Types and Equations of Waves

Q: How do you find the direction of a wave from its equation?

Look at the signs of the t and x terms in the phase. If they are opposite, as in sin(omega t minus k x), the wave moves towards positive x. If they are the same, it moves towards negative x. The speed is the coefficient of t divided by the coefficient of x.

Q: What is the relation between wave speed, frequency and wavelength?

Wave speed equals frequency times wavelength, v = f lambda, because the wave advances one wavelength during each period. It can also be written v = omega / k. The frequency is fixed by the source and the speed by the medium, so the wavelength adjusts when the wave changes medium.

Q: How is phase difference related to path difference?

Phase difference equals 2 pi divided by lambda times the path difference. Points one wavelength apart are in phase, points half a wavelength apart are in opposite phase, and points a quarter wavelength apart differ in phase by pi by 2.

Q: How are wave equations asked in JEE Main?

JEE Main usually gives an equation such as y = A sin(at minus bx) and asks for speed, wavelength, frequency, direction, phase difference or maximum particle speed. Questions on the pulse form f(ax + bt), the relation between particle velocity and slope, and speed on a string, root of T over mu, are also common.

PhysicsWavesFor JEE aspirants

A wave is a disturbance that carries energy and momentum from one place to another without carrying the medium along with it. Waves are of different types (mechanical, electromagnetic, matter waves; transverse or longitudinal), and every progressive wave has one master equation, $y = A sin (ω t - k x + ϕ)$ . This page explains the types and equations of waves: wavelength, frequency, wave speed $v = f λ$ , phase difference, particle velocity and the wave equation, the base for all of sound waves. A must-know foundation for JEE Main, JEE Advanced and NEET.

On this page1Types of waves2Transverse vs longitudinal3Wave terms4Equation of a wave5Phase difference6Particle velocity7Wave equation8Wave speed

Key Formulas - Quick Reference

★ Must learnProgressive wave towards $+ x$ : $y = A sin (ω t - k x + ϕ)$ ; towards $- x$ : $y = A sin (ω t + k x + ϕ)$
Wave number $k = \frac{2 π}{λ}$ (rad m $^{- 1}$ ); angular frequency $ω = 2 π f = \frac{2 π}{T}$
★ Must learnWave speed: $v = f λ = \frac{λ}{T} = \frac{ω}{k}$
★ Must learnPhase difference: $Δ ϕ = \frac{2 π}{λ} Δ x$ (path) and $Δ ϕ = ω Δ t = \frac{2 π}{T} Δ t$ (time)
Particle velocity: $v_{p} = \frac{\partial y}{\partial t} = A ω cos (ω t - k x + ϕ)$ ; $(v_{p})_{m a x} = A ω$
★ Must learnParticle velocity from slope: $v_{p} = - v \frac{\partial y}{\partial x}$ (wave towards $+ x$ )
Particle acceleration: $a_{p} = \frac{\partial ^{2} y}{\partial t ^{2}} = - ω^{2} y$
Wave equation: $\frac{\partial ^{2} y}{\partial t ^{2}} = v^{2} \frac{\partial ^{2} y}{\partial x ^{2}}$ ; any $y = f (x \mp v t)$ is a wave
Speed of mechanical waves: string $T / μ$ ; solid rod $Y / ρ$ ; fluid $B / ρ$ ; hanging rope $g x$

1. What Is a Wave?

Drop a stone into still water. Ripples spread outwards, but a leaf floating on the water only bobs up and down; it does not travel with the ripples. What moves outwards is the disturbance (and the energy it carries), not the water. This is the heart of every wave.

★ Must learnWave: a disturbance that travels through space (or a medium), transferring energy and momentum from one point to another without any net transport of the particles of the medium. Each particle only oscillates about its mean position, and passes the disturbance to its neighbour a little later.

The disturbance (a force) is passed from one point to the next.
The energy is passed from one point to the next, with no net displacement of the medium.
The oscillation of each particle is handed on to the neighbouring particle that follows it, a little later.
To keep a wave going continuously, the source must keep creating the disturbance.

In a mechanical wave the particles obey Newton's laws: the forces between neighbouring atoms of the medium pass the motion along, while no atom suffers any net displacement. For a mechanical wave the medium must have two properties: inertia (so particles overshoot and keep oscillating) and elasticity (so a displaced particle is pulled back and drags its neighbour). Sound, for example, is a mechanical wave produced by a vibrating source such as a guitar string, the vocal cords, the prongs of a tuning fork or the diaphragm of a loudspeaker.

1.1 Classification by medium

Type	Needs a medium?	Examples
Mechanical waves	Yes (inertia + elasticity)	Sound, waves on a string, water waves, seismic waves
Electromagnetic waves	No, travel even in vacuum at $c = 3 \times 1 0^{8} m s^{- 1}$	Light, radio waves, microwaves, X-rays
Matter waves	No; associated with moving particles	Electron beams (de Broglie waves, used in electron microscopes)

Figure 1: Types of waves. Sound is a mechanical, longitudinal wave, so it needs a medium; light is electromagnetic and crosses vacuum.

1.2 Classification by dimension

Dimension	Energy spreads along	Example
One-dimensional	a line	Wave on a stretched string
Two-dimensional	a surface	Ripples on water, vibrations of a drum membrane
Three-dimensional	all directions in space	Sound from a bell, light from a bulb

Sound spreading out from a small source is a three-dimensional wave. Far from the source, a small part of the spherical wavefront is nearly flat, so it can be treated as a plane wave travelling along one direction ( $x$ ). Every equation on this page is for such a plane wave.

2. Transverse and Longitudinal Waves

Transverse wave

Particles oscillate perpendicular to the direction of travel. Forms crests and troughs. Needs a medium with rigidity (shear elasticity): solids, a stretched string, the surface of a liquid. Can be polarised.

Longitudinal wave

Particles oscillate along the direction of travel. Forms compressions (layers crowded, pressure above normal) and rarefactions (layers spread, pressure below normal). Travels in solids, liquids and gases. Cannot be polarised.

Figure 2: A slinky carries both kinds of wave (the NCERT demonstration). (a) Moving the end sideways sends a transverse pulse: coils move perpendicular to the travel. (b) Pushing and pulling the end sends a longitudinal pulse: a compression (

C

) followed by a rarefaction (

R

), coils moving along the travel. In both, the pulse moves on; the coils return to where they were.

Figure 3: Transverse wave (crests and troughs) and longitudinal wave (compressions C, rarefactions R). In both, only the disturbance travels; each particle just oscillates about its mean position.

What pulls a string particle back? When a particle of a stretched string is displaced sideways, the string on either side is bent, and the tension gets a component towards the mean position. The particle accelerates back, overshoots because of its kinetic energy, and the cycle repeats. A longitudinal wave needs a restoring mechanism along the direction of travel instead: compression and expansion of the medium.

Why sound in air is longitudinal: a gas has no rigidity. A layer of air cannot drag its neighbour sideways, but it can push it forwards and pull it back by changing its pressure. So in fluids (gases and the bulk of liquids) only longitudinal waves travel. In a solid both types travel, and they travel at different speeds; this is why an earthquake produces a fast P-wave (longitudinal) and a slower S-wave (transverse).

Feature	Transverse	Longitudinal
Particle motion	Perpendicular to wave velocity	Parallel to wave velocity
Pattern	Crests and troughs	Compressions and rarefactions
Distance crest to next crest	$λ$	$λ$ (compression to next compression)
Density / pressure change	No change of density	Density and pressure vary periodically
Media	Solids, strings, liquid surface	Solids, liquids, gases
Polarisation	Possible	Not possible
Examples	String wave, ripples, light (EM)	Sound, a pushed spring (slinky)

Key idea

Transverse or longitudinal is about the particle direction relative to the wave direction. Sound in air: longitudinal, always.

3. Terms Used to Describe a Wave

Term	Meaning	Symbol, SI unit
Amplitude	Maximum displacement of a particle from its mean position	$A$ , m
Wavelength	Distance between two nearest particles in the same phase (crest to crest, or compression to compression); the distance the wave travels in one period	$λ$ , m
Time period	Time for one complete oscillation of a particle	$T$ , s
Frequency	Number of oscillations per second of a particle; set by the source	$f = 1/ T$ , Hz
Angular frequency	$2 π$ times the frequency	$ω = 2 π f$ , rad s $^{- 1}$
Wave number (propagation constant)	Phase change per unit length	$k = 2 π / λ$ , rad m $^{- 1}$
Wave speed	Speed with which the disturbance (a crest) moves; set by the medium	$v$ , m s $^{- 1}$

Figure 4: The

y

x

graph (a photograph of the wave) shows

λ

and

A

; the

y

t

graph (one particle filmed over time) shows

T

. Both are sine curves.

In one period $T$ each particle completes one oscillation, and in the same time the disturbance moves forward by exactly one wavelength. Therefore

★ Must learn

v = \frac{λ}{T} = f λ = \frac{ω}{k}

Exam Trick

Source fixes $f$ , medium fixes $v$ . When a wave passes from one medium into another (sound from air into water, a pulse from a thin string into a thick one), the frequency stays the same and $λ = v / f$ changes in proportion to the speed. Faster medium, longer wavelength.

4. Equation of a Progressive (Travelling) Wave

4.1 Derivation

Let the particle at the origin $x = 0$ oscillate in SHM: $y (0, t) = A sin (ω t + ϕ)$ .
The disturbance travels towards $+ x$ with speed $v$ . It reaches the particle at distance $x$ after a time $x / v$ , so this particle repeats the motion of the origin later by $x / v$ .
Hence its displacement at time $t$ equals the origin's displacement at time $t - x / v$ :
$y (x, t) = A sin [ω (t - \frac{x}{v}) + ϕ]$
Since $ω / v = k$ :
$y (x, t) = A sin (ω t - k x + ϕ)$
This is the equation of a displacement wave travelling towards $+ x$ . The position of the particle from the origin at any time is $x + y$ for a longitudinal wave, where $x$ is its mean position and $y$ its displacement.

Figure 5:

y = A sin (ω t - k x)

t = 0

and

t = T /4

. The whole shape slides

λ /4

towards

+ x

in time

T /4

, so

v = λ / T = f λ

4.2 Equivalent forms

Form	Useful when you are given
$y = A sin (ω t - k x)$	$ω$ and $k$ directly
$y = A sin ω (t - \frac{x}{v})$	the wave speed $v$
$y = A sin \frac{2 π}{λ} (v t - x)$	$λ$ and $v$
$y = A sin 2 π (\frac{t}{T} - \frac{x}{λ})$	$T$ and $λ$ (read them off directly)
$y = A sin (k x - ω t)$	same wave towards $+ x$ , shifted in phase by $π$

Direction of travel. Write the phase as $(ω t \pm k x)$ or $(k x \pm ω t)$ . If $t$ and $x$ have opposite signs the wave moves towards $+ x$ ; if they have the same sign it moves towards $- x$ .

4.3 Two ways to read the equation

Fix

x = x_{0}

: one particle

$y = A sin (ω t - k x_{0} + ϕ)$ is an SHM of amplitude $A$ and period $T$ . Its initial phase is $ϕ - k x_{0}$ : farther particles lag more.

Fix

t = t_{0}

: a snapshot

$y = A sin (ω t_{0} - k x + ϕ)$ is a sine curve in space with period $λ$ : the shape of the whole wave at that instant.

Exam Trick

Read $v$ in one line. For $y = A sin (a t \pm b x)$ , speed $v = \frac{a}{b} = \frac{coefficient of t}{coefficient of x}$ , $λ = \frac{2 π}{b}$ , $f = \frac{a}{2 π}$ . The sign rule then gives the direction. The same trick works for any pulse $y = f (a x + b t)$ : speed $b / a$ .

Figure 6: Read any wave equation in four steps: direction from the signs,

ω

and

k

from the coefficients, then

v = \frac{ω}{k}

λ

f

, and finally the particle values

A ω

and

A ω^{2}

4.4 Choosing the initial phase

If $y (0, 0) \neq = 0$ , the wave needs a phase constant: $y = A sin (ω t - k x + ϕ)$ , so $y (0, 0) = A sin ϕ$ . Every positive value of $sin ϕ$ comes from two angles ( $ϕ$ and $π - ϕ$ ). The second condition is the direction in which the particle at $x = 0$ is moving, $v_{p} (0, 0) = A ω cos ϕ$ . Find it from the snapshot: imagine the whole shape shifted a little along the direction of travel and see whether the point at $x = 0$ rises or falls.

Figure 7:

y (0, 0) = A / 2

allows

ϕ = π /4

3 π /4

. Shift the shape slightly along the wave direction: the particle at

x = 0

goes down, so

cos ϕ < 0

and

ϕ = 3 π /4

(Solved Example 1).

4.5 Any shape: the general wave function

Consider a single pulse on a long, light, uniform string, moving towards $+ x$ with constant speed $v$ . Snapshots at $t$ and $t + Δ t$ show that each particle's displacement depends on where it is and when we look: $y = f (x, t)$ . If the end $x = 0$ moves as $y (0, t) = f (t)$ , the particle at $x$ repeats that motion after a delay $x / v$ :

y (x, t) = f (t - \frac{x}{v}) = g (x - v t)

For a wave towards $- x$ , $y = f (t + \frac{x}{v})$ . The quantity $(x - v t)$ is the phase; a point of fixed phase satisfies $x - v t = constant$ , so $\frac{d x}{d t} = v$ . That is why $v$ is called the phase velocity.

Figure 8:

y = \frac{2}{( t - x /2 ) ^{2} + 1}

t = 0

(dashed) and

t = 1 s

. The shape is unchanged and the peak moves

2 m

1 s

: speed

2 m s^{- 1}

towards

+ x

(Solved Example 4).

Test for a travelling wave: $x$ and $t$ must travel together. A wave function must contain $x$ and $t$ only in the combination $(x \pm v t)$ . So $(x - v t)^{2}$ is a wave but $x^{2} - v^{2} t^{2}$ is not; $sin (k x - ω t)$ is a wave but $sin k x cos ω t$ is not a travelling wave. The function must also stay finite for all $x$ and $t$ .

4.6 $sin (k x - ω t)$ or $sin (ω t - k x)$ ?

Both describe a wave towards $+ x$ with the same speed $ω / k$ , because $A sin (k x - ω t) = A sin (ω t - k x + π)$ . They differ in phase by $π$ : at $x = 0$ , $t = 0$ the first has slope $+ k A$ (particle moving down) and the second has slope $- k A$ (particle moving up). Pick either form, but keep it throughout a problem.

5. Phase and Phase Difference

The angle $(ω t - k x + ϕ)$ is the phase of the particle at $x$ at time $t$ . It tells where the particle is and which way it is moving.

Two points on the same wave at one instant, separated by $Δ x$ : $Δ ϕ = k Δ x = \frac{2 π}{λ} Δ x$ . The particle farther along the direction of travel lags behind.
One point at two instants, separated by $Δ t$ : $Δ ϕ = ω Δ t = \frac{2 π}{T} Δ t$ .
Points $λ, 2 λ, 3 λ \dots$ apart are in phase (always same displacement and velocity).
Points $λ /2, 3 λ /2 \dots$ apart are in opposite phase (equal and opposite displacements).

Figure 9:

Δ ϕ = \frac{2 π}{λ} Δ x

(wave moving towards

+ x

P

and

R

(

λ

apart) move together;

P

and

Q

(

λ /2

apart) always move opposite;

S

lags

P

π /2

Quick Recall: tap to check

Two points are

λ /4

apart. What is their phase difference?

\frac{2 π}{λ} \cdot \frac{λ}{4} = \frac{π}{2}

In which direction does

y = A sin (k x + ω t)

travel?

Towards

- x

x

and

t

have the same sign.

What changes when sound goes from air into water:

f

v

λ

v

and

λ

increase;

f

stays the same.

6. Particle Velocity and Acceleration

Differentiate $y = A sin (ω t - k x)$ partially, keeping $x$ fixed (we follow one particle):

Particle velocity:
$v_{p} = \frac{\partial y}{\partial t} = A ω cos (ω t - k x)$
maximum $A ω$ at the mean position.
Particle acceleration:
$a_{p} = \frac{\partial ^{2} y}{\partial t ^{2}} = - A ω^{2} sin (ω t - k x) = - ω^{2} y$
so every particle is in SHM.
Slope of the wave (keep $t$ fixed):
$\frac{\partial y}{\partial x} = - A k cos (ω t - k x)$
Divide the first by the third and use $ω / k = v$ :
$v_{p} = - v \frac{\partial y}{\partial x}$

Wave velocity

v

Speed of the disturbance along the medium. Constant for a given medium. Direction: the direction of propagation.

Particle velocity

v_{p}

Velocity of one particle about its mean position. Changes with time, from $0$ to $A ω$ . Direction: perpendicular (transverse) or parallel (longitudinal) to $v$ .

Figure 10: Particle velocity

v_{p} = - v \frac{\partial y}{\partial x}

for a wave moving towards

+ x

. Crests and troughs are momentarily at rest; particles at

y = 0

move fastest (

v_{p} = A ω

Exam Trick

Walk along the wave. Walk in the direction the wave travels: where the road goes uphill, particles move down; where it goes downhill, particles move up ("uphill down, downhill up"). Crests and troughs are momentarily at rest.

Also note: $\frac{( v _{p} ) _{m a x}}{v} = \frac{A ω}{ω / k} = A k = \frac{2 π A}{λ}$ , which is also the maximum slope of the wave. For a small-amplitude wave this ratio is much less than 1.

7. The Differential Wave Equation

From Section 6, $\frac{\partial ^{2} y}{\partial t ^{2}} = - ω^{2} y$ and $\frac{\partial ^{2} y}{\partial x ^{2}} = - k^{2} y$ . Dividing,

\frac{\partial ^{2} y}{\partial t ^{2}} = \frac{ω ^{2}}{k ^{2}} \frac{\partial ^{2} y}{\partial x ^{2}} \Rightarrow \frac{\partial ^{2} y}{\partial t ^{2}} = v^{2} \frac{\partial ^{2} y}{\partial x ^{2}}

Wave equation. Any quantity

y (x, t)

that satisfies

\frac{\partial ^{2} y}{\partial t ^{2}} = v^{2} \frac{\partial ^{2} y}{\partial x ^{2}}

travels as a wave with speed

v

. Comparing a physical equation with this form gives the wave speed directly (Section 8).

JEE Advanced

Every function of $(x - v t)$ is a wave. If $y = f (x - v t)$ then $\partial^{2} y / \partial t^{2} = v^{2} f^{''}$ and $\partial^{2} y / \partial x^{2} = f^{''}$ , so the wave equation holds for any shape $f$ , not just sines. A pulse keeps its shape and slides towards $+ x$ ; $f (x + v t)$ slides towards $- x$ . A function like $y = A sin k x cos ω t$ also satisfies the wave equation, but it is not a single function of $x \pm v t$ : it is a stationary wave (sum of two opposite travelling waves), studied in the air column concept. For a test in the exam: the expression must be finite for all $x$ , $t$ and depend on $x$ and $t$ only through $(x \pm v t)$ .

8. Speed of Mechanical Waves: A First Look

The speed of a mechanical wave depends only on the medium: an elastic (restoring) property divided by an inertial property, under a square root.

Wave	Speed	Symbols
Transverse wave on a stretched string	$v = \frac{T}{μ}$	$T$ tension (N), $μ$ mass per unit length (kg m $^{- 1}$ )
Longitudinal wave in a solid rod	$v = \frac{Y}{ρ}$	$Y$ Young's modulus, $ρ$ density
Longitudinal wave in a fluid	$v = \frac{B}{ρ}$	$B$ bulk modulus, $ρ$ density
Sound in a gas (Laplace)	$v = \frac{γ P}{ρ}$	$γ = C_{p} / C_{v}$ ; equals $γ R T / M$ ; derived in the Sound Waves concept

Dimensional check. Every entry has the form $stress / density$ or $force / linear density$ , both of which have units of $m^{2} s^{- 2}$ under the root. Use this to reject wrong options in a hurry.

8.1 Derivation: speed of a transverse wave on a string

Let a small pulse move to the right with speed $v$ on a string of tension $T$ and mass per unit length $μ$ . Look at it from a frame moving with the pulse: the pulse is at rest and the string slides through it to the left with speed $v$ .

Figure 11: In the pulse's frame the segment

Δ l = 2 R θ

moves on a circle at speed

v

. The net inward force

2 T sin θ \approx 2 T θ

supplies

\frac{( μ Δ l ) v ^{2}}{R}

, giving

v = T / μ

The top of the pulse is a small arc of length $Δ l = 2 R θ$ on a circle of radius $R$ ; its mass is $m = μ Δ l = 2 μ R θ$ .
The tensions at its two ends have equal and opposite horizontal parts, which cancel. Their parts towards the centre add: $\sum F_{r} = 2 T sin θ \approx 2 T θ$ (small $θ$ ).
The segment moves on the circle at speed $v$ , so it needs a centripetal force $\frac{m v ^{2}}{R}$ :
$2 T θ = (2 μ R θ) \frac{v ^{2}}{R}$
Hence
$v = \frac{T}{μ}$
independent of the shape and size of the pulse (for small amplitudes).

JEE Advanced

A hanging rope. A uniform rope of mass $m$ and length $l$ hangs from a ceiling. At height $x$ above the free end the tension supports only the rope below: $T = \frac{m}{l} xg$ , and $μ = \frac{m}{l}$ , so

v = g x

The pulse speeds up as it climbs. Time to travel the whole rope (Solved Example 9): $d t = \frac{d x}{g x}$ , so

t = \int_{0}^{l} \frac{d x}{g x} = 2 \frac{l}{g}

Key idea

Stiffer medium, faster wave; denser medium, slower wave. Every formula has the shape

v = E / ρ

: an elastic modulus over an inertia.

Quick Recall: tap to check

What is the particle velocity at a crest of a transverse wave?

Zero. Crests and troughs are momentarily at rest; particles at

y = 0

move fastest, at

A ω

The tension in a string is made 4 times. What happens to the wave speed?

It doubles, since

v = T / μ

y = A sin k x cos ω t

a travelling wave?

No. It satisfies the wave equation but is a stationary wave, the sum of two waves moving in opposite directions.

Figure 12: Revision map: types, terms, the equation

y = A sin (ω t - k x + ϕ)

, phase, particle motion and wave speed.

9. Solved Examples

Solved Example 1

Find the equation of a wave moving towards

+ x

if, at

x = 0

and

t = 0

, the displacement is

y = \frac{A}{2}

and the snapshot rises through

x = 0

(as in Figure 7).

Solution:

Let $y = A sin (ω t - k x + ϕ)$ . At $x = 0$ , $t = 0$ : $\frac{A}{2} = A sin ϕ$ , so $ϕ = \frac{π}{4}$ or $\frac{3 π}{4}$ .

Shift the wave slightly towards $+ x$ : the particle at $x = 0$ moves down (equivalently, the slope there is positive, so $v_{p} = - v \partial y / \partial x < 0$ ). So $v_{p} = A ω cos ϕ < 0$ , i.e. $cos ϕ < 0$ , which picks $ϕ = \frac{3 π}{4}$ .

Answer: $y = A sin (ω t - k x + \frac{3 π}{4})$ .

Solved Example 2

State the direction of travel of each wave: (i)

y = A sin (ω t - k x)

(ii)

y = A sin (- k x + ω t)

(iii)

y = A sin (- k x - ω t)

(iv)

y = A sin (k x + ω t + ϕ)

Solution:

Compare the signs of the $t$ term and the $x$ term. Opposite signs: towards $+ x$ . Same signs: towards $- x$ . The phase constant does not affect the direction.

Answer: (i) and (ii) towards $+ x$ ; (iii) and (iv) towards $- x$ .

Solved Example 3

Verify that

y = \frac{2}{( x - 3 t ) ^{2} + 1}

is a solution of the linear wave equation (

x

y

in cm,

t

in s), and find the speed of the pulse.

Solution:

Write $u = x - 3 t$ and $y = f (u) = \frac{2}{u ^{2} + 1}$ . Then $\frac{\partial y}{\partial x} = f^{'} (u)$ , $\frac{\partial ^{2} y}{\partial x ^{2}} = f^{''} (u)$ , and $\frac{\partial y}{\partial t} = - 3 f^{'} (u)$ , $\frac{\partial ^{2} y}{\partial t ^{2}} = 9 f^{''} (u)$ .

So $\frac{\partial ^{2} y}{\partial t ^{2}} = 9 \frac{\partial ^{2} y}{\partial x ^{2}}$ , which is the wave equation with $v^{2} = 9$ . (Explicitly, $\frac{\partial ^{2} y}{\partial x ^{2}} = \frac{12 u ^{2} - 4}{( u ^{2} + 1 ) ^{3}}$ and $\frac{\partial ^{2} y}{\partial t ^{2}} = \frac{108 u ^{2} - 36}{( u ^{2} + 1 ) ^{3}}$ .)

Answer: it is a solution; the pulse moves at $3 cm s^{- 1}$ towards $+ x$ .

Solved Example 4

A wave pulse travels on a string at

2 m s^{- 1}

. The displacement of the particle at

x = 0

y = \frac{2}{t ^{2} + 1}

. Find (i)

y (x, t)

(ii) the shape of the pulse at

t = 0

and

t = 1 s

Solution:

(i) The particle at $x$ repeats the motion of $x = 0$ after a delay $x / v = x /2$ . Replace $t$ by $t - \frac{x}{2}$ :

y (x, t) = \frac{2}{( t - \frac{x}{2} ) ^{2} + 1}

(ii) At $t = 0$ : $y = \frac{2}{x ^{2} /4 + 1}$ , giving $y = 2$ at $x = 0$ , $y = 1$ at $x = \pm 2$ , $y = 0.4$ at $x = \pm 4$ . At $t = 1 s$ : the maximum $y = 2$ is at $x = 2$ , and $y = 1$ at $x = 0$ and $x = 4$ . The pulse has kept its shape and moved $2 m$ to the right (Figure 8). Check: $t - \frac{x}{2} = constant$ gives $\frac{d x}{d t} = 2 m s^{- 1}$ .

Solved Example 5

A sinusoidal wave travelling towards

+ x

has amplitude

15 cm

, wavelength

40 cm

and frequency

8 Hz

. The displacement at

x = 0

t = 0

15 cm

. (a) Find

k

T

ω

and

v

. (b) Find the phase constant and write the wave function.

Solution:

(a) $k = \frac{2 π}{λ} = \frac{2 π}{40} = \frac{π}{20} rad cm^{- 1}$ ; $T = \frac{1}{f} = \frac{1}{8} s$ ; $ω = 2 π f = 16 π rad s^{- 1}$ ; $v = f λ = 8 \times 40 = 320 cm s^{- 1}$ .

(b) $y = A sin (ω t - k x + ϕ)$ with $15 = 15 sin ϕ$ , so $sin ϕ = 1$ and $ϕ = \frac{π}{2}$ (only one angle, since the particle is at the extreme).

Answer: $y = (15 cm) sin (16 π t - \frac{π}{20} x + \frac{π}{2})$ , with $x$ in cm and $t$ in s; equivalently $y = 15 cos (16 π t - \frac{π x}{20}) cm$ .

Solved Example 6

A string of linear mass density

500 g m^{- 1}

passes over a pulley and supports a

20 kg

block. Find the speed of transverse waves on the string. Take

g = 10 m s^{- 2}

and assume the string's own weight does not change the tension.

Solution:

$T = m g = 20 \times 10 = 200 N$ ; $μ = 0.5 kg m^{- 1}$ .

$v = \frac{T}{μ} = \frac{200}{0.5} = 20 m s^{- 1}$ .

Answer: $20 m s^{- 1}$ .

Solved Example 7

A taut string with tension

100 N

and linear mass density

0.25 kg m^{- 1}

is fixed inside a cart. A pulse is started at its left end and runs towards the right. With what velocity must the cart move so that the pulse stays at rest relative to the ground?

Solution:

Speed of the pulse relative to the string (cart): $v = \frac{100}{0.25} = 20 m s^{- 1}$ , towards $+ x$ .

Relative velocity: $v_{P G} = v_{P C} + v_{C G}$ . For $v_{P G} = 0$ : $v_{C G} = - 20 \hat{i} m s^{- 1}$ .

Answer: the cart must move at $20 m s^{- 1}$ towards the left.

Solved Example 8

One end of a

12.0 m

long rubber tube of total mass

0.9 kg

is fixed. A cord from the other end passes over a pulley and supports a

5.0 kg

mass. The tube is struck at one end. How long does the pulse take to reach the other end? (

g = 9.8 m s^{- 2}

)

Solution:

$T = 5.0 \times 9.8 = 49 N$ ; $μ = \frac{0.9}{12} = 0.075 kg m^{- 1}$ .

$v = \frac{49}{0.075} \approx 25.6 m s^{- 1}$ ; $t = \frac{12}{25.6} \approx 0.47 s$ .

Answer: about $0.47 s$ .

Solved Example 9

A uniform rope of mass

0.1 kg

and length

2.45 m

hangs from a ceiling. (a) Find the speed of a transverse wave at

0.5 m

from the lower end. (b) Find the time a wave takes to travel the full length of the rope.

Solution:

(a) At distance $x$ from the free end, the tension equals the weight of the rope below: $T = \frac{m}{l} xg$ , and $μ = \frac{m}{l}$ , so $v = g x = 9.8 \times 0.5 \approx 2.21 m s^{- 1}$ .

(b) $d t = \frac{d x}{g x}$ , so

t = \int_{0}^{l} \frac{d x}{g x} = 2 \frac{l}{g} = 2 \frac{2.45}{9.8} = 1.0 s

Answer: (a) $2.21 m s^{- 1}$ ; (b) $1.0 s$ (the mass of the rope does not matter).

Solved Example 10

The equation of a wave is

y = 0.02 sin (30 t - 4 x)

, with

x

y

in metres and

t

in seconds. Find the amplitude, wavelength, frequency, time period, wave speed and direction, and the maximum particle speed and acceleration.

Solution:

Compare with $y = A sin (ω t - k x)$ : $A = 0.02 m$ , $ω = 30 rad s^{- 1}$ , $k = 4 rad m^{- 1}$ .

$λ = \frac{2 π}{k} = \frac{π}{2} \approx 1.57 m$ ; $f = \frac{ω}{2 π} \approx 4.77 Hz$ ; $T = \frac{2 π}{ω} \approx 0.209 s$ ; $v = \frac{ω}{k} = 7.5 m s^{- 1}$ towards $+ x$ (opposite signs of $t$ and $x$ ).

$(v_{p})_{m a x} = A ω = 0.02 \times 30 = 0.6 m s^{- 1}$ ; $(a_{p})_{m a x} = A ω^{2} = 0.02 \times 900 = 18 m s^{- 2}$ .

Answer: $A = 2 cm$ , $λ \approx 1.57 m$ , $f \approx 4.77 Hz$ , $v = 7.5 m s^{- 1}$ along $+ x$ .

Solved Example 11

Sound of frequency

500 Hz

travels in air at

330 m s^{- 1}

. (a) What is the phase difference between two points

11 cm

apart along the direction of travel? (b) By how much does the phase at one point change in

1 ms

Solution:

$λ = \frac{v}{f} = \frac{330}{500} = 0.66 m$ .

(a) $Δ ϕ = \frac{2 π}{λ} Δ x = \frac{2 π \times 0.11}{0.66} = \frac{π}{3}$ (i.e. $6 0^{\circ}$ ).

(b) $Δ ϕ = ω Δ t = 2 π (500) (1 0^{- 3}) = π$ .

Answer: (a) $π /3$ ; (b) $π$ (the point is then in opposite phase to where it started).

Solved Example 12

A wave of amplitude

2 mm

and wavelength

0.5 m

travels along

- x

with speed

100 m s^{- 1}

. At

t = 0

the particle at

x = 0

is at its mean position moving towards

+ y

. Write the equation of the wave.

Solution:

$k = \frac{2 π}{0.5} = 4 π rad m^{- 1}$ ; $f = \frac{v}{λ} = 200 Hz$ , so $ω = 400 π rad s^{- 1}$ . Travelling towards $- x$ : phase $(ω t + k x + ϕ)$ .

At $x = 0$ , $t = 0$ : $y = A sin ϕ = 0$ and $v_{p} = A ω cos ϕ > 0$ , so $ϕ = 0$ .

Answer: $y = (2 \times 1 0^{- 3} m) sin (400 π t + 4 π x)$ , with $x$ in m and $t$ in s.

Solved Example 13

A transverse wave moves along

+ x

on a string at

10 m s^{- 1}

. At some instant the slope of the string at point

P

0.02

. Find the velocity of the particle at

P

Solution:

$v_{p} = - v \frac{\partial y}{\partial x} = - (10) (0.02) = - 0.2 m s^{- 1}$ .

Answer: $0.2 m s^{- 1}$ , directed towards $- y$ (the string is going uphill at $P$ , so the particle moves down).

Solved Example 14

The maximum particle speed in a wave equals the wave speed when
(A)

λ = A

(B)

λ = 2 π A

(C)

λ = π A

(D)

λ = \frac{A}{2 π}

Solution:

$A ω = \frac{ω}{k}$ gives $A k = 1$ , that is $A \cdot \frac{2 π}{λ} = 1$ .

Answer: (B).

Solved Example 15

Which of the following represent a travelling wave?
(A)

y = A sin (15 x - 2 t)

(B)

y = A sin x cos t

(C)

y = A e^{- (x + 3 t)^{2}}

(D)

y = A (x^{2} - t)

Solution:

A travelling wave must be a finite function of $(x \pm v t)$ only.

(A) $15 x - 2 t = 15 (x - \frac{2}{15} t)$ : wave towards $+ x$ with $v = \frac{2}{15} m s^{- 1}$ . Yes. (B) Product of a function of $x$ and a function of $t$ : a stationary wave, not travelling. (C) A pulse moving towards $- x$ with speed $3 m s^{- 1}$ . Yes. (D) $x^{2} - t$ cannot be written as a function of $(x \pm v t)$ . No.

Answer: (A) and (C).

Solved Example 16

A pulse is described by

y = \frac{0.8}{( 4 x + 5 t ) ^{2} + 5}

(SI units). Find the speed and direction of the pulse.

Solution:

$4 x + 5 t = 4 (x + \frac{5}{4} t)$ , so $y$ is a function of $(x + v t)$ with $v = \frac{5}{4} m s^{- 1}$ .

Answer: $1.25 m s^{- 1}$ towards $- x$ (coefficient of $t$ divided by coefficient of $x$ ; same signs, so $- x$ ).

Practice Questions

For $y = 0.01 sin (100 π t - 0.5 π x)$ (SI), find $λ$ , $f$ and $v$ .Answer: $λ = 4 m$ , $f = 50 Hz$ , $v = 200 m s^{- 1}$ along $+ x$ .
Find the speed and direction of $y = 4 sin π (0.01 x + 2 t)$ , with $x$ , $y$ in cm and $t$ in s.Answer: $v = 2/0.01 = 200 cm s^{- 1} = 2 m s^{- 1}$ towards $- x$ .
Two points on a wave of wavelength $6 m$ are $1.5 m$ apart. Find their phase difference.Answer: $π /2$ .
A wave has $(v_{p})_{m a x} = 0.4 m s^{- 1}$ and $v = 20 m s^{- 1}$ . Find the maximum slope of the wave.Answer: $A k = 0.4/20 = 0.02$ .
Is $y = sin^{2} (x - 2 t)$ a travelling wave? If so, give its speed.Answer: Yes, a function of $(x - 2 t)$ : $2 m s^{- 1}$ along $+ x$ .
Sound of $400 Hz$ goes from air ( $340 m s^{- 1}$ ) into water ( $1480 m s^{- 1}$ ). Find the wavelengths.Answer: $0.85 m$ in air, $3.7 m$ in water; $f$ unchanged.
The tension in a string is made four times. What happens to the wave speed?Answer: It doubles ( $v \propto T$ ).

Common Mistakes to Avoid

Watch out

Thinking the medium travels with the wave. Only energy and the disturbance travel; each particle stays near its mean position.
Mixing up wave velocity and particle velocity. $v = ω / k$ is constant; $v_{p} = A ω cos (ω t - k x)$ keeps changing.
Getting the direction wrong: $y = A sin (k x + ω t)$ travels towards $- x$ (same signs), not $+ x$ .
Reading $λ$ as the coefficient of $x$ . The coefficient is $k = 2 π / λ$ ; take care whether the equation has a $2 π$ outside the bracket.
Changing the frequency when a wave enters a new medium. The source sets $f$ ; only $v$ and $λ$ change.
Forgetting units: in $y = 5 sin 2 π (t /0.04 - x /40)$ with $x$ in cm, $λ = 40 cm$ , not $40 m$ .
Saying sound can be transverse in air. Gases have no rigidity, so sound in air is always longitudinal.
Calling $y = A sin k x cos ω t$ a travelling wave. It satisfies the wave equation but is a stationary wave.

Frequently Asked Questions

What is the difference between transverse and longitudinal waves?

In a transverse wave the particles of the medium move perpendicular to the direction of travel, forming crests and troughs, as on a string. In a longitudinal wave they move along the direction of travel, forming compressions and rarefactions, as in sound. Only transverse waves can be polarised.

Why can sound not travel through vacuum?

Sound is a mechanical wave. It needs a medium with inertia and elasticity so that each layer can push the next one and pass the disturbance on. Vacuum has no particles, so there is nothing to compress or rarefy and sound cannot travel, while light, an electromagnetic wave, can.

How do you find the direction of a wave from its equation?

Look at the signs of the $t$ and $x$ terms in the phase. If they are opposite, as in $sin (ω t - k x)$ , the wave moves towards positive $x$ . If they are the same, as in $sin (ω t + k x)$ , it moves towards negative $x$ . The speed is the coefficient of $t$ divided by the coefficient of $x$ .

What is the relation between wave speed, frequency and wavelength?

Wave speed equals frequency times wavelength, $v = f λ$ , because the wave advances one wavelength during each period. It can also be written $v = ω / k$ . The frequency is fixed by the source and the speed by the medium, so the wavelength adjusts when the wave changes medium.

What is the difference between wave velocity and particle velocity?

Wave velocity is the constant speed at which the disturbance moves through the medium, $v = ω / k$ . Particle velocity is the changing velocity of an individual particle about its mean position, $v_{p} = A ω cos (ω t - k x)$ , which is related to the slope of the wave by $v_{p} = - v \partial y / \partial x$ .

How is phase difference related to path difference?

Phase difference equals $2 π / λ$ times the path difference, $Δ ϕ = 2 π Δ x / λ$ . Points one wavelength apart are in phase, points half a wavelength apart are in opposite phase, and points a quarter wavelength apart differ in phase by $π /2$ .

How are wave equations asked in JEE Main?

JEE Main usually gives an equation such as $y = A sin (a t - b x)$ and asks for speed, wavelength, frequency, direction, phase difference or maximum particle speed. Questions on the pulse form $f (a x + b t)$ , the relation $v_{p} = - v \partial y / \partial x$ , and speed on a string $T / μ$ are also common.

Which wave concepts are important for NEET?

For NEET, learn the difference between transverse and longitudinal waves, the relations $v = f λ$ and $v = ω / k$ , reading $A$ , $λ$ , $f$ and direction from $y = A sin (ω t - k x)$ , phase difference $2 π Δ x / λ$ , and speed of waves on a string. Most questions are direct substitution.

Previous year questions on Types and Equations of Waves

10 questions from past papers, each with a step-by-step solution.

Ready to master Waves?

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Practice questions →Take a mock test

1. What Is a Wave?

1.1 Classification by medium

1.2 Classification by dimension

2. Transverse and Longitudinal Waves

3. Terms Used to Describe a Wave

4. Equation of a Progressive (Travelling) Wave

4.1 Derivation

4.2 Equivalent forms

4.3 Two ways to read the equation

4.4 Choosing the initial phase

4.5 Any shape: the general wave function

4.6 sin(kx−ωt) or sin(ωt−kx)?

5. Phase and Phase Difference

6. Particle Velocity and Acceleration

7. The Differential Wave Equation

8. Speed of Mechanical Waves: A First Look

8.1 Derivation: speed of a transverse wave on a string

9. Solved Examples

Common Mistakes to Avoid

Frequently Asked Questions

What is the difference between transverse and longitudinal waves?

Why can sound not travel through vacuum?

How do you find the direction of a wave from its equation?

What is the relation between wave speed, frequency and wavelength?

What is the difference between wave velocity and particle velocity?

How is phase difference related to path difference?

How are wave equations asked in JEE Main?

Which wave concepts are important for NEET?

Previous year questions on Types and Equations of Waves

Ready to master Waves?

4.6 $sin (k x - ω t)$ or $sin (ω t - k x)$ ?