Types of Waves
• Mechanical
• Electromagnetic
• Matter
• Transverse vs Longitudinal Waves
• Demo: Slinky
Wavelength and Frequency
• Wave on a string is characterized by its
shape: y = y(x,t)
• Sinusoidal Wave: y(x,t) = A sin (kx-ωt)
• Wavenumber and wavelength are related:
k = 2π/λ
• Modulation in space and time!
• Snapshot vs graph
• Simulation: Wave on a String
Speed of a traveling wave
• Snapshots of a wave at different times show
that the wave pattern is moving a distance at
a time, whereas the parts of the medium are
just oscillating up and down (or left and
right in a longitudinal wave)
• Condition for point A to have same
displacement  phase must be the same
 kx – ωt = const
 Derivative w.r.t. time: v = ω/k = λf
Lecture 33: Energy & Waves
Example of a wavefunction:
y(x,t) = 0.52m sin(45x/m – 1.57 t/s)
•
•
•
•
•
•
•
•
Amplitude
Frequency
Period
Wavelength
Velocity
Displacement at fixed place
Displacement at fixed time
Displacement at fixed place and time
Example of a wavefunction:
y(x,t) = 0.52m sin(45x/m – 1.57 t/s)
•
•
•
•
•
•
•
•
Amplitude = 0.52m
Frequency: ω=1.57/s = π/2 Hz  f = 1/4 Hz
Period T=1/f=4 s
Wavelength: k=45/m = 2π/λ  λ= 2π/45 m = 0.14m
Velocity = ω/k = 0.035m/s
Displacement at fixed place: y(0,t)= – 0.52m sin(1.57 t/s)
Displacement at fixed time: y(x,0)= 0.52m sin(45x/m)
Displacement at fixed place and time
y(x=1m,t=1s)= 0.52m sin(45-1.57)= – 0.27 m
Which of the following describes
a wave moving in the negative x
direction?
•
•
•
•
y(x,t) = 0.52m sin(45x/m – 1.57 t/s)
z(x,t) = - 0.52m sin(- 45x/m + 1.57 t/s)
u(x,t) = - 0.52m cos(45x/m – 1.57 t/s)
v(x,t) = 0.52m sin(45x/m + 1.57 t/s)
Communication between parts of
the string
• Demo: The falling green slinky
• What happens to the lower part?
–
–
–
–
Falls immediately
Rises
Stays put
Impossible to tell
Wave speed on a string
• Has to depend on properties defining the
physical situation: mass per length μ,
tension T
• Dimensional analysis suggests that
v = C √T/μ
Newton II: C=1
Example: T= 50N, mass density 500 g/m
 v = √50Nm/0.5kg = (√100) m/s = 10 m/s
Velocities: wave velocity vs.
transverse velocity
• Velocity of the wave is the velocity of a
pulse along the medium, e.g. stretched
string
• transverse velocity is the velocity of a tiny
bit of the medium perpendicular to the
string (one oscillator going up and down)
Energy & Power of a Wave
• Little bit of string with mass dm moves like
an oscillator:
– KE: transverse velocity
– PE: stretching of the string, not the “spring”
• Forces due to tension in string do work to
transfer energy along the string, e.g. parts of
the string with x=0, v=0 are moving and
elongated
Rate of energy Transmission
• KE: dK = ½ dm u2 = ½ μdx u2
• u= ∂ y(x,t)/∂t
• With y = A sin (kx-ωt) we get that KE is
proportional to the square of a cosine
• The average of cos2t over a period is 1/2 , so
(dK/dt) = ¼ μ v ω2A2
The average of the potential energy is the
same, so P = 2 (dK/dt) = ½ μ v ω2A2
Example
• String specs: 500g/m, T=50N
• Oscillation specs: f =100Hz, A = 1cm
• Average rate at which the string transports
energy:
• ½ μ v ω2A2 = ½ (0.5kg/m)(50Nm/0.5kg)1/2
(2π 100 Hz 0.01m)2
which is roughly 100 W, so 100J per sec.
Wave Equation
• Consider little piece of string, tension
pulling on both ends in slightly different
directions
• Derive wave equation from this:
•
𝜕2 𝑦
𝜕𝑥 2
=μ/T
𝜕2 𝑦
𝜕𝑡 2
• This means that the second space and the
second time derivative of the displacement
are related by the square of v
Lecture 34: Superposition,
Interference, Reflections &
Standing Waves
• Physics Coffee today 3:30pm, Sci 205
• Starry Monday tonight 7 pm, Sci 237
“Comets”
• Don’t forget to vote  Evaluations
• Lab tomorrow?
Superposition
•
•
•
•
Total wave = sum of individual waves
Waves do not affect each other!
Particles collide – Waves interfere
Simulation: Wave on a string
– See that pulses do not affect one another
– Try different pulses
Interference
• Constructive and destructive
• Use sin(α+β)+sin(α–β)=2sinα cosβ
• Here:
sin(kx+ωt)+sin(kx– ωt)=2sinkx cos ωt
= A cos ωt
• Also partial interference
• Constructive superposition
http://www2.biglobe.ne.jp/~norimari/science
• Destructive superposition
http://www2.biglobe.ne.jp/~norimari/science
Standing Waves
– Demo: standing waves on a string
– Simulation: Two waves on a String
• Standing waves are the result of
interference of two traveling waves, going
in opposite direction
• Do the math: [sin(α+β)+sin(α–β)=2sinα cosβ) ]
• y1(x,t) = A sin (kx – ωt)
• y2 (x,t) = A sin (kx + ωt)
• ytotal(x,t) = y1(x,t) + y2(x,t) = 2A sin (kx)cos(ωt)
Standing Waves
• ytotal(x,t) = y1(x,t) + y2(x,t) = 2A sin (kx)cos(ωt)
• Nodes and antinodes
• Amplitude varies with position
• Can find positions of maximal amplitude as
function of wavelength
A string clamped at both ends is plucked so that it
vibrates in a standing mode between two extreme
positions a and b. When the string is at position c, the
instantaneous (transverse) velocity of points along the
string
is zero everywhere
is positive (upward) everywhere
is negative (downward) everywhere
depends on location along string
A string clamped at both ends is plucked so that it
vibrates in a standing mode between two extreme
positions a and b. When the string is at position b,
the instantaneous velocity of points along the string
A. is zero everywhere
B. is positive (upward) everywhere
C. is negative (downward) everywhere
D. depends on location along string
Reflections at a Boundary
•
•
•
•
Two types: open and closed
See lab
At closed end pulse is inverted
At open end pulse is not inverted
Standing Waves and Resonance
• If both ends are fixed only certain
wavelengths produce a standing wave
• L = λ/2, 2 λ/2, 3 λ/2, …, n λ/2
• Resonant wavelength or frequencies
• λ/2 is the fundamental mode or first
harmonic
• Simulation
Sound Wave
• A longitudinal wave: Simulation
• Wavefronts
– Oscillations due to sound wave have same value
• Rays
– Perpendicular to wavefronts, direction of front
Speed of Sound
• Cf. stretched string: v = v(elastic property
(T), inertial property(mass density))
• v = √B/ρ
• Bulk modulus B= – Δp/ (ΔV/V)
– How much does volume change as pressure on
material changes?
• Density ρ
Speed of Sound in different
materials
• Air (0C): 331 m/s
• Air (20C): 343 m/s
• Hydrogen: 1284 m/s
• Water: 1482 m/s
• Steel: 5941 m/s
Lecture 35: Sound, Beats &
Doppler Effect
Traveling Sound Waves
• Analogous to traveling transverse wave
• Note that air elements travel longitudinally
– Name displacement: s(x,t) [in x direction!]
• Also can interfere & superimpose, as with
any wave
Interference
• The interference at a point P depends on the
path length difference
• Fully constructive interference: ΔL= nλ, n
integer
• Fully destructive interference: ΔL= nλ/2, n
odd
Intensity & Sound
Level
• I = P/A = ½ ρv(ωA)2
– Derivation via dK = ½ dm v2
– See: transverse wave P = 2 (dK/dt) = ½ μ v ω2A2
• Difference: divide by Area, μ --> ρ
• Falls off with distance like surface area of a
sphere: I=P/4πr2
Decibel Scale
• Sound level is: β=(10dB) log I/I0
• I0 = 10-12 W/m2 is a standard reference
intensity
• If the sound intensity is I0, then
β=(10dB) log I0/I0 = (10dB) 0 = 0
• Noise due to air flow in intro lab: 55 dB
• Conversation: 60 dB
• Pain threshold: 120 dB
Beats
• Interference of waves that are very close in
frequency
• The ear perceives them as wave of average
frequency, but with intensity that is modulated
with difference of frequencies (beats!)
• So two waves of 440Hz & 446 Hz are perceived
as wave of 443 Hz with beats of 6 Hz
 Tuning of musical instruments
• Simulation
Doppler Effect
• Simulation, cf. also the related applet to see
how the wavelength of a moving source
changes
• f ’ = f (v±vD)/(v±vS)
• v: speed of sound
• vD; detector relative to air
• vS; source relative to air
Example: f ’ = f (v±vD)/(v±vS)
•
•
•
•
v = 330 m/s (speed of sound)
vD= 10 m/s ; detector relative to air
vS= 20 m/s; source relative to air
Source and detector move towards each
other, f= 440 Hz (A)
• Rule: choose signs such that f ‘ > f if they
move towards each other
• So choose f ’ = f (v+vD)/(v-vS) = 440 Hz
(340 m/s)/(310m/s)= 482Hz  higher pitch
Hubble’s
Law
v = Hd
with
H ≈70 km/s/Mpc
(1 Mpc = 3.26 Mlyr = 3.1 × 1019 km)
The Expanding Universe
Shock Waves
• Happens when source is traveling faster
than the speed of sound
• See previous simulation
Shock Waves
The four figures below represent sound waves
emitted by a moving source. Which picture(s)
represent(s) a source moving at less than the speed
of sound?
Three observers, A, B and C, are listening to a moving
source of sound. The diagram shows the location of the
wave crests of the moving source with respect to the three
observers. Which of the following is true?
A. The wave fronts move faster at C than at A and B.
B. The frequency of the sound is highest at A.
C. The frequency of the sound is highest at B.
D. The frequency of the sound is highest at C.