# Is a given expression a travelling wave?

• shanepitts
In summary, to determine if an expression is a traveling wave, one can use the characteristic wave equation δ^(2)ψ/δt^(2)=v^(2)δ^(2)ψ/δx^(2) and see if the expression matches the equation. Additionally, one can also check if the expression follows the definition of a traveling wave, which is if it maintains its basic shape while changing position. For the most generic definition, the expression should obey the equation A(x,t) = A(x+deltaX, t+deltaT).

#### shanepitts

How can one compute and/or tell if a given expression is a traveling wave? I thought of using the characteristic wave equation δ^(2)ψ/δt^(2)=v^(2)δ^(2)ψ/δx^(2).

An example of an expression would be ψ(x,t)=A(x-t). Should I partially derive this expression twice with respect to x and then to t and see if they match with the wave equation? Moreover, would I be correct in assuming that if they do match then this expression is a traveling wave?

(This is not a homework assignment)

Characteristic wave equation: $$\frac{\partial^2}{\partial t^2}\psi = v^2\frac{\partial^2}{\partial x^2}\psi$$
... just checking.

You can compute if a particular expression is a traveling wave by following the definition of a traveling wave.
i.e. if it holds it's basic shape while changing position - then it's a traveling wave.

If you start out with some expression, then you can prove that it is a wave by seeing if it is a solution to the wave equation.
To see if it is a traveling wave - you need to use the definition of a traveling wave.

Similarly you can check to see if your example is a general form of a traveling wave.

shanepitts
I think the OP question is more generic. That is, how do you mathematically establish whether a given solution is a traveling wave? There are a lot of solutions to the above wave equation that I wouldn't call a wave

I think the most generic definition of a traveling wave is if it (within certain bounds) obeys the equation A(x,t) = A(x+deltaX, t+deltaT). Meaning, after a time of deltaT the shape is the same but has moved by deltaX.
For the regular sine function (the most basic solution to the above mentioned wave equation) this equation for example should hold, as the sine function "travels" over time.

shanepitts

## 1. What is a travelling wave?

A travelling wave is a type of wave that propagates through a medium by transferring energy from one point to another. It is characterized by its ability to maintain its shape and speed as it moves through the medium.

## 2. How is a travelling wave different from other types of waves?

A travelling wave is different from other types of waves, such as standing waves or transverse waves, because it moves through a medium rather than staying in one place or oscillating up and down.

## 3. What are the main properties of a travelling wave?

The main properties of a travelling wave include its amplitude, wavelength, frequency, and speed. Amplitude is the maximum displacement of the wave, wavelength is the distance between two consecutive points with the same displacement, frequency is the number of complete cycles per second, and speed is the rate at which the wave moves through the medium.

## 4. How is a travelling wave described mathematically?

A travelling wave can be described mathematically using a sinusoidal function, such as the sine or cosine function. The general equation for a travelling wave is y(x,t) = A sin(kx - ωt), where A is the amplitude, k is the wave number, x is the position, ω is the angular frequency, and t is the time.

## 5. What are some examples of travelling waves?

Some common examples of travelling waves include ocean waves, sound waves, and electromagnetic waves. These waves all propagate through a medium, whether it be water, air, or a vacuum, and transfer energy from one point to another.