# Periodic Wave Solutions of the Wave Equation

• quasar987
In summary, any "spacially periodic" function of the form f(x,t) = X(x)cos(wt) is a solution of the wave equation, with the restriction that X(x) must be a harmonic function. This restricts the possible values of the constant k in the argument of the cosine function to k = \omega / v, where v is the wave speed and \omega is an arbitrary constant. However, there are cases where this form does not satisfy the wave equation, such as when f(x,t) = cos(wt).
quasar987
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Every "spacially periodic" function [i.e. s.t there exist P s.t. f(x+P,t) = f(x,t)] of the form f(x,t) = X(x)cos(wt) is a solution of the wave equation.

Last edited:
True...

quasar,

False.

For one thing, if f(x,t) is periodic (and it doesn't have to be periodic), then there's a strict relation between the periodicity in x and in t. In other words, in an example using your format, if f(x,t) = sin(kx)cos(wt), then w/k = v.

It's not too hard to see what the wave equation is saying about f(x,t) if you think about it. Within a multiplicative constant, the two partial derivatives are the same. That means f has to depend on x and t in very similar ways. I think the most general form for f(x,t) that satisfies the W.E. is f(kx-wt). Certainly any function of that form will work. Although that's not really what you were asking.

Uh oh!

James R., I posted before I saw yours. Why do you say it's true?

Hmm... Here are my thoughts. I think I might have changed my mind!

At first, I thought that any function of the form f(x,t) = X(x)cos wt will describe a standing wave, and so it is necessarily a solution of the wave equation.

But then...

The wave equation, in 1 dimension, is:

$$\frac{\partial^2 f}{\partial x^2} - \frac{\partial^2 f}{v^2 \partial t^2} = 0$$

where v is a constant (the wave speed).

Using the function given, we have:

$$\frac{\partial^2 f}{\partial x^2} = \frac{d^2 X}{dx^2}\cos \omega t$$
$$\frac{\partial^2 f}{\partial t^2} = -\omega^2 X(x)\cos \omega t$$

Therefore, we require:

$$\frac{d^2 X}{dx^2} + \frac{X}{v^2} = 0$$

This restricts X(x) to harmonic functions of the form:

$$X(x) = A \sin kx + B \cos kx$$

where A and B are arbitrary constants, but k is restricted:

$$k=\omega / v$$

So, it seems that the most general functions of the given form which satisfy the wave equation are:

$$f(x,t) = [A \sin kx + B \cos kx]\cos \omega t$$

with the above restriction on k.

Does that seem right?

You don't you just work out the simplest case: $f(x,t)=\cos(wt)$.
We clearly have $f(x+P,t)=f(x,t)$ for any t (and any P).
The wave equation clearly doesn't hold in this case (unless $\omega=0$, but I understand $\omega$ can be arbitrary).

## What is the wave equation?

The wave equation is a mathematical equation that describes the propagation of waves through a medium. It is a partial differential equation that relates the second derivative of a wave to its velocity and other physical properties of the medium.

## What are periodic wave solutions?

Periodic wave solutions are solutions to the wave equation that repeat themselves at regular intervals in both time and space. This means that the shape of the wave remains the same as it travels through the medium.

## How are periodic wave solutions related to waves in real life?

Many natural phenomena, such as sound waves and electromagnetic waves, can be described by periodic wave solutions. These solutions help us understand and predict the behavior of waves in real life.

## What is the significance of the wave equation and its periodic solutions?

The wave equation and its periodic solutions have a wide range of applications in various fields such as physics, engineering, and mathematics. They allow us to model and analyze different types of waves, making it easier to understand and manipulate them for practical purposes.

## Are there any limitations to the use of periodic wave solutions in the real world?

While periodic wave solutions are useful for describing many natural phenomena, there are some cases where they may not accurately represent real-life waves. For example, waves that encounter obstacles or travel through nonlinear media may not behave in a purely periodic manner.

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