Wave Equation

The wave equation tells you how a wave changes in space and time. In the standard one-dimensional model with constant wave speed $v$, it is

$$\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial x^2}$$

Here, $u(x,t)$ is the wave quantity. Depending on the problem, it could mean the displacement of a string, a small pressure change in sound, or another wave amplitude.

What The Wave Equation Means

The left side measures how the wave value accelerates in time at one point. The right side measures how curved the wave shape is in space.

That link is the main idea. If a part of the wave is curved, that curvature drives how the disturbance evolves, which is why the shape can travel.

When The 1D Wave Equation Applies

The equation above is not a universal formula for every wave. It is the common $1$D constant-speed form, so the conditions matter.

It works well for small transverse waves on an idealized stretched string and for simple sound models in a uniform medium. If the medium changes with position, the geometry is more complicated, or the motion is not well approximated as one-dimensional, the equation usually changes too.

Worked Example: Check A Traveling Sine Wave

Take

$$u(x,t) = A \sin(kx - \omega t)$$

This describes a right-moving sinusoidal wave with amplitude $A$, wave number $k$, and angular frequency $\omega$.

Differentiate twice with respect to time:

$$\frac{\partial^2 u}{\partial t^2} = -\omega^2 A \sin(kx - \omega t)$$

Differentiate twice with respect to position:

$$\frac{\partial^2 u}{\partial x^2} = -k^2 A \sin(kx - \omega t)$$

Now put both results into the wave equation:

$$\frac{\partial^2 u}{\partial t^2} = v^2 \frac{\partial^2 u}{\partial x^2}$$

That gives

$$-\omega^2 A \sin(kx - \omega t) = v^2 \left(-k^2 A \sin(kx - \omega t)\right)$$

So the sine wave is a solution only if

$$\omega^2 = v^2 k^2$$

For a positive wave speed, this becomes

$$v = \frac{\omega}{k}$$

This is the useful check to remember: a traveling sine wave does satisfy the wave equation, but only when $\omega$, $k$, and $v$ match correctly.

Common Wave Equation Mistakes

• Treating the simple form as universal. It assumes a constant wave speed in a suitable $1$D model.
• Forgetting that $u$ depends on both position and time. That is why partial derivatives appear.
• Mixing up the wave's motion with the material's motion. On a string, the pattern travels along the string while each point mainly moves up and down.
• Assuming any sine wave works automatically. In this model, the parameters must satisfy $v = \omega/k$.

Where The Wave Equation Is Used

The wave equation appears whenever a small disturbance travels through a medium or field in a wave-like way. Introductory physics uses it for vibrating strings and sound, and related forms appear in electromagnetism and other parts of physics.

Try A Similar Check

Take

$$u(x,t) = 3 \sin(2x - 6t)$$

Differentiate twice with respect to $x$ and twice with respect to $t$, then test whether it satisfies the wave equation with $v = 3$. If you want to try your own version after that, change the $6$ to another value and see which wave speed makes the equation work.