Chapter 16
Waves (I)
What determines
the tones of
strings on a guitar?
Key contents
Types of Waves
Wave Variables
The Speed of a Traveling Wave
Energy and Power of a Wave Traveling along a String
The Wave Equation
The Superposition of Waves
Interference of Waves
Standing Waves and Resonance
16.1 Types of Waves
Mechanical waves
Electromagnetic waves
Gravitational waves
Matter waves
# Waves are propagating ‘disturbance’ without
matter transportation, but energy and momentum
are transported.
16.1 Types of Waves
In a transverse wave, the displacement
of every such oscillating element
along the wave is perpendicular to the
direction of travel of the wave, as
indicated in Fig. 16-1.
# EM waves in vacuum are also called
transverse waves, because the
direction of fields is perpendicular to
the propagation direction.
In a longitudinal wave the motion of
the oscillating particles is parallel to
the direction of the wave’s travel, as
shown in Fig. 16-2.
# In this chapter, we will focus on transverse mechanical waves.
16.2 Wave Variables
16.2 Wave Variables
The amplitude ym of a wave is the magnitude
of the maximum displacement of the elements
from their equilibrium positions as the
wave passes through them.
The phase of the wave is the argument
(kx –wt) of the sine function. As the wave
sweeps through a string element at a particular
position x, the phase changes linearly with
time t.
The wavelength l of a wave is the distance
parallel to the direction of the wave’s
travel between repetitions of the shape of the
wave (or wave shape). It is related to the
angular wave number, k, by
:
The period of oscillation T of a wave is the
time for an element to move through one full
oscillation. It is related to the angular frequency,
w, by
The frequency f of a wave is defined as 1/T
and is related to the angular frequency w by
A phase constant f in the wave function:
y =ym sin(kx –wt+ f). The value of f can be
chosen so that the function gives some other
displacement and slope at x = 0 when t = 0.
16.3 The Speed of a Traveling Wave
Example, Transverse Wave
Example, Transverse Wave, Transverse Velocity, and Acceleration
16.4 Wave Speed on a Stretched String
Dimension analysis : what factors should come in?
Tension [t] = ML/T2
Density [r] = M/L3
Area of cross section [A] = L2
t
t
Þv=
=
rA
m
 Restoring force factor
 Inertia factor
16.4 Wave Speed on a Stretched String
The speed of a wave along a stretched ideal string depends only on the tension
and linear density of the string and not on the frequency of the wave.
A small string element of length Dl within the pulse
is an arc of a circle of radius R and subtending an
angle 2q at the center of that circle. A force with a
magnitude equal to the tension in the string, t, pulls
tangentially on this element at each end. The
horizontal components of these forces cancel,
but the vertical components add to form a radial
restoring force . For small angles,
If m is the linear mass density of the string, and Dm
the mass of the small element,
The element has an acceleration:
Therefore,
16.5 Energy and Power of a Wave Traveling along a String
dU = t (dl - dx) = t ( dx 2 + dy 2 - dx)
æ ¶y ö
1
» t dx ç ÷ = dK
è ¶x ø
2
2
The average power, which is the average rate at
which energy of both kinds (kinetic energy and
elastic potential energy) is transmitted by the wave,
is:
Example, Transverse Wave:
16.6 The Wave Equation
A travelling wave is always in the following form:
y(x, t) = f (x ± vt)
Such functions are solutions of the wave equation:
# It is a linear partial differential equation; when y1 and y2 are
solutions, any linear combination of y1 and y2 (like ay1+by2) is
also a solution.
16.7 The Superposition of Waves
Overlapping waves algebraically
add to produce a resultant wave
(or net wave).
Overlapping waves do not in any
way alter the travel of each other.
They interfere but do not interact.
# The principle of linear superposition
is valid only when the amplitude is
small.
16.8 Interference of Waves
If two sinusoidal waves of the same amplitude and wavelength
travel in the same direction along a stretched string, they interfere
to produce a resultant sinusoidal wave traveling in that direction.
16.8 Interference of Waves
16.8 Interference of Waves
Example, Transverse Wave:
16.9: Standing Waves
16.9: Standing Waves
If two sinusoidal waves of the same amplitude and wavelength travel
in opposite directions along a stretched string, their interference with
each other produces a standing wave.
16.9: Standing Waves
The amplitude is zero when kx =np, for n =0,1,2, . . . .
Since k =2p/l, we get
x = n l/2, for n =0,1,2, . . . (nodes),
as the positions of zero amplitude or the nodes.
The amplitude has a maximum value of 2ym when
kx = 1/2p, 3/2p, 5/2p, . . .=(n+1/2) p, for n =0,1,2, . . . .
That is,
x = (n+1/2) l/2, for n 0,1,2, . . . (antinodes),
as the positions of maximum amplitude or the antinodes.
16.10: Standing Waves and Resonance
* Reflection at a boundary
16.10: Standing Waves and Resonance
For certain frequencies, the
interference produces a standing
wave pattern (or oscillation mode)
with nodes and large antinodes like
those in Fig. 16-19.
Such a standing wave is said to be
produced at resonance, and the
string is said to resonate at these
certain frequencies, called resonant
frequencies.
Fig. 16-19 Stroboscopic photographs reveal
(imperfect) standing wave patterns on a string
being made to oscillate by an oscillator at the
left end. The patterns occur at certain
frequencies of oscillation. (Richard
Megna/Fundamental Photographs)
16.10: Standing Waves and Resonance
The frequencies associated with these modes are often labeled f1,
f2, f3, and so on. The collection of all possible oscillation modes is
called the harmonic series, and n is called the harmonic number of
the nth harmonic.
Example, Standing Waves and Harmonics:
Figure 16-22 shows a pattern of resonant oscillation of a
string of mass m =2.500 g and length L =0.800 m and that
is under tension t =325.0 N. What is the wavelength l of
the transverse waves producing the standing-wave pattern,
and what is the harmonic number n? What is the frequency f
of the transverse waves and of the oscillations of the
moving string elements? What is the maximum magnitude
of the transverse velocity um of the element oscillating at
coordinate x =0.180 m ? At what point during the
element’s oscillation is the transverse velocity maximum?
For the transverse velocity,
Calculations:
We need:
By counting the number of loops (or half-wavelengths) in
Fig. 16-22, we see that the harmonic number is n=4.
Also,
But ym =2.00 mm, k =2p/l =2p/(0.400
m), and w= 2pf =2p (806.2 Hz).
Then the maximum speed of the
element at x =0.180 m is
Homework:
Problems 13, 23, 27, 50, 58