© 2017 Pearson Education, Inc. Slide 16-1

Sinusoidal Waves

A wave source at x = 0 that oscillates with simple

harmonic motion (SHM) generates a sinusoidal

wave.

© 2017 Pearson Education, Inc. Slide 16-2

Above is a history graph for a sinusoidal wave, showing

the displacement of the medium at one point in space.

Each particle in the medium undergoes simple

harmonic motion with frequency f, where f = 1/T.

The amplitude A of the wave is the maximum value of

the displacement.

Sinusoidal Waves

© 2017 Pearson Education, Inc. Slide 16-3

Above is a snapshot graph for a sinusoidal wave,

showing the wave stretched out in space, moving to the

right with speed v.

The distance spanned by one cycle of the motion is

called the wavelength λ of the wave.

Sinusoidal Waves

© 2017 Pearson Education, Inc. Slide 16-4

A wave on a string is traveling

to the right. At this instant, the

motion of the piece of string

marked with a dot is

QuickCheck

A. Up.

B. Down.

C. Right.

D. Left.

© 2017 Pearson Education, Inc. Slide 16-5

or, in terms of frequency:

The distance spanned by one cycle of the motion is

called the wavelength λ of the wave. Wavelength is

measured in units of meters.

During a time interval of exactly one period T, each

crest of a sinusoidal wave travels forward a

distance of exactly one wavelength λ.

Because speed is distance divided by time, the

wave speed must be

Sinusoidal Waves

© 2017 Pearson Education, Inc. Slide 16-6

The Mathematics of Sinusoidal Waves

Define the wave number and angular frequency:

Wave function:

This wave travels at a speed v = ω/k.

𝜔 =2𝜋𝑣

λ=2𝜋

𝑇

𝑦(𝑥, 𝑡) = 𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡 + ϕ𝑜)

Particle velocity and acceleration in a sinusoidal wave

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𝜕2𝑦(𝑥, 𝑡)

𝜕𝑥2=

1

𝑣2𝜕2𝑦(𝑥, 𝑡)

𝜕𝑡2

Wave equation:

All wave behavior obeys the wave equation. Likewise,

any physical behavior that satisfies the wave equation

can be modeled as a wave.

© 2017 Pearson Education, Inc. Slide 16-8

Which of the following

equations satisfy the wave

equation?

QuickCheck

A.

B.

C.

D. Both A and B.

𝜕2𝑦(𝑥, 𝑡)

𝜕𝑥2=

1

𝑣2𝜕2𝑦(𝑥, 𝑡)

𝜕𝑡2

Acos(kx+ωt); .

Acos(kx+ωt); .

𝑦(𝑥, 𝑡) = 𝐴𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡)

𝑦(𝑥, 𝑡) = 𝐴𝑐𝑜𝑠(𝑘𝑥 − 𝜔𝑡)

𝑦 𝑥, 𝑡 = 𝐴𝑐𝑜𝑠 𝑘𝑥 + 𝐴𝑐𝑜𝑠(𝜔𝑡)

© 2016 Pearson Education, Inc.

Example 1 - A water wave traveling in a straight line on a lake is described by

the equation

y(x,t) = 2.75cos(0.410x+6.20t) cm

(a)How much time does it take for one complete wave pattern to go past a

fisherman in a boat at anchor?

(b) What horizontal distance does the wave crest travel in that time?

(c) What is the wave number?

(d) What is the number of waves per second that pass the fisherman?

(e) How fast does a wave crest travel past the fisherman?

(f) What is the maximum speed of his cork floater as the wave causes it to bob

up and down?

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In-class Activity 1 - Transverse waves on a string have wave speed 8.00

m/s, amplitude 0.0700 m, and wavelength 0.320 m. The waves travel in

the -x direction, and at t = 0 the x = 0 end of the string has its maximum

upward displacement.

(a) Find the frequency, period, and wave number of these waves.

(b) Write the wave function describing this wave.

© 2017 Pearson Education, Inc.

If wave 1 displaces a particle in the medium by y1

and wave 2 simultaneously displaces it by y2, the net

displacement of the particle is y1 + y2.

The Principle of Superposition

Slide 17-11

It can be easily shown that the superposition of waves

still satisfies the wave equation.

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The figure shows the

superposition of two waves

on a string as they pass

through each other.

The principle of

superposition comes into

play wherever the waves

overlap.

The solid line is the sum at

each point of the two

displacements at that

point.

The Principle of Superposition

Slide 17-12

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QuickCheck 17.1

Two wave pulses on a

string approach each

other at speeds of

1 m/s. How does the

string look at t = 3 s?

Slide 17-13

© 2017 Pearson Education, Inc.

QuickCheck 17.2

Two wave pulses on a

string approach each

other at speeds of

1 m/s. How does the

string look at t = 3 s?

Slide 17-14

FIGURE 16.20

Constructive interference of two identical waves produces a wave with twice the

amplitude, but the same wavelength.

FIGURE 16.21

Destructive interference of two identical waves, one with a phase shift of 180°(𝜋 rad) ,

produces zero amplitude, or complete cancellation.

Superposition of two waves

with identical amplitudes,

wavelengths, and frequency,

but that differ in a phase shift.

The resultant wave has a

modified amplitude and

phase shift but in other ways

it is similar to the original

waves.

𝑦𝑛𝑒𝑡(𝑥, 𝑡) = 2𝐴𝑐𝑜𝑠(ϕ𝑜

2)𝑠𝑖𝑛(𝑘𝑥 − 𝜔𝑡 +

ϕ𝑜

2)

Standing waves on a string

• Waves traveling in opposite directions on a taut string

interfere with each other.

• The result is a standing wave pattern that does not move on

the string.

• Destructive interference occurs where the wave

displacements cancel, and constructive interference occurs

where the displacements add.

• At the nodes no motion occurs, and at the antinodes the

amplitude of the motion is greatest.

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Standing Waves

Time snapshots of two sine waves.

The red wave is moving in the −x-

direction and the blue wave is

moving in the +x-direction. The

resulting wave is shown in black.

Consider the resultant wave at the

points x = 0 m, 3 m, 6 m, 9 m, 12

m, 15 m and notice that the resultant

wave always equals zero at these

points, no matter what the time is.

These points are known as fixed

points (nodes).

In between each two nodes is an

antinode, a place where the medium

oscillates with an amplitude equal to

the sum of the amplitudes of the

individual waves.

FIGURE 16.27

𝑦 𝑥, 𝑡 = 2𝐴𝑠𝑖𝑛 𝑘𝑥 cos(𝜔𝑡)

The sine function dictates the position of the standing waves

while the cosine function expresses how the shape

oscillates with time.

Example 2 – A guitar string is plucked and creates a standing sinusoidal

wave with amplitude 0.750 mm and frequency 440 Hz. The wave velocity

is 143 m/s.

(a) Find the equation of the standing wave.

(b) Locate the nodes

(c) Find the maximum speed and acceleration of the string.

Standing waves on a string

• This is a time exposure of a

standing wave on a string.

• This pattern is called the

second harmonic.

© 2016 Pearson Education, Inc.

Standing waves on a string

• As the frequency of the

oscillation of the right-hand

end increases, the pattern of

the standing wave changes.

• More nodes and antinodes

are present in a higher

frequency standing wave.

© 2016 Pearson Education, Inc.

Normal modes

• For a taut string fixed at both

ends, the possible wavelengths

are and the possible

frequencies are fn = n v/2L =

nf1, where n = 1, 2, 3, …

• f1 is the fundamental

frequency, f2 is the second

harmonic (first overtone), f3 is

the third harmonic (second

overtone), etc.

• The figure illustrates the first

four harmonics.

© 2016 Pearson Education, Inc.

Example 3 - Adjacent antinodes of a standing wave on a string are 15.0 cm

apart. A particle at an antinode oscillates in simple harmonic motion with

amplitude 0.850 cm and period 0.0750 s. The string lies along the +x-axis and

is fixed at x = 0.

(a) How far apart are the adjacent nodes? How far from antinodes?

(b) Find the wavelength, amplitude, and speed of the standing wave.

(c) Find the wavelength, amplitude, and speed of the traveling waves.

(c) Find the max speed of the string.

In-class Activity #2 – A standing wave on a wire has an amplitude of 2.40

mm, an angular frequency of 934 rad/s, and wave number 0.750π rad/m.

The left end of the wire is at x = 0. At what distances from the left end are

(a) the nodes of the standing wave?

(b) the antinodes of the standing wave?

(c) What is the node to antinode distance?