PES 2130 Fall 2014, Spendier
Lecture 11/Page 1
Lecture today: Chapter 16 Waves-1
1) The Principle of Superposition for Waves
2) Interference of Waves
3) Reflection of wave pulses
4) Standing Waves and Resonance
Announcements:
- HW 4 due Monday
Transverse velocity and wave equation:
The wave equation
Speed of a transverse wave on a stretched string
√
µ … Mass per unit length
τ …. Tension of string
Average power of a transverse wave
v … wave speed
ω … angular frequency (angular speed)
ym … amplitude (magnitude) of wave
Principle of superposition: When several effects occur simultaneously, their net effect is
the sum of the individual effects.
Interference. When two or more waves are present at the same position and time, then
the waves are said to interfere.
Example of interference:
Let one wave traveling along a stretched string be given by
y1(x, t) = ym sin(kx - ωt)
and another, shifted from the first, by
y2(x, t) = ym sin(kx - ωt + ϕ).
These waves have the same angular frequency ω (and thus the same frequency f ), the
same angular wave number k (and thus the same wavelength λ), and the same amplitude
ym. They both travel in the positive direction of the x axis, with the same speed. They
differ only by a constant angle ϕ, the phase constant. These waves are said to be out of
phase by ϕ or to have a phase difference of ϕ, or one wave is said to be phase-shifted
from the other by ϕ.
PES 2130 Fall 2014, Spendier
Lecture 11/Page 2
Resulting wave is the sum
y(x, t) = ym sin(kx - ωt) + ym sin(kx - ωt + ϕ) = ym [sin(kx - ωt) + sin(kx - ωt + ϕ)]
Use the following trig identity:
sin(α) + sin(β) = 2 sin(
) cos(
)
applying this relation
α = kx – ωt
β = kx – ωt+ϕ
y(x, t) =2 ym sin(
–
y(x, t) =2 ym cos(-ϕ/2) sin(
since
cos(-ϕ/2)= cos(ϕ/2)
y(x, t) =2 ym cos(ϕ/2) sin(
–
–
) cos(
–
–
–
)
)
)
The resultant wave is also a sinusoidal wave traveling in the direction of increasing x.
New amplitude: 2 ym cos(ϕ/2)
New phase constant (new phase shift): ϕ/2
Let’s look at different values for ϕ:
1) cos(ϕ/2) = 0
This is the case when:
ϕ/2 = π/2
ϕ = π rad = 1800
then y(x, t) = 0
And we say that the waves “fully destructively interfered”. This also happens at 3 π,
5π, … (5400, 9000). The Two waves are out of phase and will produce a flat string.
PES 2130 Fall 2014, Spendier
2) ϕ = 0:
The two waves are in phase:
y(x, t) =2 ym cos(0/2) sin( –
y(x, t) =2 ym sin(
–
Lecture 11/Page 3
)
)
We end up with a resulting wave that has the same frequency and wavelength as the two
input waves, but with twice the amplitude (magnitude). And we say that the waves “fully
constructively interfered”.
3) ϕ =
:
When interference is neither fully constructive nor fully destructive, it is called
intermediate interference. The amplitude of the resultant wave is then intermediate
between 0 and 2ym.
Example with two pulses (We will look at this more when we study sound waves):
In the left-hand set of three panels (going down), we see two wave pulses with opposite
amplitudes approaching each other. When they combine, their amplitudes cancel and
there is apparently nothing left, since the rope is now a straight line. But this is an
illusion: the energy and momentum of the waves cannot be disposed of that
easily. Though the rope may be straight for an instant, it cannot stay that way. It is
accelerating as it changes into and out of a straight line, and both waves reemerge from
the straight line and continue onward with no change whatever. The right-hand side of
the graphic also shows two wave pulses colliding, except this time the pulse amplitudes
are in the same direction, so when they meet the amplitudes add and temporarily create
one large pulse.
PES 2130 Fall 2014, Spendier
Lecture 11/Page 4
In two or three dimensions, overlapping waves can create complex interference patterns,
as can be seen in the wave patterns at a beach. Even with only two wave sources, and
only looking in two dimensions, the interference patterns are complex.
(Read about phasors yourself)
Reflection of wave pulses
Up to this point we have been discussing waves that propagate continuously in the
same/opposite direction.
What happens when a wave strikes the boundary of its medium? All parts of the wave
will be reflected.
Example: Echo – your voice is reflected by an object (building, cliff) far away from you
Let’s look at wave reflection for a transverse wave on a string. When the wave reaches
the boundary of its medium there are two different ways a wave can be reflected (or two
different boundary conditions)
1) Fixed boundary: The string is attached to the wall and cannot move
In Fig. 16-18a, the string is fixed at its left end. When the pulse arrives at that end, it
exerts an upward force on the support (the wall). By Newton’s third law, the support
exerts an opposite force of equal magnitude on the string. This second force generates a
pulse at the support, which travels back along the string in the direction opposite that of
the incident pulse. In a “hard” reflection of this kind, there must be a node at the support
because the string is fixed there. The reflected and incident pulses must have opposite
signs, so as to cancel each other
2) Free boundary: The string is attached to a ring that can slide freely (no friction)
up and down the pole.
In Fig. 16-18b, the left end of the string is fastened to a light ring that is free to slide
without friction along a rod. When the incident pulse arrives, the ring moves up the rod.
As the ring moves, it pulls on the string, stretching the string and producing a reflected
pulse with the same sign and amplitude as the incident pulse. Thus, in such a “soft”
reflection, the incident and reflected pulses reinforce each other, creating an antinode at
the end of the string; the maximum displacement of the ring is twice the amplitude of
either of these two pulses.
PES 2130 Fall 2014, Spendier
Lecture 11/Page 5
Standing wave on a string
We just discussed the reflection of a wave pulse on a string when it arrives at a boundary
point (either fixed or free end). Now let’s look at what happens when a sinusoidal wave is
reflected by a fixed point of a string
The figure shows a string fixed at its
right end. Its left end is moved up and
down in simple harmonic motion to
produce a wave that travels to the
right; the wave reflected from the
fixed end travels to the left. The
resulting motion when these two
waves combine no longer looks like
two waves traveling in opposite
direction. The string appears to be
subdivide into a number of segments.
Wave patterns such as this one are
called standing waves because the
wave patterns do not move left or
right; the locations of the maxima
and minima do not change.
PES 2130 Fall 2014, Spendier
Lecture 11/Page 6
Let’s analyze this motion of the resulting wave by the principle of superposition:
The outstanding feature of the resultant wave is that there are places along the string,
called nodes, where the string never moves. Halfway between adjacent nodes are
antinodes, where the amplitude of the resultant wave is a maximum.
Math:
y1(x, t) = ym sin(kx - ωt)
(wave traveling to right)
y2(x, t) = ym sin(k + ωt).
(wave traveling to left)
After Applying a trigonometric relation one can show that
y(x, t) = y1(x, t) + y2(x, t) = [2ym sin kx] cos ωt.
This equation does not describe a traveling wave because it is not of the form of a wave
function.
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.
PES 2130 Fall 2014, Spendier
Lecture 11/Page 7
In a traveling sinusoidal wave, the amplitude of the wave is the same for all string
elements. That is not true for a standing wave, in which the amplitude varies with
position.
Amplitude of standing wave:
The quantity 2ym sin kx in the brackets of Eq. 16-60 can be viewed as the amplitude of
oscillation of the string element that is located at position x. However, since an amplitude
is always positive and sin kx can be negative, we take the absolute value of the quantity
2ym sin kx to be the amplitude at x.
Nodes of standing wave (points were string never moves)
sin kx = 0
kx = 0, π, 2 π, 3 π, ….
Position
x = 0, π/k, 2 π/k, 3 π/k, ….
using k = 2 π/λ
x = 0, λ/2, λ, 3 λ/2, ….
or
, where n = 0,1,2,3,….
position of nodes:
Antinodes of a standing wave (points where the amplitude of the resultant wave is a
maximum):
sin kx = ± 1
kx = π/2, 3π/2, 5 π/2, ….
x = λ/4, 3 λ/4, 5 λ/4
position of antinodes:
(
)
, where n = 0,1,2,3,….
Important:
For only 1 boundary, any wavelength will form a standing wave!!!
PES 2130 Fall 2014, Spendier
Lecture 11/Page 8
Standing waves and resonance
In most applications (violin string, suspension bridge, optics: laser cavity) there are two
fixed boundaries. In this case there are multiple reflections from each end and the math
can get VERY difficult.
Let a string be stretched between two clamps separated by a fixed distance L. To find
expressions for the resonant frequencies of the string, we note that a node must exist at
each of its ends, because each end is fixed and cannot oscillate.
First harmonic:
The simplest pattern that one antinode, which is at the center of the string. Note that half
a wavelength spans the length L, which we take to be the string’s length. Thus, for this
pattern, λ/2 = L. This condition tells us that if the left-going and right-going traveling
waves are to set up this pattern by their interference, they must have the wavelength
λ = 2L.
Second harmonic:
This pattern has three nodes and two antinodes: λ = L
(this the frequency of the first harmonic)
Third harmonic:
This pattern has four nodes and two antinodes: L=3 λ/2
Thus, a standing wave can be set up on a string of length L by a wave with a wavelength
equal to one of the values
, where n = 1,2,3,….
The resonant frequencies that correspond to these wavelengths follow from
√
Which gives
√
PES 2130 Fall 2014, Spendier
Lecture 11/Page 9
and
√
Here v is the speed of traveling waves on the string.
The last equation tells us that the resonant frequencies are integer multiples of the lowest
resonant frequency, f = v/2L, which corresponds to n = 1. The oscillation mode with that
lowest frequency is called the fundamental mode or the first harmonic. The second
harmonic is the oscillation mode with n = 2, the third harmonic is that with n = 3, and so
on. The collection of all possible oscillation modes is called the harmonicseries, and n is
called the harmonic number of the nth harmonic.
DEMO: String
When the frequency of a string is driven at a frequency that matches one of these
resonant frequencies, then a standing wave forms and the largest amplitude motion is
seen,