Progressive Waves and Their Equation – Propagation of Mechanical Waves (Senior 5 Physics) | Ganzaa

Propagation of Mechanical Waves

Lessons

Senior 5 · Physics

Propagation of Mechanical Waves

4 lessons

Lesson 2 of 4

Progressive Waves and Their Equation

25 min

The stadium wave

When a crowd in a stadium makes a "wave", each spectator does something very simple: stands up and sits down once. Nobody runs round the stadium. The wave travels because each person copies the neighbour on one side a moment later. Physicists describe every progressive wave in the same way: a particle further from the source does exactly what the source did, but later. Turning that sentence into an equation gives a single formula for the displacement of any particle at any time.

Deriving the equation of a progressive wave

Let the source at $x = 0$ oscillate with SHM: $y = A\sin\omega t$ . The wave travels in the $+ x$ direction at speed $v$ , so the disturbance takes a time $\frac{x}{v}$ to reach a particle at position $x$ . That particle repeats the source's motion after this delay:

y(x, t) = A\sin\omega\left(t - \frac{x}{v}\right) = A\sin\left(\omega t - kx\right)

where the wave number $k = \frac{\omega}{v}$ . Since $\omega = 2\pi f$ and $v = f\lambda$ ,

k = \frac{2\pi f}{f\lambda} = \frac{2\pi}{\lambda} \qquad\text{and}\qquad y = A\sin 2\pi\left(\frac{t}{T} - \frac{x}{\lambda}\right)

A wave travelling in the $- x$ direction has $y = A\sin(\omega t + kx)$ : a particle at positive $x$ is reached earlier.
The wave speed is $v = \frac{\omega}{k}$ : read $\omega$ and $k$ from the coefficients of $t$ and $x$ .
Some books write $y = A\sin(kx - \omega t)$ . This is the same wave travelling in $+ x$ , with the opposite sign of $y$ : the two forms differ only by a phase of $π \pi$ .

The equation contains two graphs. Fix $t$ and you get a snapshot, $y$ against $x$ , repeating every $\lambda$ . Fix $x$ and you get the history of one particle, $y$ against $t$ , repeating every $T$ . Check which variable is on the horizontal axis before reading a wavelength or a period from a graph.

Definition

The equation of a progressive wave travelling in the $+ x$ direction is $y = A\sin(\omega t - kx)$ , where $A$ is the amplitude, $\omega = 2\pi f$ the angular frequency and $k = \frac{2\pi}{\lambda}$ the wave number (in $\mathrm{rad\,m^{-1}}$ ). The wave speed is $v = \frac{\omega}{k}$ . Example: $y = 0.010\sin(100\pi t - 2\pi x)$ has $f = 50\,\mathrm{Hz}$ , $\lambda = 1.0\,\mathrm{m}$ and $v = 50\,\mathrm{m\,s^{-1}}$ .

Worked example 1: reading the equation

A wave on a rope is described by $y = 0.020\sin(40\pi t - 0.50\pi x)$ , in SI units. Find (a) its amplitude, frequency, wavelength, speed and direction; (b) the greatest speed of a rope particle; (c) the displacement of the particle at $x = 1.0\,\mathrm{m}$ when $t = 0.010\,\mathrm{s}$ .

Solution

(a) $A = 0.020\,\mathrm{m}$ . $\omega = 40\pi$ , so $f = 20\,\mathrm{Hz}$ . $k = 0.50\pi$ , so $\lambda = \frac{2\pi}{0.50\pi} = 4.0\,\mathrm{m}$ . $v = f\lambda = 80\,\mathrm{m\,s^{-1}}$ (or $\frac{\omega}{k} = \frac{40\pi}{0.50\pi} = 80$ ). The signs of the $t$ and $x$ terms differ, so it travels in the $+ x$ direction.

(b) $u_{\max} = A\omega = 0.020 \times 40\pi = 2.5\,\mathrm{m\,s^{-1}}$ , much less than the wave speed of $80\,\mathrm{m\,s^{-1}}$ .

(c) Phase $= 40\pi(0.010) - 0.50\pi(1.0) = 0.40\pi - 0.50\pi = -0.10\pi$ . $y = 0.020\sin(-0.10\pi) = 0.020 \times (-0.309) = -6.2 \times 10^{-3}\,\mathrm{m}$ .

Phase difference between two points

At the same instant, two particles a distance $\Delta x$ apart have phases $\omega t - kx_1$ and $\omega t - kx_2$ . Their phase difference is

\Delta\phi = k\,\Delta x = \frac{2\pi\,\Delta x}{\lambda}

$\Delta x = \lambda, 2\lambda, 3\lambda, \ldots$ : $\Delta\phi = 2\pi, 4\pi, \ldots$ , the particles move in phase.
$\Delta x = \frac{\lambda}{2}, \frac{3\lambda}{2}, \ldots$ : $\Delta\phi = \pi, 3\pi, \ldots$ , the particles move in antiphase.

Similarly, one particle at two times $\Delta t$ apart has a phase difference $\omega\,\Delta t = \frac{2\pi\,\Delta t}{T}$ .

Particle velocity and wave velocity

Differentiating with respect to $t$ at fixed $x$ gives the velocity of a particle of the medium:

u = \frac{\partial y}{\partial t} = A\omega\cos(\omega t - kx), \qquad u_{\max} = A\omega

This is a completely different velocity from the wave speed $v = \frac{\omega}{k}$ : $u$ is how fast a bit of rope moves up and down, and it changes all the time; $v$ is how fast the pattern moves along, and it is constant. A neat way to find each particle's direction is to sketch the wave a moment later, shifted slightly in the direction of travel: every particle moves towards the new curve.

Energy carried by a wave

Each particle performs SHM, so its energy is proportional to $A^2\omega^2$ (Unit 1). The power carried along a string of mass per unit length $μ \mu$ is $P = \frac{1}{2}\mu v\omega^2A^2$ , and the intensity (power per unit area) of any mechanical wave is likewise proportional to $A^{2} f^{2}$ . For a point source sending sound equally in all directions, the power spreads over a sphere of area $4\pi r^2$ , so $I \propto \frac{1}{r^2}$ and the amplitude falls as $\frac{1}{r}$ .

Worked example 2: phase difference

For the wave in Worked example 1 ( $\lambda = 4.0\,\mathrm{m}$ ), find (a) the phase difference between particles 0.50 m apart, (b) the smallest separation of two particles moving in antiphase.

Solution

(a) $\Delta\phi = \frac{2\pi \times 0.50}{4.0} = \frac{\pi}{4}\,\mathrm{rad}$ (45°).

(b) Antiphase needs $\Delta\phi = \pi$ : $\Delta x = \frac{\lambda}{2} = 2.0\,\mathrm{m}$ .

Worked example 3: writing the equation

A sound wave of amplitude $3.0 \times 10^{-6}\,\mathrm{m}$ and frequency 500 Hz travels at $340\,\mathrm{m\,s^{-1}}$ in the $- x$ direction. Write its equation, taking $y = 0$ and increasing at $x = 0$ , $t = 0$ .

Solution

$\omega = 2\pi \times 500 = 1000\pi = 3.14 \times 10^3\,\mathrm{rad\,s^{-1}}$ ; $k = \frac{\omega}{v} = \frac{1000\pi}{340} = 9.24\,\mathrm{rad\,m^{-1}}$ . Travelling in $- x$ : $y = 3.0 \times 10^{-6}\sin(1000\pi t + 9.24x)$ . Here $y$ is the displacement of the air layers along the direction of travel, since sound is longitudinal; the same equation describes both kinds of wave.

Warning

Common mistakes. (1) Reading $\lambda$ from a $y$ – $t$ graph, or $T$ from a $y$ – $x$ graph. (2) Writing $k = \lambda$ or $k = \frac{\lambda}{2\pi}$ : $k = \frac{2\pi}{\lambda}$ . (3) Equating the particle speed $A\omega$ with the wave speed $\frac{\omega}{k}$ . (4) Getting the direction wrong: opposite signs of $\omega t$ and $k x$ mean $+ x$ ; the same sign means $- x$ . (5) A calculator in degrees: the phase is in radians.

Tip

Think of $\omega t - kx$ as "time-phase minus distance-phase". Every metre further from the source puts a particle $k$ radians behind; every second later puts it $\omega$ radians ahead. The pattern stays still for an observer who moves so that the two cancel, at $\frac{dx}{dt} = \frac{\omega}{k} = v$ .

The equation of a wave (Khan Academy). Watch how the snapshot and the motion of one point are combined into one formula. Khan Academy writes the phase as (kx − ωt) form; as this lesson explains, it describes the same wave travelling in +x.

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Practice

A wave is $y = 0.050\sin(10\pi t - 2\pi x)$ (SI). Find its amplitude, frequency, wavelength, speed and direction.
Two points on a wave of wavelength 60 cm are 15 cm apart. Find their phase difference.
Sound of frequency 660 Hz travels at $330\,\mathrm{m\,s^{-1}}$ . Find the wavelength and the smallest distance between two points vibrating in antiphase.
Write the equation of a wave of amplitude 2.0 cm, period 0.10 s and wavelength 0.40 m travelling in the $+ x$ direction (take $y = 0$ at $x = 0, t = 0$ , increasing).
For the wave in question 4, find the greatest particle speed and the wave speed.
Show that $y = A\sin(\omega t - kx)$ has the same value at $(x + v\Delta t,\ t + \Delta t)$ as at $(x, t)$ when $v = \frac{\omega}{k}$ . What does this show?
The amplitude of a sound wave from a point source is halved. By what factor does its intensity change? If the listener moves from 2.0 m to 8.0 m from the source, by what factor does the intensity change?
Ishimwe says: "In a wave on a rope, each bit of rope moves along at the wave speed." Evaluate this, using the numbers of Worked example 1.

Show answersHide answers

$A = 0.050\,\mathrm{m}$ ; $f = 5.0\,\mathrm{Hz}$ ; $\lambda = \frac{2\pi}{2\pi} = 1.0\,\mathrm{m}$ ; $v = 5.0\,\mathrm{m\,s^{-1}}$ ; $+ x$ direction.
$\Delta\phi = \frac{2\pi \times 15}{60} = \frac{\pi}{2}\,\mathrm{rad}$ .
$\lambda = \frac{330}{660} = 0.50\,\mathrm{m}$ ; antiphase at $\frac{\lambda}{2} = 0.25\,\mathrm{m}$ .
$\omega = \frac{2\pi}{0.10} = 20\pi$ , $k = \frac{2\pi}{0.40} = 5\pi$ : $y = 0.020\sin(20\pi t - 5\pi x)$ .
$u_{\max} = 0.020 \times 20\pi = 1.26\,\mathrm{m\,s^{-1}}$ ; $v = \frac{20\pi}{5\pi} = 4.0\,\mathrm{m\,s^{-1}}$ .
New phase $\omega(t + \Delta t) - k(x + v\Delta t) = \omega t - kx + (\omega - kv)\Delta t = \omega t - kx$ . The whole pattern moves forward a distance $v\Delta t$ in time $\Delta t$ : it travels at speed $v$ .
$I \propto A^2$ : a quarter. $I \propto \frac{1}{r^2}$ : $\left(\frac{2.0}{8.0}\right)^2 = \frac{1}{16}$ .
Wrong. The bits of rope oscillate up and down about fixed positions, with speed varying from 0 to $A\omega = 2.5\,\mathrm{m\,s^{-1}}$ ; it is the pattern that moves along at $80\,\mathrm{m\,s^{-1}}$ . The two speeds have different directions and sizes.

Key points

$y = A\sin(\omega t - kx)$ travels in $+ x$ ; $y = A\sin(\omega t + kx)$ in $- x$ ; $k = \frac{2\pi}{\lambda}$ , $v = \frac{\omega}{k}$ .
Phase difference between points: $\Delta\phi = \frac{2\pi\Delta x}{\lambda}$ .
Particle velocity $u = A\omega\cos(\omega t - kx)$ , $u_{\max} = A\omega$ , is not the wave speed.
Intensity $\propto A^2f^2$ ; from a point source $I \propto \frac{1}{r^2}$ .

Sources and credits

16.2 Mathematics of Waves and 16.4 Energy and Power of a Wave, from University Physics Volume 1 by William Moebs, Samuel J. Ling and Jeff Sanny, OpenStax, licensed under CC BY 4.0. Adapted by Ganzaa.
Video: The equation of a wave, Khan Academy, embedded from YouTube.

The two diagrams, the worked examples and the practice exercises are written by Ganzaa.

1The stadium wave

2Deriving the equation of a progressive wave

3Phase difference between two points

4Particle velocity and wave velocity

5Energy carried by a wave

The stadium wave

Deriving the equation of a progressive wave

Phase difference between two points

Particle velocity and wave velocity

Energy carried by a wave