Chapter 2 Wave Motion
Chapter 2 Wave Motion
Chapter 2 Wave Motion
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PART 1
Oscillations and
Mechanical Waves
Chapter 2
Wave Motion
Types of Waves
• Example of a wave
– A pebble hits the water’s surface.
– The resulting circular wave moves outward from the creation point.
– Electromagnetic waves
• No medium required.
• Examples are light, radio waves, x-rays
Introduction
General Features of Waves
• In wave motion, energy is transferred over a
distance.
• Matter is not transferred over a distance.
Introduction
Mechanical Wave Requirements
Section 16.1
Pulse on a String
• The wave is generated by a
flick on one end of the string.
• The string is under tension.
• A single bump is formed and
travels along the string.
– The bump is called a pulse.
Section 16.1
Pulse on a String
• The hand is the source of the disturbance.
• The string is the medium through which the pulse travels.
– Individual elements of the string are disturbed from their
equilibrium position.
– The elements are connected together so they influence each
other.
Section 16.1
Transverse Wave
• A wave is a periodic disturbance traveling
through a medium.
• A traveling wave or pulse that causes the
elements of the disturbed medium to
move perpendicular to the direction of
propagation is called a transverse wave.
– To create the wave, you would move the
end of the string up and down repeatedly.
Section 16.1
Longitudinal Wave
• S waves
– “S” stands for secondary
– Slower, at 4 – 5 km/s
– Transverse
Section 16.1
Traveling Pulse, cont.
Section 16.1
Traveling Pulse, final
• For a pulse traveling to the right
– y (x, t) = f (x – vt)
Section 16.1
Sinusoidal Waves
Section 16.2
Sinusoidal Waves, cont
• The wave moves toward the right.
– In the previous diagram, the brown wave represents the
initial position.
– As the wave moves toward the right, it will eventually
be at the position of the blue curve.
Section 16.2
Terminology: Amplitude and
•
Wavelength
The crest of the wave is the
location of the maximum
displacement of the element from
its normal position.
– This distance is called the
amplitude, A.
Section 16.2
Terminology: Period and
Frequency, cont
Section 16.2
Terminology, Example
• The wavelength, l, is
40.0 cm
• The amplitude, A, is
15.0 cm
• The wave function can
be written in the form
y = A cos(kx – t).
Section 16.2
Speed of Waves
• Waves travel with a specific speed.
– The speed depends on the properties of the medium being
disturbed.
2
y ( x, t ) A sin x vt
– This is for a wave moving to the right.
– For a wave moving to the left, replace x – vt
– with x + vt.
Section 16.2
Wave Function, Another Form
v=l/T
• The wave function can then be expressed as
x t
y ( x, t ) A sin 2
T
• This form shows the periodic nature of y.
– y can be used as shorthand notation for y(x, t).
Section 16.2
Wave Equations
• We can also define the angular wave number (or just wave
number), k. 2
k
• The angular frequency can also be defined.
2
2 ƒ
T
• The wave function can be expressed as y = A sin (k x – wt).
• The speed of the wave becomes v = l ƒ.
• If x ¹ 0 at t = 0, the wave function can be generalized to y = A
sin (k x – wt + f) where f is called the phase constant.
Wave Equations, cont.
Section 16.2
Sinusoidal Wave on a String
• To create a series of
pulses, the string can be
attached to an oscillating
blade.
• The wave consists of a
series of identical
waveforms.
• The relationships between
speed, velocity, and period
hold. Section 16.2
Sinusoidal Wave on a String, 2
• Each element of the string
oscillates vertically with simple
harmonic motion.
– For example, point P
dv y
ay
dt x constant
– ay, max = w2 A
– a is a maximum at y = ±A
tension T
v
mass/length
Section 16.3
Reflection of a Wave, Fixed End
• When the pulse reaches the
support, the pulse moves back
along the string in the opposite
direction.
• This is the reflection of the pulse.
Section 16.4
Transmission of a Wave, 2
Section 16.4
Transmission of a Wave, 4
•Conservation of energy governs the pulse
– When a pulse is broken up into reflected and transmitted parts
at a boundary, the sum of the energies of the two pulses must
equal the energy of the original pulse.
Section 16.4
Energy in Waves in a String
• Waves transport energy when they propagate through a medium.
• We can model each element of a string as a simple harmonic
oscillator.
– The oscillation will be in the y-direction.
= ¼mw2A 2l
•This gives a total energy of
– El = Kl + Ul = ½mw2A 2l
Section 16.5
Power Associated with a Wave
• The power is the rate at which the energy is being transferred:
1
E 2 A
2 2
1
P 2 A2v
T T 2
• The power transfer by a sinusoidal wave on a string is
proportional to the
– Square of the frequency
– Wave speed
Section 16.6
Linear Wave Equation Applied
to a Wave on a String
• The string is under tension T.
• Consider one small string
element of length Dx.
• The net force acting in the y
direction is
Fy T (tanB tan A )
• .
– This uses the small-angle
approximation that sin θ ≈ tan
θ.
Section 16.6
Linear Wave Equation Applied to
Wave on a String, cont.
2 y y x B y x A
T t 2 x
2y 2 y
2
T t 2
x
Section 16.6
Linear Wave Equation, General
• The equation can be written as
2y 1 2y
2 2
x 2
v t
• This applies in general to various types of
traveling waves.
– y represents various positions.
• For a string, it is the vertical displacement of the elements of the string.
• For a sound wave propagating through a gas, it is the longitudinal position
of the elements of the gas from the equilibrium position.
• For electromagnetic waves, it is the electric or magnetic field components.
Section 16.6