The equation of a travelling wave

A formula for a moving pattern

Picture a long rope that carries a smooth wave, made by a hand that moves up and down in simple harmonic motion. We want one formula that tells us the height $y$ of the rope at any place $x$ and at any time $t$. For a wave moving towards $+x$, that formula is

Each letter has a meaning.

• $a$ is the amplitude, the largest displacement of a particle from its rest position.
• $kx - \omega t + \phi$ is the phase. It tells us where in its cycle the particle at $x$ is at the time $t$.
• $\phi$ is the phase constant, the phase at $x = 0$ and $t = 0$.
• $k$ is the angular wave number, in rad/m.
• $\omega$ is the angular frequency, in rad/s.

Wavelength and angular wave number

Freeze the wave at one moment, like a photograph. This is a snapshot. Along the rope, the shape repeats every wavelength. So the phase must grow by $2\pi$ when $x$ grows by $\lambda$:

We want $k$, so $k = \dfrac{2\pi}{\lambda}$. A wave of wavelength $0.5\ \text{m}$ has $k = \dfrac{2\pi}{0.5\ \text{m}} = 12.6\ \text{rad/m}$.

Period and angular frequency

Now watch one particle of the rope, at a fixed $x$. It moves up and down in simple harmonic motion. A graph of its $y$ against time is a history graph. The motion repeats every period, so the phase must change by $2\pi$ in the time $T$:

We want $\omega$, so $\omega = \dfrac{2\pi}{T} = 2\pi f$.

The speed of the wave

Follow one crest as it moves. A crest is a place of fixed phase. So for the crest, $kx - \omega t$ stays the same. As $t$ grows, $x$ must grow too, to keep the phase fixed. The crest moves a distance $\Delta x$ in the time $\Delta t$, with $k\Delta x = \omega\Delta t$. So the wave speed is

What if the formula has $kx + \omega t$? Then $x$ must fall as $t$ grows, to keep the phase fixed. The wave moves towards $-x$. With $kx - \omega t$ it moves towards $+x$.

Two different speeds

Do not mix up the speed of the wave with the speed of a particle. The wave speed $v = \omega/k$ is the speed of the pattern along the rope. It is the same everywhere, and it does not change with time.

A particle of the rope moves up and down. Its velocity is the rate of change of $y$ at a fixed $x$:

It changes all the time. Its largest value is $a\omega$, when the particle passes its rest position. At a crest or a trough the particle stops for an instant.

Worked example. A wave is $y = 0.03\sin(2.5x - 40t)$, with $x$ and $y$ in metres and $t$ in seconds. Find its wavelength, frequency, speed and the largest speed of a particle.

From the formula, $a = 0.03\ \text{m}$, $k = 2.5\ \text{rad/m}$ and $\omega = 40\ \text{rad/s}$.

• Wavelength: $\lambda = \dfrac{2\pi}{2.5\ \text{rad/m}} = 2.51\ \text{m}$.
• Frequency: $f = \dfrac{40\ \text{rad/s}}{2\pi} = 6.37\ \text{Hz}$.
• Speed: $v = \dfrac{40\ \text{rad/s}}{2.5\ \text{rad/m}} = 16\ \text{m/s}$, towards $+x$.
• Largest particle speed: $a\omega = 0.03\ \text{m} \times 40\ \text{rad/s} = 1.2\ \text{m/s}$.

So the pattern races along at $16\ \text{m/s}$, while each particle never moves faster than $1.2\ \text{m/s}$.

Practise with new numbers

Try it in Python

The program reads $a$, $k$ and $\omega$ from the formula of the worked example and works out everything else.

Common mistakes

• Taking $k$ as the wavelength. $k = 2\pi/\lambda$, in rad/m. A larger $k$ means a shorter wavelength.
• Taking the particle speed as the wave speed. The wave speed is $\omega/k$. The particle speed is at most $a\omega$, and it keeps changing.
• Getting the direction wrong. $kx - \omega t$ moves towards $+x$. $kx + \omega t$ moves towards $-x$.
• Using degrees. The phase $kx - \omega t$ is in radians.

Remember these

• $y = a\sin(kx - \omega t + \phi)$ for a wave moving towards $+x$.
• $k = \dfrac{2\pi}{\lambda}$ and $\omega = \dfrac{2\pi}{T} = 2\pi f$.
• $v = \dfrac{\omega}{k} = f\lambda$.
• A snapshot shows $y$ against $x$. A history graph shows $y$ against $t$ for one particle.
• Largest particle speed: $a\omega$. It is not the wave speed.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A wave is $y = 0.05\sin(\pi x - 10\pi t)$, in SI units. Find its amplitude, wavelength and frequency.

2. Find the speed of the wave in question 1, and say which way it moves.

3. A wave has a wavelength of $4\ \text{m}$. What is the phase difference between two points of the medium $1\ \text{m}$ apart along the wave?

4. At the instant a crest passes a particle of a rope, what is the velocity of that particle?