phy 102: waves & quanta topic 4 standing waves john cockburn (j.cockburn@... room e15)
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PHY 102: Waves & Quanta
Topic 4
Standing Waves
John Cockburn (j.cockburn@... Room E15)
•Wave reflection at boundaries
•Principle of superposition, interference
•Standing waves on a string
•Normal modes
Reflection of a wave pulse at a boundary
Fixed end Free end
Pulse incident from right is reflected from the boundary at left
HOW the pulse is reflected depends on the boundary conditions
For fixed end, reflected pulse is inverted
For free (in transverse direction) end, reflected pulse is same way up.
time
Frictionless sliding ring
Reflection of a wave pulse at a boundary
Behaviour at interface can be modelled as sum of two pulses moving in opposite directions at the interface:
Transverse displacement always 0 at interface
“fixed end” “free end”
Transverse force
always 0 at interface
Principle of superposition
When 2 (or more) waves overlap in time/space, the total effect is just the algebraic sum of the individual wave functions:
),(..........),(),(),(),( 321 txytxytxytxytxy ntotal
2
2
22
2 ),(1),(
t
txy
vx
txy
(must be so, because wave equation is linear: if y1(x,t) and y2(x,t) are both solutions, for example, then y1+y2 must also be a solution)
Formation of standing wave on a string
Pink line represents wave travelling from right to left along the string.
Blue line represents wave travelling from left to right.(wave reflection at boundaries)
•Black line = sum of left and right-travelling waves = STANDING WAVE
•Constructive interference of waves at ANTINODE of standing wave (max displacement)
•Destructive interference of waves at NODE of standing wave (zero displacement)
•Distance between successive nodes/antinodes = λ/2
Mathematical formulation of standing wave
Wave moving right to left (pink wave)
)cos(),(1 tkxAtxy
Wave moving left to right (blue wave)
)cos(),(2 tkxAtxy
Total wave function (black wave):
Mathematical formulation of standing wave
)sin()sin2(),( tkxAtxy
Mathematical formulation of standing wave
)sin()sin2(),( tkxAtxy
amplitude dependson position
Zero y-displacement (node) when sin(kx) = 0
Maximum y-displacement (y=2A) when sin(kx)=+/- 1……..
Comparison between standing wave and travelling wave
Travelling wave
particles undergo SHM
all particles have same amplitude
all particles have same frequency,
adjacent particles have different phase
Standing wave
particles undergo SHM
adjacent particles have different amplitude
all particles have same frequency
all particles on same side of a node have same phase. Particles on opposite sides of
node are in antiphase
Some very basic physics of stringed instruments……….
The fundamental frequency determines the pitch of the note.
the higher harmonics determine the “colour” or “timbre” of the note.
(ie why different instruments sound different)
Fundamental wavelength = 2L
From v = fλ,
f1= v/2L
So, for a string of fixed length, the pitch is determined by the wave velocity on the string…..
Example Calculation
The string length on standard violin is 325mm. What tension is required to tune a steel “A” string (diameter =0.5mm) to correct pitch (f=440Hz)?
Density of steel = 8g cm
Changing the “harmonic content”
string plucked here
Plucking a string at a certain point produces a triangular waveform, that can be built up from the fundamental plus the higher harmonics in varying proportions.
Plucking the string in a different place (or even in a different way) gives a different waveform and therefore different contributions from higher harmonics (see Fourier analysis)
This makes the sound different, even though pitch is the same…………………
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