The equation of a travelling wave

Writing down a travelling wave

Narrated video. Transcript:

Move one end of a long string up and down in simple harmonic motion. A sinusoidal wave travels along the string towards plus x.

To describe it completely, we need the displacement y of every element of the string, at every position x, at every time t.

First freeze time at t equals zero. The snapshot is a sine curve: y equals a sin of k x plus φ.

Now let time run. The whole curve slides to the right without changing shape. To slide it, we subtract ω t inside the sine.

So a harmonic wave travelling towards plus x is y of x and t equals a sin of k x minus ω t plus φ.

a is the amplitude: the largest displacement of any element from its mean position. It is always positive. Notice it is the height of a crest, not the crest-to-trough distance.

The whole quantity k x minus ω t plus φ is called the phase. It tells us where each element is in its cycle of up-and-down motion at that moment.

At x equals zero and t equals zero, the phase is just φ. That is why φ is called the initial phase angle. Choosing a different origin simply changes φ.

Flip the sign in front of ω t, and the curve slides the other way. y equals a sin of k x plus ω t plus φ is a wave travelling towards minus x.

Wavelength, period and frequency

Narrated video. Transcript:

Freeze the wave again. The wavelength λ is the smallest distance between two points in the same phase: crest to the next crest, or trough to the next trough.

At t equals zero, y equals a sin k x. Move along by one wavelength and the shape must repeat, so k λ equals 2π. This gives k equals 2π over λ.

k is called the angular wave number, or propagation constant. Its SI unit is radian per metre.

Now do the opposite. Fix your eye on one element, at x equals zero, and watch it in time. y equals a sin of minus ω t. It simply moves up and down in simple harmonic motion.

It returns to the same state after one period T, so ω T equals 2π. The angular frequency is ω equals 2π over T, in radians per second.

The frequency ν is the number of oscillations per second: ν equals 1 over T, which is ω over 2π. Its unit is the hertz.

Here is a worked example. A wave on a string is y equals 0.005 sin of 80 x minus 3 t, in SI units. The amplitude is 0.005 metres, or 5 millimetres.

k equals 80 radians per metre, so λ equals 2π over 80, which is 0.0785 metres, about 7.85 centimetres.

ω equals 3 radians per second, so the period T equals 2π over 3, about 2.09 seconds, and the frequency is about 0.48 hertz.

Where is the element at x equals 30 centimetres when t equals 20 seconds? The phase is 80 times 0.3 minus 3 times 20, which is minus 36 radians. The sine of that is about 0.99, so y is about 4.96 millimetres, almost at a crest.

Try it: build a travelling wave

Interactive simulation. Things to try:

• The λ bracket rides along between two crests. Change λ and watch k = 2π/λ in the readouts.
• The lower graph is the marked particle’s y against t: a sine curve of period T = 1/ν.
• Change φ: the whole wave shifts, and the marked particle starts at a different point in its cycle.
• Tick “Travel towards −x”. This is y = a sin(kx + ωt + φ).

Half a wavelength apart

Two particles on a string carrying a harmonic wave are exactly λ/2 apart. How do their motions compare?

1. They move exactly in step: phase difference 0
2. They are a quarter cycle apart: phase difference π/2
3. They always move in opposite directions: phase difference π
4. It depends on the amplitude of the wave

Read the equation

A wave is described by y = 0.03 sin(4πx − 50πt), with x and y in metres and t in seconds. What are its wavelength and frequency?

1. λ = 0.5 m, ν = 25 Hz
2. λ = 4π m, ν = 50π Hz
3. λ = 2 m, ν = 0.04 Hz
4. λ = 0.25 m, ν = 50 Hz

Summary

• Wave towards +x: y(x, t) = a sin(kx − ωt + φ)
• Wave towards −x: y(x, t) = a sin(kx + ωt + φ)
• Angular wave number: k = 2π/λ
• Angular frequency and frequency: ω = 2π/T = 2πν

• a is the amplitude (always positive); kx − ωt + φ is the phase; φ is the initial phase angle.
• Wavelength λ: the smallest distance between two points in the same phase. Points λ/2 apart move in opposite directions.
• Every element of the medium performs SHM with period T, the same for all elements.
• Read k and ω from the equation, then use λ = 2π/k and ν = ω/2π.