KAHIBARO
Table of Contents
A Rule That Waves Must Follow
A wave is not just any moving shape. In physics, a wave must evolve in space and time in a very specific way. The mathematical rule that connects how the wave changes with position and how it changes with time is called the wave equation.
The wave equation is important because it describes a huge range of physical systems, such as waves on strings, sound waves in air, and even electromagnetic waves in more advanced physics. In this chapter, we focus on the basic mathematical form of the equation and what it means.
The Basic One-Dimensional Wave Equation
For a wave traveling along one spatial direction, usually called the $x$ direction, we often describe the disturbance by a function $y(x,t)$. This could mean the displacement of a string from its rest position at position $x$ and time $t$.
The standard one-dimensional wave equation is
$$
\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}
$$
where $v$ is the wave speed.
This equation says that the curvature of the wave shape in space is related to how the displacement accelerates in time. If the shape is strongly curved at some place, that part of the medium tends to accelerate more.
The one-dimensional wave equation is
$$
\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}
$$
A function represents a traveling wave with speed $v$ if it satisfies this equation.
Meaning of the Derivatives
The symbol $\partial$ is used because $y$ depends on more than one variable, position and time.
The quantity
$$
\frac{\partial^2 y}{\partial x^2}
$$
measures how curved the wave is as a function of position. A straight section has zero spatial curvature. A peak or trough has nonzero curvature.
The quantity
$$
\frac{\partial^2 y}{\partial t^2}
$$
measures how the displacement changes in time with acceleration at a fixed position.
So the wave equation is telling us that wave motion is not random. The time evolution at each point is controlled by the shape of the wave around that point.
A Traveling Wave Solution
A very common traveling wave has the form
$$
y(x,t) = f(x - vt)
$$
This represents a shape $f$ moving to the right with speed $v$ without changing form.
A left-moving wave has the form
$$
y(x,t) = f(x + vt)
$$
These both satisfy the wave equation.
A sinusoidal example is
$$
y(x,t) = A\sin(kx - \omega t)
$$
For this to satisfy the wave equation, the quantities $k$, $\omega$, and $v$ must obey
$$
v = \frac{\omega}{k}
$$
This connects the wave equation to the usual sinusoidal wave form.
Any function of the form
$$
y(x,t) = f(x - vt)
$$
or
$$
y(x,t) = f(x + vt)
$$
satisfies the one-dimensional wave equation and represents a traveling wave of speed $v$.
Right-Moving and Left-Moving Waves
The sign inside the function tells the direction of motion.
If the wave is
$$
y(x,t) = f(x - vt)
$$
then as time increases, the same value of the argument is found at larger $x$. The shape moves to the right.
If the wave is
$$
y(x,t) = f(x + vt)
$$
then the shape moves to the left.
This is an important result because the wave equation itself allows motion in both directions.
Checking a Sinusoidal Wave
Let us test the sinusoidal wave
$$
y(x,t) = A\sin(kx - \omega t)
$$
Take the second derivative with respect to position:
$$
\frac{\partial^2 y}{\partial x^2} = -k^2 A\sin(kx - \omega t)
$$
Take the second derivative with respect to time:
$$
\frac{\partial^2 y}{\partial t^2} = -\omega^2 A\sin(kx - \omega t)
$$
Substituting into the wave equation gives
$$
-k^2 A\sin(kx - \omega t) = \frac{1}{v^2}\left(-\omega^2 A\sin(kx - \omega t)\right)
$$
which becomes
$$
k^2 = \frac{\omega^2}{v^2}
$$
so
$$
v = \frac{\omega}{k}
$$
This shows that the sinusoidal wave is a solution only when the wave speed has the correct relation to angular frequency and wave number.
General Solution
The most general solution of the one-dimensional wave equation is a sum of a right-moving part and a left-moving part:
$$
y(x,t) = f(x-vt) + g(x+vt)
$$
This is very useful. It means that wave motion in one dimension can be built from two traveling pieces, one moving right and one moving left.
This is the mathematical basis for many situations where waves reflect and overlap.
The general one-dimensional solution is
$$
y(x,t) = f(x-vt) + g(x+vt)
$$
where $f$ and $g$ are arbitrary functions determined by the physical situation.
What Determines the Wave Speed
The wave equation contains the speed $v$, but the equation itself does not tell us what $v$ is. The value of $v$ depends on the physical medium.
For example, for a stretched string,
$$
v = \sqrt{\frac{T}{\mu}}
$$
where $T$ is the tension and $\mu$ is the mass per unit length.
For other media, the form of $v$ is different. The important point here is that once the speed is known, the wave equation predicts how disturbances move.
Wave Equation in More Than One Dimension
In more than one spatial dimension, the idea is similar. The wave equation becomes
$$
\nabla^2 \psi = \frac{1}{v^2}\frac{\partial^2 \psi}{\partial t^2}
$$
Here, $\psi$ is the wave function and $\nabla^2$ is the Laplacian operator, which describes curvature in space.
In three Cartesian coordinates,
$$
\nabla^2 \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2}
$$
This form is used for waves spreading through space, not just along a line.
Physical Interpretation
The wave equation expresses a balance between spatial shape and temporal response. If part of the medium is bent or compressed relative to nearby parts, restoring effects act on it. These restoring effects produce acceleration, and the disturbance moves through the medium.
In this sense, the wave equation is the mathematical translation of a physical idea, local deformation causes local acceleration.
Summary Table
| Quantity | Meaning |
|---|---|
| $y(x,t)$ or $\psi(x,t)$ | Wave disturbance |
| $v$ | Wave speed |
| $\frac{\partial^2 y}{\partial x^2}$ | Spatial curvature |
| $\frac{\partial^2 y}{\partial t^2}$ | Time acceleration of the disturbance |
| $f(x-vt)$ | Right-moving wave |
| $f(x+vt)$ | Left-moving wave |
| $v=\omega/k$ | Speed relation for a sinusoidal wave |
Final Key Ideas
The wave equation is the central mathematical rule for wave motion. It connects the shape of a disturbance in space to its evolution in time. In one dimension, its standard form is
$$
\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}
$$
Its solutions include right-moving and left-moving traveling waves, and sinusoidal waves satisfy it when
$$
v = \frac{\omega}{k}
$$
Key facts to remember:
$$
\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 y}{\partial t^2}
$$
$$
y(x,t) = f(x-vt), \quad y(x,t) = f(x+vt)
$$
$$
y(x,t) = A\sin(kx-\omega t), \quad v=\frac{\omega}{k}
$$
The wave equation is the mathematical condition that a disturbance must satisfy to propagate as a wave with speed $v$.