Longitudinal Wave Examples: 46 Problems with Answers

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

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

Concept Recap

A wave in which the particles of the medium oscillate parallel to the direction of wave propagation, creating alternating regions of compression (high pressure) and rarefaction (low pressure).

A slinky pushed back and forth: compressions and stretches travel along it.

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: Longitudinal Wave asks what oscillates, what travels, and which wave quantity is being measured.

Common stuck point: Students often know a formula related to longitudinal wave 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 longitudinal wave in a spring has compressions

0.4 \text{ m}

apart. The wave frequency is

5 \text{ Hz}

. What is the wave speed?

Answer

v = 2 m/s

First step

The distance between consecutive compressions is the wavelength:

\lambda = 0.4 \text{ m}

Full solution

2
Wave speed: $v = f λ = 5 \times 0.4 = 2 m/s$
3
In a longitudinal wave, particles vibrate parallel to the wave's direction of travel.

In a longitudinal wave, particles oscillate back and forth along the same direction the wave travels. Compressions (high density) and rarefactions (low density) propagate through the medium.

Example 2

medium

Sound travels at

340 \text{ m/s}

in air and

5100 \text{ m/s}

in steel. A

1000 \text{ Hz}

sound wave enters a steel rail. What is the wavelength in each medium?

Example 3

medium

A pipe organ produces a sound at

440\text{ Hz}

. Sound speed in air is

343\text{ m/s}

. Find the wavelength.

Example 4

hard

A sound wave's pressure variation has amplitude

\Delta P = 0.02\text{ Pa}

. The intensity is

I = (\Delta P)^2/(2\rho v)

for

\rho = 1.2\text{ kg/m}^3,\; v = 343\text{ m/s}

. Compute

I

Practice Problems

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

Example 1

medium

An ultrasound device emits sound at

2 \times 10^6 \text{ Hz}

in human tissue (

v = 1540 \text{ m/s}

). What is the wavelength? Why is this frequency useful for medical imaging?

Example 2

hard

An earthquake produces both P-waves (

v_P = 6000 \text{ m/s}

, longitudinal) and S-waves (

v_S = 3500 \text{ m/s}

, transverse). A seismograph detects the P-wave

20 \text{ s}

before the S-wave. How far away is the earthquake?

Example 3

easy

In a longitudinal wave, the particles oscillate in which direction relative to wave travel?

Example 4

easy

Longitudinal waves consist of which regions: crests and troughs, or compressions and rarefactions?

Example 5

easy

Pushing and pulling a slinky along its length produces what type of wave?

Example 6

easy

Is sound a longitudinal or transverse wave?

Example 7

easy

Can longitudinal waves be polarised?

Example 8

easy

Can longitudinal waves travel through solids and liquids, or only gases?

Example 9

easy

In a longitudinal wave, a compression is a region of high or low particle density?

Example 10

easy

Which is longitudinal: a sound wave in air or a light wave?

Example 11

medium

A longitudinal sound wave has compressions 0.5 m apart and a frequency of 680 Hz. Find the wave speed.

Example 12

medium

Why does sound travel faster in steel than in air, given both are longitudinal waves?

Example 13

medium

A slinky longitudinal wave repeats its compression pattern every 0.2 s. Find its frequency and, if compressions are 0.3 m apart, the wave speed.

Example 14

medium

A student draws a sound wave as a sine curve with crests and troughs. What is conceptually wrong, and how should it be represented?

Example 15

medium

Seismic P-waves are longitudinal. If a P-wave travels at 6000 m/s and has frequency 2 Hz, find its wavelength.

Example 16

medium

Explain why a longitudinal wave needs a medium with particles that can be compressed.

Example 17

medium

A longitudinal wave has speed 1500 m/s in water and frequency 3000 Hz. Find the distance between adjacent compressions.

Example 18

medium

A longitudinal wave has frequency 5 Hz and wavelength 0.6 m. Find its speed.

Example 19

medium

Name two examples of longitudinal waves and one feature they share.

Example 20

challenge

A longitudinal wave on a slinky has compressions 0.4 m apart and travels 8 m in 5 s. Find the speed, wavelength, and frequency.

Example 21

challenge

In a longitudinal wave the distance from a compression to the next rarefaction is 0.15 m. If the frequency is 1000 Hz, find the wave speed.

Example 22

challenge

Sound (longitudinal) travels at 343 m/s in air and 1480 m/s in water at the same frequency 740 Hz. Find the wavelength in each and the ratio water-to-air.

Example 23

easy

A sound wave has frequency

500\text{ Hz}

and wavelength

0.68\text{ m}

in air. Find its speed.

Example 24

easy

A sound wave of frequency

200\text{ Hz}

travels at

340\text{ m/s}

in air. Find the wavelength.

Example 25

easy

A sound wave has period

2\text{ ms}

and travels at

340\text{ m/s}

in air. Find its wavelength.

Example 26

medium

Two compressions in a sound wave are

0.2\text{ m}

apart and arrive at a microphone

0.5\text{ ms}

apart. Find the wave speed and frequency.

Example 27

medium

Sound moves from air (

v=340\text{ m/s}

) into water (

v=1480\text{ m/s}

) at a fixed frequency of

500\text{ Hz}

. Find the wavelengths in each medium.

Example 28

medium

An echo from a wall returns to a clapper in

0.6\text{ s}

(sound speed

340\text{ m/s}

). How far away is the wall?

Example 29

medium

Why does sound travel faster in steel than in air, even though steel is much denser?

Example 30

medium

An ultrasound machine sends a

5\text{ MHz}

longitudinal pulse into tissue (

v \approx 1540\text{ m/s}

). Find the wavelength.

Example 31

medium

A speaker produces a sound wave of wavelength

0.5\text{ m}

moving at

340\text{ m/s}

. Find the frequency.

Example 32

medium

A bat emits an ultrasonic pulse at

40\text{ kHz}

. It hears the echo

0.012\text{ s}

later. How far is the moth (sound speed

340\text{ m/s}

Example 33

medium

In a closed organ pipe of length

L

, the fundamental wavelength is

4L

. If

L = 0.85\text{ m}

and sound speed is

340\text{ m/s}

, find the fundamental frequency.

Example 34

hard

Sound in helium travels at

972\text{ m/s}

340\text{ m/s}

in air. If a singer's voice has fundamental

200\text{ Hz}

in air, what fundamental will their voice produce when speaking helium (vocal-tract length unchanged)?

Example 35

hard

An earthquake P-wave (longitudinal) at

6.0\text{ km/s}

and an S-wave (transverse) at

3.5\text{ km/s}

start together. A seismograph

300\text{ km}

away reports the P-wave arrival. How long after, does the S-wave arrive?

Example 36

hard

Sound in air at

20°\text{C}

343\text{ m/s}

. The speed scales as

\sqrt{T}

(in Kelvin). Find the sound speed at

0°\text{C}

Example 37

hard

An open organ pipe of length

L = 0.5\text{ m}

supports its fundamental at

\lambda_1 = 2L = 1.0\text{ m}

. Find the fundamental frequency (sound speed

340\text{ m/s}

) and the second harmonic frequency.

Example 38

hard

Why is it incorrect to call a sound wave's high-pressure region a 'crest'?

Example 39

hard

An observer hears a

1000\text{ Hz}

horn from a train moving toward them at

30\text{ m/s}

(sound speed

340\text{ m/s}

). Find the observed frequency.

Example 40

challenge

A standing longitudinal wave in a pipe has nodes

0.4\text{ m}

apart. The driving frequency is

425\text{ Hz}

. Find the wave speed in the gas.

Example 41

challenge

A sound wave at

1000\text{ Hz}

enters a steel beam (

v=5100\text{ m/s}

). After traveling

20.4\text{ m}

in the beam, how many full wavelengths fit, and what is the corresponding phase delay in radians?

Example 42

challenge

Show that a longitudinal wave's intensity drops as

1/r^2

from a point source in 3D, by power-conservation reasoning.

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Related Concepts

Waves Sound Pressure Wave

Background Knowledge

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

waves