KAHIBARO
Table of Contents
What a wave function means in this chapter
In the study of mechanical waves, a wave function is a mathematical expression that tells us how the disturbance of the medium changes with position and time. It is the tool that describes the shape of a wave and how that shape moves.
For a wave on a string, the disturbance might be the vertical displacement of the string from its resting position. For a sound wave, the disturbance could be the pressure variation in air. In each case, the wave function gives the value of that disturbance at any point and at any instant.
If we use the symbol $y(x,t)$, then $y$ is the displacement, $x$ is position, and $t$ is time. The notation means that the displacement depends on both where you look and when you look.
A wave function is not the wave itself. It is a mathematical description of the disturbance produced by the wave.
Position and time dependence
A mechanical wave changes in space and in time. If you freeze time, the wave function shows the shape of the wave along the medium. If you fix a position, the wave function shows how that point moves as time passes.
For example, suppose at time $t = 0$ the shape of a pulse on a string is known. As the pulse travels, the same shape may appear at different positions later. The wave function keeps track of this motion.
A very simple traveling pulse can be written as
$$
y(x,t) = f(x - vt)
$$
where $f$ represents the shape of the pulse and $v$ is the wave speed.
This means that the whole shape moves to the right with speed $v$. If instead the pulse moves to the left, we write
$$
y(x,t) = f(x + vt)
$$
For traveling waves,
$$
y(x,t) = f(x - vt)
$$
describes motion in the positive $x$ direction, and
$$
y(x,t) = f(x + vt)
$$
describes motion in the negative $x$ direction.
A sinusoidal wave function
One of the most important wave functions is the sinusoidal wave, because many wave phenomena can be modeled or approximated by sine or cosine functions.
A common form is
$$
y(x,t) = A \sin(kx - \omega t)
$$
Here, $A$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency.
This equation tells us the displacement of the medium at every position $x$ and time $t$. The wave has a repeating pattern in space and a repeating motion in time.
Another equally valid form is
$$
y(x,t) = A \cos(kx - \omega t)
$$
The choice between sine and cosine depends on the starting condition of the wave.
Meaning of the symbols
The symbols in the wave function each have a clear physical meaning.
| Symbol | Meaning | Typical unit |
|---|---|---|
| $y(x,t)$ | Disturbance or displacement | m |
| $A$ | Amplitude | m |
| $x$ | Position | m |
| $t$ | Time | s |
| $k$ | Wave number | rad/m |
| $\omega$ | Angular frequency | rad/s |
The wave number and angular frequency are related to wavelength $\lambda$ and frequency $f$ by
$$
k = \frac{2\pi}{\lambda}
$$
and
$$
\omega = 2\pi f
$$
Using these, the wave speed is
$$
v = \frac{\omega}{k}
$$
For a sinusoidal traveling wave,
$$
y(x,t) = A \sin(kx - \omega t)
$$
with
$$
k = \frac{2\pi}{\lambda}, \qquad \omega = 2\pi f, \qquad v = \frac{\omega}{k} = f\lambda
$$
Right-moving and left-moving sinusoidal waves
The sign inside the wave function tells the direction of motion.
A wave moving to the right can be written as
$$
y(x,t) = A \sin(kx - \omega t)
$$
A wave moving to the left can be written as
$$
y(x,t) = A \sin(kx + \omega t)
$$
This sign matters because it tells us how points of equal phase move through space.
If the quantity inside the sine function stays constant, then the wave crest keeps its identity as it moves. For the right-moving case,
$$
kx - \omega t = \text{constant}
$$
which gives
$$
x = \frac{\omega}{k} t + \text{constant}
$$
So the crest moves in the positive $x$ direction with speed $v = \omega/k$.
Reading a wave function
A wave function lets us answer practical questions. If we know the equation
$$
y(x,t) = 0.02 \sin(4x - 10t)
$$
then we can read several facts from it immediately.
The amplitude is $A = 0.02 \, \text{m}$. The wave number is $k = 4 \, \text{rad/m}$. The angular frequency is $\omega = 10 \, \text{rad/s}$.
From these we find
$$
\lambda = \frac{2\pi}{4} = \frac{\pi}{2} \, \text{m}
$$
$$
f = \frac{10}{2\pi} = \frac{5}{\pi} \, \text{Hz}
$$
and
$$
v = \frac{\omega}{k} = \frac{10}{4} = 2.5 \, \text{m/s}
$$
Because the form is $kx - \omega t$, the wave moves to the right.
Snapshot and time history
There are two useful ways to interpret a wave function.
A snapshot means looking at the wave at one fixed time. Then $t$ is treated as constant, and the function shows how displacement varies with position.
A time history means watching one fixed point in the medium. Then $x$ is treated as constant, and the function shows how displacement varies with time.
For the same wave function, these two views are different slices of the same mathematical description.
Example of a pulse
Not all wave functions are sinusoidal. A wave can have almost any shape.
For instance, a triangular pulse, a rectangular pulse, or a localized bump can all be represented by some function $f(x \mp vt)$. The important idea is that the shape is carried along through the medium.
If a pulse at $t = 0$ is
$$
y(x,0) = f(x)
$$
then after time $t$ it becomes
$$
y(x,t) = f(x - vt)
$$
for motion to the right.
This is a very general description of a traveling disturbance.
Visualizing a traveling wave
This drawing shows a snapshot of the wave at one instant. The height of the curve gives the displacement $y$ at each position $x$. The maximum height is the amplitude $A$, and the horizontal repetition distance is the wavelength $\lambda$.
Initial phase
A more general sinusoidal wave function includes an initial phase constant:
$$
y(x,t) = A \sin(kx - \omega t + \phi)
$$
The constant $\phi$ shifts the wave horizontally or changes the starting point of oscillation. It does not change the amplitude, wavelength, frequency, or speed. It only changes where in its cycle the wave begins.
This is useful when the wave does not start exactly at zero displacement when $x = 0$ and $t = 0$.
Units and consistency
The quantity inside a sine or cosine function must be dimensionless. That is why $kx$ and $\omega t$ must each have no units overall.
Since $x$ has units of meters, $k$ must have units of radians per meter. Since $t$ has units of seconds, $\omega$ must have units of radians per second.
This is an important consistency check when reading or building a wave function.
In a wave function such as
$$
y(x,t) = A \sin(kx - \omega t + \phi)
$$
the argument of the sine must be dimensionless.
What the wave function does and does not tell us
The wave function tells us the disturbance at each place and time. It describes the motion of the medium caused by the wave. It does not by itself explain why the wave exists or how the medium produces it. Those ideas belong to the broader discussion of wave dynamics and the wave equation.
In this chapter, the key point is that a wave function is the mathematical language of wave motion. Once the function is known, the wave can be analyzed quantitatively.
Summary formulas
| Wave type | Wave function |
|---|---|
| Right-moving pulse | $y(x,t) = f(x - vt)$ |
| Left-moving pulse | $y(x,t) = f(x + vt)$ |
| Right-moving sinusoidal wave | $y(x,t) = A \sin(kx - \omega t + \phi)$ |
| Left-moving sinusoidal wave | $y(x,t) = A \sin(kx + \omega t + \phi)$ |
Core ideas of the wave function:
$$
y(x,t) = \text{disturbance at position } x \text{ and time } t
$$
$$
y(x,t) = f(x - vt) \text{ or } f(x + vt)
$$
$$
y(x,t) = A \sin(kx \mp \omega t + \phi)
$$
with
$$
k = \frac{2\pi}{\lambda}, \qquad \omega = 2\pi f, \qquad v = \frac{\omega}{k}
$$