Chapter 14

Waves and Sound

Chapter 14: Waves and SoundA wave is a disturbance that propagates from

one place to another.

For example:• water waves• sound waves• waves on a string• EM waves (no medium)• What is an EM wave?

Notice that the wave can move a great distance even though the medium moves very little.

Types of Waves

• Transverse wave - The medium is displaced in a direction perpendicular to the direction of the wave motion.– Radio waves, Light– Some sound waves in solids

• Longitudinal wave - The medium is displaced in a direction parallel to the wave motion– Sound wave in air

Sound Waves

Properties of Waves•A pure tone is a single sine (or cosine) wave. •A complex tone can be constructed as a superposition (sum) of sine waves of different frequency.•If the wave has amplitude A, then every point in the medium undergoes simple harmonic motion of amplitude A and period T.•The wavelength is the distance over which the wave repeats itself.

Wave Velocity

The wave velocity describes how fast the wave travels:

v = /T = f

In general, the velocity of a wave will depend only on the properties of the medium. The velocity does not change with the frequency. The product of frequency and wavelength will always be the same in the same medium.

Waves on a StringConsider a string with a tension equal to F and a mass per unit length of . The wave speed is:

F

v

Waves on a string are inverted when they reflect from a fixed end and and are not inverted when they reflect from a free end. (Pulse, not sine wave)

fixed end free end

ExampleA string is fixed at one end, passes over a massless pulley, and is attached to a 5-g hanging mass. The mass of 1 m of the string is 2 g. If a 10 Hz wave of amplitude 5 cm passes along the string, (a) what is its wavelength? (b) How long does it take to travel a distance of 1 m from point A to B?

m

AB

(a) 0.49 m(b) 0.2 s

Sound• Sound is a longitudinal pressure wave.• The speed of sound in air is 343 m/s at standard

temperature and pressure. It increases with increasing temperature.

• The speed of sound in solids is greater than in gasses. It is greater for stiffer materials.

• Sound waves originate from vibrating objects:Audible: 20 20,000 HzInfrasonic: < 20 HzUltrasonic: > 20 kHz

• The pitch of a sound wave is determined by the frequency of the sound.

Notice that sound waves carry energy. We define the intensity I as the rate at which energy E flows through a unit area A perpendicular to the direction of travel of the wave.

timeArea

Energy

Area

PowerIIntensity

Sound Intensity

All waves have intensity defined in this way.

Human Perception of SoundThe loudness of a sound depends on its intensity. A sound perceived as roughly twice as loud as another actually has an intensity that is 10 times greater.

We measure loudness by a logarithmic scale of the intensity level of a wave:

= 10 log (I/I0), where I0 = 10-12 W/m2

is dimensionless but we give it a name, decibels (dB).3dB is a factor 2 change in intensityEvery 10dB is a factor 10 change in intensity20 dB is a factor 100 change in intensity

The Doppler EffectThe observed frequency of sound changes if there is

relative motion between the the observer and the source of the sound wave. Example: As a train moves away from you, the pitch (frequency) of the whistle is lower than when it comes toward you.

ffv

uv

u

s

o

1

1

f’ = shifted frequency, f = unshifted frequencyuo = observer speed, us = source speedv = wave speed

+ means that the source is moving away from the observer

+ means that the observer is moving toward the source

ProblemA train on one track moves in the same direction as a second train on the adjacent track. The first train, which is ahead of the second train and moves with a speed of 29 m/s, blows a horn whose frequency is 125 Hz. If the frequency heard on the second train is 131 Hz, what is its speed? [answer = 46.9 m/s]

ffv

uv

u

s

o

1

1

Superposition and Interference of Waves

If two or more waves are moving through a medium, the resultant wave is found by adding together the displacements of the individual waves, point by point.

As a result of superposition, waves can interfere. Interference does not mean that waves are destroyed; they will pass through each other.

Constructive Interference Destructive Interference

Standing Waves

If two waves with equal amplitude and frequency, but traveling in opposite directions, interfere, the result is a standing wave.

• nodes - the zeroes of a standing wave (no displacement)

• antinodes - the points of maximum displacement

A string fixed at both ends

• fundamental frequency - the lowest frequency, also called the first harmonic,

f1 = v/2L and 1 = 2L

• second harmonic - the next allowed frequency, f2 = 2f1.

• In general

fn = nf1 and n = 2L/n

for the nth harmonic on a string of length L. v is the velocity of waves on the string.

Standing Waves on a String: both ends fixed.

Standing Waves in Air ColumnsIn a pipe that is open at one end, there must be an antinode at the open end and a node at the closed end.

f1 = v/4L

fn = nf1

n = 4L/n

for n = 1,3,5…

Problem

A guitar string 60 cm long vibrates with a standing wave that has three antinodes. (a) Which harmonic is this? (b) What is the wavelength of this wave? (c) If this harmonic is excited with a frequency of 600 Hz, what is the frequency of the fundamental?

(a) 3 (b) 0.4m (c) f1 = 200 Hz

Problem 2

Find the fundamental frequency for a pipe of length 0.5 m which is open at both ends. (343 Hz)

Woodwind Instruments

• Oboe, Flute, Recorder = Pipe open at each end Frequencies: fn = n v / (2L), n=1,2,… Fundamental f1 = v / (2L),

• Clarinet = Pipe closed at mouth piece, open at bell Frequencies fn = (2n-1) v / (4L), n= 1, 2, … Fundamental f1 = v / (4L), Clarinet sounds 1 octave below flute, for same L

• Change pitch: Changing length L Change velocity v (change temperature).

Questions

1. Why do big instruments tend to play low pitch notes?

2. What happens to the pitch of a note played on a guitar string if you increase the tension in the string?