Amplitude Examples: 66 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Amplitude.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

The maximum displacement of a wave from its equilibrium (rest) position, measuring the wave's strength or intensity.

How 'tall' the wave is measured from the center line — bigger amplitude carries more energy and produces stronger effects.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Amplitude asks what oscillates, what travels, and which wave quantity is being measured.

Common stuck point: Students often know a formula related to amplitude but skip the recognition step: Am I describing a repeating disturbance using wavelength, frequency, amplitude, speed, medium, or superposition? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I describing a repeating disturbance using wavelength, frequency, amplitude, speed, medium, or superposition?

Worked Examples

Example 1

easy

A speaker produces a sound wave. The air molecules vibrate

0.002 \text{ m}

on either side of their equilibrium position. What is the amplitude of the sound wave?

The amplitude A is the maximum displacement from the rest position (equilibrium line).

Answer

A = 0.002 m

First step

Amplitude is the maximum displacement from the equilibrium position.

Full solution

2
The molecules vibrate $0.002 \text{ m}$ from equilibrium, so the amplitude is $A = 0.002 \text{ m}$ .
3
The total peak-to-peak distance of vibration is $2A = 0.004 \text{ m}$ .

Amplitude is the maximum displacement of a point on a wave from its rest position. For sound waves, larger amplitude means louder sound. Amplitude is always measured from equilibrium, not peak-to-peak.

Example 2

medium

Two identical waves with amplitude

A = 3 \text{ cm}

overlap perfectly in phase. What is the resulting amplitude? What if they are perfectly out of phase (shifted by half a wavelength)?

Constructive interference: two in-phase waves of amplitude 3 cm add to give amplitude 6 cm.

Example 3

easy

A mass on a spring is pulled

0.15 \text{ m}

from equilibrium and released. Ignoring damping, what is the amplitude of the resulting oscillation?

Example 4

medium

When the amplitude of a wave on a string is doubled, by what factor does the energy carried by the wave change?

Example 5

medium

A mass-spring oscillator has

m = 0.5 \text{ kg}

k = 200 \text{ N/m}

, and amplitude

A = 0.10 \text{ m}

. What is the maximum speed of the mass?

Example 6

medium

Two coherent waves with amplitudes

A_1 = 3 \text{ cm}

and

A_2 = 4 \text{ cm}

overlap in phase. What is the resulting amplitude?

Constructive interference: waves of amplitude 3 cm and 4 cm add in phase to give 7 cm.

Example 7

hard

A sound speaker doubles its amplitude. The original sound level was

70 \text{ dB}

. What is the new sound level?

Example 8

hard

A damped oscillator follows

A(t) = A_0 e^{-t/\tau}

with

\tau = 4 \text{ s}

. What fraction of the original amplitude remains after

12 \text{ s}

Example 9

hard

Resonance: a driven oscillator's steady-state amplitude near resonance is

A_{\text{res}} = F_0 / (b\omega_0)

where

b

is the damping coefficient. If

F_0 = 5 \text{ N}

b = 0.5 \text{ kg/s}

, and

\omega_0 = 10 \text{ rad/s}

, find

A_{\text{res}}

Example 10

hard

Two speakers emit identical sound waves of amplitude

A

. At a point, the path-length difference is

\lambda/4

. What is the resulting pressure amplitude at that point?

Example 11

hard

An LC circuit oscillates with charge

q(t) = (4 \text{ μC}) \cos(2000\, t)

. What is the amplitude of the current?

Example 12

challenge

A spherical sound source emits with constant power

P

. The displacement amplitude at distance

r_1 = 2 \text{ m}

A_1

. What is the displacement amplitude

A_2

r_2 = 8 \text{ m}

, ignoring absorption?

Example 13

medium

A simple harmonic oscillator follows

y(t) = 0.04 \sin(8\pi t) \text{ m}

. State its amplitude and the maximum displacement from equilibrium.

Example 14

medium

A pressure wave in air has pressure amplitude

\Delta p_{\max} = 0.02 \text{ Pa}

. The atmospheric pressure is

10^5 \text{ Pa}

. What fraction of atmospheric pressure does the wave's pressure variation represent?

Example 15

medium

A wave on a string is described by

y(x,t) = 0.05 \sin(2x - 6t) \text{ m}

. Identify the amplitude, wave number, and angular frequency.

Example 16

hard

Two coherent sound sources produce waves of amplitudes

A_1 = 3

and

A_2 = 4

that meet

90^\circ

out of phase. What is the resultant amplitude?

Example 17

hard

A child on a swing follows SHM with amplitude

A = 1.5 \text{ m}

and period

T = 3 \text{ s}

. Compute the maximum speed of the child at the bottom of the swing.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

A wave on a string has amplitude

0.1 \text{ m}

and frequency

2 \text{ Hz}

. The string has a linear mass density of

0.05 \text{ kg/m}

and the wave speed is

10 \text{ m/s}

. What is the power transmitted by the wave? Use

P = \frac{1}{2}\mu\omega^2 A^2 v

Example 2

hard

A seismograph records an earthquake wave with amplitude

2 \text{ mm}

. A second earthquake produces waves with

4

times the energy. What is the amplitude of the second earthquake's waves?

Example 3

easy

A wave oscillates between +4 cm and -4 cm from equilibrium. What is its amplitude?

The wave oscillates between +4 cm and −4 cm. Read the amplitude as the distance from equilibrium to a peak.

Example 4

easy

A wave's peak-to-trough height is 10 cm. Find its amplitude.

The full peak-to-trough height is 10 cm. What is the amplitude?

Example 5

easy

Is amplitude measured vertically (perpendicular to travel) or horizontally (along travel) on a transverse wave?

Amplitude is measured vertically (perpendicular to travel); wavelength is measured horizontally (along travel).

Example 6

easy

Wave A has amplitude 2 cm and wave B has amplitude 6 cm (same wave type). Which carries more energy?

Wave A has amplitude 2 cm; wave B has amplitude 6 cm. Which carries more energy?

Example 7

easy

For a sound wave, which property does amplitude control: loudness or pitch?

Example 8

easy

A wave's equilibrium line is at displacement 0. Its highest point reaches +7 mm. State the amplitude.

The wave's highest point is +7 mm above equilibrium. The amplitude equals this peak displacement.

Example 9

easy

Does increasing amplitude change a wave's speed in a given medium?

Example 10

easy

Two ocean waves have amplitudes 1 m and 3 m. Which produces a stronger effect on a buoy?

Wave with amplitude 1 m vs wave with amplitude 3 m. Which produces a stronger effect on a buoy?

Example 11

medium

Wave energy is proportional to amplitude squared. If amplitude triples, by what factor does the energy increase?

When amplitude triples (A → 3A), by what factor does the wave energy increase?

Example 12

medium

A sound's amplitude is halved. By what factor does its energy (proportional to amplitude squared) change?

When amplitude is halved, by what factor does the wave energy change?

Example 13

medium

On a displacement-time graph a wave rises to +6, drops to -6, and the graph oscillates about 0. A second wave on the same axes peaks at +2. Compare their amplitudes and relative energy.

Wave 1 peaks at +6; wave 2 peaks at +2. Read off both amplitudes, then compare their energies.

Example 14

medium

A wave's amplitude is 5 cm. After traveling through a damping medium, its amplitude drops to 1 cm. What fraction of its original energy remains (energy proportional to amplitude squared)?

Amplitude drops from 5 cm to 1 cm after passing through the damping medium. What fraction of energy remains?

Example 15

medium

Why can two sound waves have the same pitch but different loudness?

Example 16

medium

A pendulum swings with amplitude 8 cm. Friction reduces the amplitude by 25% each swing. What is the amplitude after one swing?

Example 17

medium

A wave has amplitude 3 units and another identical-type wave has 12 units. The second's energy is how many times the first's (energy proportional to amplitude squared)?

Wave with amplitude 3 vs amplitude 12. How many times greater is the energy of the second?

Example 18

medium

A wave's amplitude doubles. By what factor does its energy (proportional to amplitude squared) increase?

When amplitude doubles, by what factor does the wave energy increase?

Example 19

medium

A wave with amplitude 10 cm passes through a medium that reduces amplitude to 4 cm. What fraction of the original energy remains (energy proportional to amplitude squared)?

Example 20

challenge

A wave starts with amplitude 16 cm. Each second its amplitude halves due to damping. After how many seconds is the amplitude 2 cm, and what fraction of the original energy remains then?

Example 21

challenge

Two speakers play the same note in phase, each producing amplitude 3 units at a point. Find the combined amplitude and how the combined energy compares to one speaker alone (energy proportional to amplitude squared).

Example 22

challenge

A wave's energy is 100 J at amplitude A. To raise its energy to 225 J in the same medium, what amplitude (as a multiple of A) is needed (energy proportional to amplitude squared)?

Example 23

easy

A pendulum swings

8 \text{ cm}

to the right and

8 \text{ cm}

to the left of its rest position. What is its amplitude?

Example 24

easy

An oscilloscope shows a sinusoidal voltage signal with peak-to-peak value

12 \text{ V}

. What is the amplitude of the signal?

Oscilloscope trace with peak-to-peak value 12 V. What is the amplitude?

Example 25

medium

A guitar string vibrates with displacement

y(t) = (4 \text{ mm}) \sin(2\pi \cdot 440 t)

. What is the amplitude and frequency?

Example 26

medium

A simple pendulum of length

1 \text{ m}

has amplitude

0.05 \text{ m}

(small angle). What is its maximum angular displacement from vertical, in radians?

Example 27

medium

A radio carrier wave has amplitude

200 \text{ mV}

at the antenna and

20 \text{ mV}

at the receiver. By what factor has the wave amplitude decreased? By what factor has the carried power decreased?

Example 28

medium

A tuning fork is struck and its amplitude decays exponentially. After

5 \text{ s}

, its amplitude is half the initial value. What is its amplitude after

10 \text{ s}

as a fraction of the initial value?

Example 29

medium

A vertical mass-spring system has amplitude

0.20 \text{ m}

and the mass is

0.40 \text{ kg}

k = 100 \text{ N/m}

. What is the maximum kinetic energy of the mass?

Example 30

hard

Two coherent waves of amplitude

A

meet with a phase difference of

\pi/3

radians. What is the resulting amplitude?

Example 31

hard

A transverse wave on a string has

y(x, t) = (0.02 \text{ m})\sin(5x - 10t)

in SI units. Find the amplitude and the maximum transverse speed of a particle on the string.

Example 32

hard

A laser beam has intensity

I = 500 \text{ W/m}^2

. Using

I = \frac{1}{2}c\varepsilon_0 E_0^2

with

c = 3\times 10^8 \text{ m/s}

and

\varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m}

, find the electric field amplitude

E_0

Example 33

hard

A standing wave on a

2 \text{ m}

string in the fundamental mode has antinode displacement amplitude

A_0 = 5 \text{ mm}

. What is the displacement amplitude at

x = 0.5 \text{ m}

from one end?

Example 34

easy

A pendulum swings between

+8 \text{ cm}

and

-8 \text{ cm}

from rest. State its amplitude.

Pendulum displacement vs time. The swing reaches ±8 cm from rest.

Example 35

easy

On a graph of displacement vs. time, the curve oscillates between

+0.6 \text{ m}

and

-0.6 \text{ m}

. What is the amplitude of the motion?

Example 36

easy

A wave on a rope has amplitude

5 \text{ cm}

. What is its peak-to-trough vertical distance?

A rope wave with amplitude 5 cm. What is the peak-to-trough distance?

Example 37

medium

Wave

A

has amplitude

2 \text{ cm}

and wave

B

has amplitude

6 \text{ cm}

(same medium, same frequency). How does the energy carried by wave

B

compare to that carried by wave

A

Wave A has amplitude 2 cm; wave B has amplitude 6 cm (same medium, same frequency). Compare their energies.

Example 38

medium

A sound wave's intensity is increased by a factor of

16

. By what factor does its amplitude increase?

Example 39

medium

Two identical sound waves with amplitudes

A

each combine in phase. What is the amplitude of the result? What is the intensity of the result compared to a single wave?

Two identical waves of amplitude A combine in phase. The result has amplitude 2A and intensity 4I.

Example 40

medium

A damped oscillator's amplitude drops from

10 \text{ cm}

5 \text{ cm}

in one period. By what factor does its mechanical energy drop in that period?

Amplitude drops from 10 cm to 5 cm in one period. By what factor does the mechanical energy drop?

Example 41

medium

A wave with amplitude

A

has its frequency doubled while

A

stays the same. By what factor does the wave power (proportional to

A^2 f^2

) change?

Example 42

hard

A spherical wave radiates from a point source isotropically. At

r_1 = 2 \text{ m}

its amplitude is

A_1 = 0.1

. Ignoring absorption, what is the amplitude at

r_2 = 10 \text{ m}

Example 43

hard

A string carries a transverse wave with

\mu = 0.02 \text{ kg/m}

v = 8 \text{ m/s}

, frequency

f = 5 \text{ Hz}

, and amplitude

A = 0.04 \text{ m}

. Compute the time-averaged power transmitted,

P = \tfrac{1}{2}\mu \omega^2 A^2 v

Example 44

hard

A seismic wave's amplitude drops to

1/3

of its initial value after travelling a certain distance. What fraction of its initial energy remains?

Example 45

hard

Sound from a point source has intensity

I_1 = 8 \times 10^{-6} \text{ W/m}^2

at distance

r_1 = 5 \text{ m}

. Compute the intensity at

r_2 = 20 \text{ m}

, and the ratio of amplitudes

A_2 / A_1

Example 46

medium

A microphone records a sound wave with displacement amplitude

5 \times 10^{-8} \text{ m}

. If the loudness is increased so the new displacement amplitude is

5 \times 10^{-7} \text{ m}

, by what factor has the intensity changed?

Example 47

medium

A guitar string vibrates with amplitude

2 \text{ mm}

at its midpoint. If you pluck it harder so the amplitude becomes

4 \text{ mm}

, by what factor does the energy of vibration change? Does the pitch change?

Example 48

challenge

A damped oscillator's amplitude decays as

A(t) = A_0 e^{-\gamma t}

. If the amplitude falls to half its initial value in

4 \text{ s}

, find

\gamma

, and predict the amplitude (as a fraction of

A_0

) after

12 \text{ s}

Example 49

challenge

Three coherent sources produce waves at the same location with equal amplitudes

A

but phases

0

2\pi/3

, and

4\pi/3

. What is the resultant amplitude?

← Back to Amplitude Practice Mode

Related Concepts

Waves Intensity Energy

Background Knowledge

These ideas may be useful before you work through the harder examples.

waves