentiable function can be used to model the growth of a population or the change in temperature over time. Waves can be observed in nature, such as ocean waves or sound waves, and can also be artificially created, such as in radio or light waves.

## 5. How are continuously differentiable functions and waves used in different fields of science?

Continuously differentiable functions are used in fields such as physics, engineering, and economics to model and predict various phenomena. Waves have a wide range of applications in fields such as acoustics, optics, and geology.

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