a wave source at x = 0 that oscillates with simple

© 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.


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


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


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.


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


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

© 2016 Pearson Education, Inc.

𝜕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.


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)


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?


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.


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.


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


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


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)

a wave source at x = 0 that oscillates with simple

Documents